Startseite Mathematik Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
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Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters

  • Khursheed J. Ansari , Mohammad Izadi EMAIL logo und Samad Noeiaghdam
Veröffentlicht/Copyright: 18. März 2024
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Demonstratio Mathematica
Aus der Zeitschrift Demonstratio Mathematica Band 57 Heft 1

Abstract

This study is devoted to designing two hybrid computational algorithms to find approximate solutions for a class of singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters. In our approaches, the time discretization is first performed by the well-known Rothe method and Taylor series procedures, which reduce the underlying model problem into a sequence of boundary value problems (BVPs). Hence, a matrix collocation technique based on novel shifted Delannoy functions (SDFs) is employed to solve each BVP at each time step. We show that our proposed hybrid approximate techniques are uniformly convergent in order O ( Δ τ s + M − 1 2 ) for s = 1 , 2 , where Δ τ is the time step and M is the number of SDFs used in the approximation. Numerical simulations are performed to clarify the good alignment between numerical and theoretical findings. The computational results are more accurate as compared with those of existing numerical values in the literature.

Keywords: collocation points; convergent analysis; Delannoy functions; stability; Taylor expansion; Rothe method; singularly perturbed problem
MSC 2010: 65M70; 65N12; 41A10; 35B25

1 Introduction

There are lots of studies focusing on the numerical solutions of singularly perturbed convection–diffusion–reaction equations. The singular perturbations have appeared in control theory [1], hydrodynamics, fluid or gas dynamics, in the modeling of semiconductor devices [2], etc. For more details, see [3]. Numerical treatments of differential equations with small parameters typically lead not to satisfactory results. In fact, in the presence of these parameters, we have boundary layer regions in which the calculated solution varies rapidly in order to satisfy the boundary condition. Such phenomena give rise to numerical cumbersomes and make the investigation of convergence analysis very difficult [4,5,7].

In this work, we aimed to solve 1-D singularly perturbed second-order parabolic partial differential equations with two hybrid approaches. To be more precise, we study the following convection–diffusion–reaction (CDR) model problems with two small parameters given by [6,7]:

(1) ∂ w ∂ τ ( x , τ ) − ε ∂ 2 w ∂ x 2 ( x , τ ) − μ B ( x , τ ) ∂ w ∂ x ( x , τ ) + A ( x , τ ) w ( x , τ ) = h ( x , τ ) , ( x , τ ) ∈ Ω ,

where Ω ≔ Ω x × ( 0 , T ] and Ω x ≔ ( 0 , L ) with L > 0 . Here, we assume that the coefficients A ( x , τ ) , B ( x , τ ) , and h ( x , τ ) are sufficiently smooth real-valued functions. We suppose further A ( x , τ ) satisfies

(2) A ( x , τ ) > 0 , x ∈ Ω x .

Moreover, 0 < μ ≪ 1 and 0 < ε ≪ 1 as the perturbation parameters are two small constant parameters. With (1), the following initial condition is given as:

(3) w ( x , 0 ) = g ( x ) , x ∈ Ω ¯ x ≔ [ 0 , L ] ,

and subject to the following two boundary conditions:

(4) w ( 0 , τ ) = a 0 ( τ ) , w ( L , τ ) = a L ( τ ) , τ ∈ [ 0 , T ] ,

where a 0 ( τ ) and a L ( τ ) are two given functions. We emphasize that the existence and uniqueness of the solution w ( x , τ ) of the aforementioned initial boundary value problem (BVP) are guaranteed if the data are sufficiently smooth and under some appropriate compatibility conditions (see [6,7,10] for details). On the other hand, this unique solution w ( x , τ ) exhibits two boundary layers at two extreme spatial points x = 0 , L . To the best of our knowledge, only a few methods have been developed for (1) with initial and boundary conditions (3)–(4). In [6], a parameter-uniform computational algorithm was constructed, which is based on an upwind finite difference scheme and a suitable piecewise uniform mesh. Besides, a scheme for the numerical treatments of (1)–(4) is based on a moving mesh-adaptive strategy, which adapts meshes to boundary layers proposed in [7]. The authors also proved that the proposed method is ε -uniform convergent of order O ( M − 1 + q + Δ τ ) under two restrictive conditions ε ⩾ μ 2 and M − q < C Δ τ . Here, M is the number of spatial grid points, Δ τ is the time step, and 0 < q < 1 is a real number. In [8], the authors proposed a parameter-uniform computational scheme based on higher-order finite difference method (FDM) for the model (1). Finally, a mesh-free procedure based on local radial basis function and FDM was investigated in [9].

Over the past decades, rapid strides have been made to develop various analytical and numerical procedures for solving time-dependent singular perturbation problems and some related model problems. Among various available approaches, let us mention the followings. The method is based on implicit Euler in time and upwind scheme on a nonuniform spatial mesh proposed in [11]. The sinc-Galerkin procedure is presented in [5], and the technique of Richardson extrapolations was developed in [13]. In [12], the authors used the Rothe in temporal direction and B-spline collocation scheme on a Shishkin mesh in space. Similarly, a combination of Crank-Nicolson and exponentially fitted operator finite difference scheme as well as the hybrid Crank-Nicolson/redefined cubic B-spline finite element method (FEM) developed in [14,15] respectively. Moreover, an upwind method based on the fitted Shishkin mesh was first proposed in [16]. Second, a hybrid approach comprises the midpoint upwind and the central difference method developed in [16]. The authors in [17] first discretized the time by the implicit-trapezoidal technique, and then, a combination of the midpoint upwind and central difference methods is utilized for the spatial variable. The streamline diffusion FEM was investigated in [18–20] and the finite element schemes on Bakhalov-type meshes considered in [21,22]. An exponentially fitted approach with regard to the space direction and the Crank-Nicolson scheme with respect to the time variable was proposed in [23] on a uniform mesh. Furthermore, we may point out some other effective methods for solving these kinds of model problems given in [24–32].

The primary aim of this manuscript is to examine two combined computational schemes to attain approximate solutions of (1)–(4) accurately and more straightforwardly than to existing numerical procedures. First, the temporal discretization is carried out via the Rothe horizontal method of line and the Taylor expansion series formula. Hence, a matrix collocation technique based on a novel set of orthogonal polynomials, which is called shifted Delannoy functions (SDFs), is applied to the spatial variable. Also, the convergence and error analysis of the proposed methods in each direction are investigated in detail. Hence, we prove that our proposed techniques are uniformly convergent with complexity O ( Δ τ s + 1 M ) , where Δ τ is the time step and M is the number of SDFs. Here, s = 1 corresponds to the Rothe approach, while s = 2 is related to the Taylor method.

The plan of this manuscript is given as follows. In Section 2, some fundamental facts of Delannoy polynomials together with their proof of convergence in the (weighted) L 2 and L ∞ are given. Time marching algorithms based on the Rothe and Taylor series approaches are given next in Section 3. Then, an investigation of the stability together with error estimations is considered. Section 4 is devoted to the implementations of two hybrid Rothe/Taylor-Delannoy methods in detail. In Section 5, the validations of the presented techniques are shown by comparison with existing numerical models and computational experiments. Finally, the conclusion is brought in Section 6. Throughout this study, the constants C and C 1 are positive independent of ε , M , and R as the number of time intervals. They may not have the same value at each occurrence.

2 Delannoy functions: A shifted version

Many enumeration problems can be expressed in terms of Delannoy numbers D m , k , where m , k ∈ N . We refer to [33–35] for more information. These numbers are related to the number of lattice paths from ( 0 , 0 ) to ( m , m ) in which the only allowed steps (jumps) are ( 1 , 0 ) , ( 0 , 1 ) , and ( 1 , 1 ) . The (central) Delannoy numbers are given by the formula:

D m , k = ∑ j = 0 m m j k j 2 j .

Based on the aforementioned description, we have the following definition related to D m ≡ D m , m :

Definition 2.1

The Delannoy polynomials are defined as:

(5) D m ( z ) = ∑ j = 0 m m j m + j j z j , m ∈ N .

Clearly, we have D 0 ( z ) = 1 and D 1 ( z ) = 1 + 2 z . Using the fact that D m ( z ) coincides with the Legendre polynomials L m ( 2 z + 1 ) , we obtain the following recursive relation:

(6) D m ( z ) = 2 m − 1 m ( 2 z + 1 ) D m − 1 ( z ) − m − 1 m D m − 2 ( z ) , m ⩾ 2 .

Also, the following orthogonality condition holds

(7) ∫ − 1 0 D n ( z ) D m ( z ) w ( z ) d z = 1 2 m + 1 δ m n ,

with respect to the weight function w ( z ) ≡ 1 . A simple calculation shows that these new polynomials satisfy the following differential equation:

(8) z ( z + 1 ) D m ″ ( z ) + ( 2 z + 1 ) D m ′ ( z ) = m ( m + 1 ) D m ( z ) .

Using the fact that the roots of the Legendre polynomials are located in ( − 1 , 1 ) [36, Thm. 62], we have the following observation:

Lemma 2.2

All zeros of Delannoy polynomials are negative, simple, and lie in ( − 1 , 0 ) .

In applications, we are mainly interested in using the Delannoy functions on a positive interval [ 0 , L ] . Thus, the shifted version of Delannoy polynomials in Definition 2.1 is defined next:

Definition 2.3

The SDFs on [ 0 , L ] are defined as:

(9) D m ⋆ ( x ) = D m ( z ) , z = x L − 1 , x ∈ [ 0 , L ] .

In the explicit form, according to Definition 2.1, we may write

(10) D m ⋆ ( x ) = ∑ j = 0 m L − j m j m + j j ( x − L ) j , m ∈ N .

Furthermore, the set of Delannoy polynomials { D m ⋆ ( x ) } m = 0 ∞ are satisfied in the relation:

(11) ∫ 0 L D n ⋆ ( x ) D m ⋆ ( x ) w ( x ) d x = L 2 m + 1 δ m n .

2.1 Convergence and error analysis

A function f ( x ) ∈ L 2 ( Ω x ) can be represented as a linear combination of SDFs. Thus, we have

(12) f ( x ) = ∑ m = 0 ∞ d m D m ⋆ ( x ) ,

with the coefficients d m , m ⩾ 1 being explicitly as:

(13) d m = 2 m + 1 L ∫ 0 L D m ⋆ ( x ) f ( x ) d x , m = 1 , 2 , … .

To proceed, we need the following results to estimate an upper bound for the coefficients d m given in (13).

Lemma 2.4

The following results are valid for the SDFs:

  1. D m ⋆ ( 0 ) = ( − 1 ) m , D m ⋆ ( L ) = 1 , and ∣ D m ⋆ ( x ) ∣ < 1 for 0 < x < L ,

  2. D m ⋆ ( x ) = L 2 ( 2 m + 1 ) d d x D m + 1 ⋆ ( x ) − d d x D m − 1 ⋆ ( x ) ,

  3. ∣ D m ⋆ ( x ) ∣ ⩽ L 2 π 2 m x ( L − x ) , for x ∈ ( 0 , L ) .

Proof

The proof of i) is obtained by using the fact that the Legendre polynomials satisfy L m ( 1 ) = 1 and L m ( − 1 ) = ( − 1 ) m . Also, we have ∣ L m ( z ) ∣ < 1 for − 1 < z < 1 . The second one is deduced from the following relation [36]:

(14) L m ( z ) = 1 2 m + 1 ( L m + 1 ′ ( z ) − L m − 1 ′ ( z ) ) ,

and then using the transformation z = 2 x ∕ L − 1 . To prove (iii), we apply the aforementioned change of variable to the following inequality, which is proved for the Legendre polynomials [36, Thm. 61]:

□ ∣ L m ( z ) ∣ < π 2 m ( 1 − z 2 ) 1 2 , − 1 < z < 1 .

It should be noted that the case (iii) provided an upper bound for the SDPs on [ 0 , L ] . We use this fact to prove that the infinite expansion series (12) is uniformly convergent.

Theorem 2.5

Suppose that a given function f ∈ L 2 ( Ω x ) ∩ C 2 ( Ω ¯ x ) can be written as (12) and M 2 = max x ∈ Ω x ∣ f ( 2 ) ( x ) ∣ . Then, the series in (12) is uniformly convergent. Moreover, the following upper bound is valid for the coefficients d m in (13) as:

(15) ∣ d m ∣ ⩽ C m − 3 2 , m > 3 ,

where C = 1 2 π π L 2 M 2 .

Proof

We start with relation (ii) in Lemma 2.4 to arrive at:

(16) I ≔ ∫ 0 L D m ⋆ ( x ) f ( x ) d x = L 2 ( 2 m + 1 ) ∫ 0 L d d x D m + 1 ⋆ ( x ) f ( x ) d x − ∫ 0 L d d x D m − 1 ⋆ ( x ) f ( x ) d x .

Applying the integration by parts (IP) to both integrals on the right-hand side and using the relations (i) in Lemma 2.4 reveal that

∫ 0 L d d x D s ⋆ ( x ) f ( x ) d x = f ( L ) + ( − 1 ) s + 1 f ( 0 ) − ∫ 0 L D s ⋆ ( x ) f ′ ( x ) d x , s = m ± 1 .

By inserting the former relations into (16) and canceling the similar terms, we find out

(17) I = L 2 ( 2 m + 1 ) ∫ 0 L D m − 1 ⋆ ( x ) f ′ ( x ) d x − ∫ 0 L D m + 1 ⋆ ( x ) f ′ ( x ) d x .

Using the relation (ii) in Lemma 2.4 for m → m ± 1 in the aforementioned relation (17), we obtain

I = L 2 ( 2 m + 1 ) ∫ 0 L L 2 ( 2 m − 1 ) d d x D m ⋆ ( x ) − d d x D m − 2 ⋆ ( x ) f ′ ( x ) d x − ∫ 0 L L 2 ( 2 m + 3 ) d d x D m + 2 ⋆ ( x ) − d d x D m ⋆ ( x ) f ′ ( x ) d x .

By repeating the IP in the last terms and then simplifying, we obtain

I = L 2 4 ( 4 m 2 − 1 ) ∫ 0 L ( D m − 2 ⋆ ( x ) − D m ⋆ ( x ) ) f ″ ( x ) d x − L 2 4 ( 2 m + 1 ) ( 2 m + 3 ) ∫ 0 L ( D m ⋆ ( x ) − D m + 2 ⋆ ( x ) ) f ″ ( x ) d x .

Therefore, using the triangle inequality, we obtain

∣ d m ∣ = 2 m + 1 L ∣ I ∣ ⩽ L M 2 4 ( 2 m − 1 ) ∫ 0 L ( ∣ D m − 2 ⋆ ( x ) ∣ + ∣ D m ⋆ ( x ) ∣ ) d x + L M 2 4 ( 2 m + 3 ) ∫ 0 L ( ∣ D m ⋆ ( x ) ∣ + ∣ D m + 2 ⋆ ( x ) ∣ ) d x .

Now, we apply the third inequality (iii) in Lemma 2.4 to arrive at

∣ d m ∣ ⩽ L M 2 4 ( 2 m − 1 ) ∫ 0 L 1 x ( L − x ) L 2 π 2 ( m − 2 ) + L 2 π 2 m d x + L M 2 4 ( 2 m + 3 ) ∫ 0 L 1 x ( L − x ) L 2 π 2 m + L 2 π 2 ( m + 2 ) d x ⩽ 2 L 2 M 2 4 ( 2 m − 1 ) π 2 ( m − 2 ) ∫ 0 L d x x ( L − x ) .

It is not a difficult task to show that

∫ 0 L d x x ( L − x ) = π .

By utilizing the obvious inequality 2 m − 1 > 2 m − 4 , we obtain

(18) ∣ d m ∣ ⩽ 1 2 L 2 M 2 π 2 m − 4 3 ∕ 2 , m ⩾ 3 .

Clearly, for m > 3 , we have 2 m − 4 > m . Thus, the proof is complete.□

Instead of infinite series solution given in (12), we consider practically a truncated series solution that consists of ( M + 1 ) Delannoy functions in the form:

(19) f ( x ) ≈ f M ( x ) ≔ ∑ m = 0 M d m D m ⋆ ( x ) .

Let us denote the error between two consecutive approximations f M and f M + 1 by:

e M ( x ) ≔ f M + 1 ( x ) − f M ( x ) .

Furthermore, ‖ g ‖ 2 represents the (weighted) L 2 norm on Ω x with regard to the weight function w ( x ) = 1 . The next theorem gives us an error estimation for the error e M .

Theorem 2.6

Under the hypothesis of Theorem 2.5, the following error estimate holds

‖ e M ( x ) ‖ 2 = O ( M − 2 ) .

Proof

According to the definition of error and (19), we obtain

‖ e M ( x ) ‖ 2 = ‖ f M + 1 ( x ) − f M ( x ) ‖ 2 = ∑ m = 0 M + 1 d m D m ⋆ ( x ) − ∑ m = 0 M d m D m ⋆ ( x ) 2 = ‖ d M + 1 D M + 1 ⋆ ( x ) ‖ 2 = ∣ d M + 1 ∣ ‖ D M + 1 ⋆ ( x ) ‖ 2 .

Now, we utilize the orthogonality condition (11) and the result of Theorem 2.5 to obtain

□ ‖ e M ( x ) ‖ 2 = ∣ d M + 1 ∣ L 2 M + 3 < C ( M + 1 ) − 3 2 L 1 2 2 M + 3 = O ( M − 2 ) .

Finally, we derive an upper bound for the (global) error between the infinite series expansion of f ( x ) in (12) and its truncated series expansion f M ( x ) in (19). To continue, we define E M ( x ) = f ( x ) − f M ( x ) .

Theorem 2.7

Suppose that f ( x ) satisfies the hypothesis of Theorem 2.5  and E M ( x ) = ∑ m = M + 1 ∞ d m D m ⋆ ( x ) denotes the global error. Then, we have an error estimate for E M ( x ) in the L ∞ norm as:

(20) ‖ E M ‖ ∞ ⩽ C 1 M − 1 2 , C 1 ≔ 2 C .

Proof

By utilizing Lemma 2.4, part (i) together with the result of Theorem 2.5 reveals that

∣ E M ( x ) ∣ ⩽ ∑ m = M + 1 ∞ ∣ d m ∣ ∣ D m ⋆ ( x ) ∣ ⩽ ∑ m = M + 1 ∞ ∣ d m ∣ ⩽ C ∑ m = M + 1 ∞ 1 m 3 2 .

Now, we use the well-known integral test, which says that

∑ m = M + 1 ∞ 1 m 3 2 ⩽ ∫ M ∞ y − 3 2 d y = 2 M .

Thus, the proof is completed by taking maximum over all x ∈ Ω x .□

3 Time advancement methodologies

Here, we propose two time-marching schemes to discretize the parabolic CDR equation (1). To this end, we first employ the Rothe method [37,38] as a first-order accurate procedure and we then consider the Taylor series formula to obtain a second-order accuracy in time (see [39–41]).

Let { τ r } r = 0 R be a partitioning of the interval [ 0 , T ] into R uniform subintervals R r ≔ [ τ r − 1 , τ r ] for r = 1 , 2 , … , R . Here, the grid points and the time step are given by:

τ 0 = 0 < τ 1 = Δ τ < … < τ R = R Δ τ = T , Δ τ = τ r − τ r − 1 .

In the sequel, we use the symbol w r to show the approximate solution at time level τ r for r = 1 , 2 , … , R , i.e., w r ( x ) ≔ w ( x , τ r ) .

3.1 Rothe first-order approach

The idea of Rothe technique consists of replacing the time derivative at the fixed points τ r = r Δ τ by the following difference formula:

(21) ∂ w ∂ τ ( x , τ r ) ≈ w r ( x ) − w r − 1 ( x ) Δ τ .

Moreover, the other x -derivatives can be expressed as:

w r ′ ( x ) ≔ ∂ w ∂ x ( x , τ r ) , w r ″ ( x ) ≔ ∂ 2 w ∂ x 2 ( x , τ r ) .

By replacing all terms in relation (1) by the aforementioned approximations at time level τ r , we obtain the following BVPs:

w r ( x ) − w r − 1 ( x ) Δ τ − ε w r ″ ( x ) − μ B r ( x ) w r ′ + A r ( x ) w r ( x ) = h r ( x ) , w r ( 0 ) = a 0 ( τ r ) ≕ a 0 r , w r ( L ) = a L ( τ r ) ≕ a L r , r = 1 , 2 , … , R ,

where A r ( x ) = A ( x , τ r ) , B r ( x ) = B ( x , τ r ) , and h r ( x ) = h ( x , τ r ) . After some manipulations, we rewrite the foregoing BVPs elegantly as:

(22) ( 1 + Δ τ A r ( x ) ) w r ( x ) − Δ τ μ B r ( x ) w r ′ ( x ) − ε Δ τ w r ″ ( x ) = Δ τ h r ( x ) + w r − 1 ( x ) , x ∈ Ω x , w r ( 0 ) = a 0 r , w r ( L ) = a L r ,

for r = 1 , 2 , … , R and using the starting function w 0 ( x ) = g ( x ) as the given initial condition for the model Problem (1). If we solve the BVPs (22) subsequently, the functions w 1 ( x ) , w 2 ( x ) , … w R ( x ) are obtained. Hence, one can construct the so-called Rothe functions W 1 ( x , τ ) in the whole domain Ω as:

W 1 ( x , τ ) = w r − 1 ( x ) + w r ( x ) − w r − 1 ( x ) Δ τ ( τ − τ r − 1 ) ,

on the subinterval R r for r = 1 , 2 , … , R . Clearly, the constructed W 1 ( x , τ ) is a piecewise function with respect to τ for every fixed x . By repeating the aforementioned approach with time steps Δ τ ∕ 2 , Δ τ ∕ 4 , Δ τ ∕ 8 , … , we obtain a sequence of related Rothe functions { W m ( x , τ ) } m = 1 ∞ . Our expectation is that this sequence is convergent towards the true solution w ( x , τ ) of the original equation (1).

3.2 Taylor second-order approach

To proceed, we first substitute τ = τ r at the original equation (1) to obtain

(23) ∂ w r ∂ τ = ε ∂ 2 w r ∂ x 2 + μ B r ( x ) ∂ w r ∂ x − A r ( x ) w r + h r ( x ) .

By means of the Taylor formula for w r , we obtain

(24) ∂ w r ∂ τ = w r + 1 − w r Δ τ − Δ τ 2 ∂ 2 w r ∂ τ 2 + O ( Δ τ 2 ) .

We now differentiate (1) with regard to time variable τ followed by evaluating the resulting equation at τ r to obtain

∂ 2 w r ∂ τ 2 = ε ∂ 2 ∂ x 2 ∂ w r ∂ τ + μ ∂ B r ∂ τ ∂ w r ∂ x + B r ( x ) ∂ ∂ x ∂ w r ∂ τ − ∂ A r ∂ τ w r ( x ) − A r ( x ) ∂ w r ∂ τ + ∂ h r ∂ τ .

We now exploit the difference quotient h r + 1 − h r Δ τ instead of ∂ h r ∂ τ in the former relation. After multiplying both sides of the resulting relation by Δ τ , we obtain

(25) Δ τ ∂ 2 w r ∂ τ 2 = ε ∂ 2 w r + 1 ∂ x 2 − ∂ 2 w r ∂ x 2 + μ ( B r + 1 − B r ) ∂ w r ∂ x + μ B r ( x ) ∂ w r + 1 ∂ x − ∂ w r ∂ x − ( A r + 1 − A r ) w r ( x ) − A r ( x ) ( w r + 1 − w r ) + h r + 1 − h r .

We now come back to relation (24). On the left-hand side, we use relation (23) for the term ∂ w r ∂ τ . On the right-hand side, we place Δ τ ∂ 2 w r ∂ τ 2 via relation (25). After simplifying and then multiplying by 2 Δ τ , we arrive at the time discretized equation for (1) with second-order accuracy. Consequently, we obtain the following differential equation of second order as:

(26) σ 2 , r − 1 ( x ) w r ″ ( x ) + σ 1 , r − 1 ( x ) w r ′ ( x ) + σ 0 , r − 1 ( x ) w r ( x ) = H r − 1 ( x ) , x ∈ Ω x ,

for r = 1 , 2 , … , R . Here, the related coefficients are given by:

σ 2 , r − 1 ( x ) ≔ − ε Δ τ , σ 1 , r − 1 ( x ) ≔ − μ Δ τ B r − 1 ( x ) , σ 0 , r − 1 ( x ) ≔ 2 + Δ τ A r − 1 ( x ) , H r − 1 ( x ) ≔ [ 2 − Δ τ A r ( x ) ] w r − 1 ( x ) + Δ τ ( h r + h r − 1 ) + Δ τ ε w r − 1 ″ ( x ) + μ Δ τ B r ( x ) w r − 1 ′ ( x ) .

Contrary to the Rothe method, in order to solve the system of linear equations (26), we require to have the initial condition w 0 ( x ) = g ( x ) as well as the first- and second-order derivatives. At every time step τ r , the boundary conditions (4) at x = 0 , L are converted as:

(27) w r ( 0 ) ≔ a 0 r = a 0 ( τ r ) , w r ( L ) ≔ a L r = a L ( τ r ) , r = 1 , 2 , … , R .

3.3 Stability and error estimate in time

We now investigate the stability and convergence analysis of the discretized equation (26) and will prove its second-order accuracy. Similar techniques can be applied to establish the first-order accuracy of model (22). To proceed, we introduce the linear operator ℒ ε related to (26) as:

ℒ ε [ w r ] ( x ) ≔ − ε d 2 d x 2 w r ( x ) − μ B r − 1 ( x ) d d x w r ( x ) + 2 Δ τ + A r − 1 ( x ) w r ( x ) .

Thus, we rewrite the relation (26) in a compact form:

(28) ℒ ε [ w r ] ( x ) = H r − 1 ( x ) , x ∈ Ω x ,

together with the following initial and boundary conditions:

(29) w 0 ( x ) = g ( x ) , w r ( 0 ) = a 0 r , w r ( L ) = a L r ,

for r = 1 , 2 , … , R , and we also have

H r − 1 ( x ) ≔ 2 Δ τ − A r ( x ) w r − 1 ( x ) + ε w r − 1 ″ ( x ) + μ B r ( x ) w r − 1 ′ ( x ) + h r ( x ) + h r − 1 ( x ) .

As mentioned earlier, the symbol w r ( x ) ≈ w ( x , τ r ) denotes the approximate solution of (28) at time level τ r .

Next, we state a theorem on the maximum principle for the operator ℒ ε as follows. The following maximum principle [42] holds for ℒ ε .

Lemma 3.1

(Semi-discrete maximum principle) Let assume that function v ∈ C 2 ( Ω ¯ x ) satisfies v ( 0 ) ⩾ 0 and v ( L ) ⩾ 0 . If ℒ ε [ v ] ( x ) ⩾ 0 for all x ∈ Ω x , then v ( x ) ⩾ 0 for all x ∈ Ω ¯ x .

Proof

The proof is done by contradiction. Let suppose that x ⋆ be a point such that v ( x ⋆ ) ≔ min x ∈ Ω ¯ x v ( x ) and v ( x ⋆ ) < 0 . By the assumptions, x ⋆ ≠ 0 , L . This implies that v ′ ( x ⋆ ) = 0 and v ″ ( x ⋆ ) > 0 . Now, we have

ℒ ε [ v ] ( x ⋆ ) = − ε v ″ ( x ⋆ ) + 2 Δ τ + A r ( x ⋆ ) v ( x ⋆ ) < 0 ,

since A r ( x ⋆ ) > 0 due to the constraint (2) we have assumed. This contradicts the hypothesis, and one concludes that v ( x ⋆ ) > 0 . Therefore, our conclusion is that v ( x ) ⩾ 0 for all x ∈ Ω ¯ x .□

Due to the fact that the operator ℒ ε in (28) satisfies in a maximum principle, the immediate conclusion is that the semi-discretized method (28)–(29) is stable in the following sense [11].

Corollary 3.2

(Stability) The following stability estimate holds

(30) ‖ ℒ ε − 1 ‖ ∞ ⩽ C ,

where C is a constant (independent of Δ τ and ε ).

The uniform convergence properties of semi-discretized method (28)–(29) is our next aim. To this end, we define the local truncation error (LTE) between w r ( x ) and true solution w ( x , τ r ) as:

e r ( x ) ≔ w ( x , τ r ) − w r ( x ) , x ∈ Ω x .

Lemma 3.3

(Local error estimate [LEE]) If there is a constant C such that

∂ k w ( x , τ ) ∂ τ k ⩽ C , ∀ ( x , τ ) ∈ Ω ¯ x × [ 0 , T ] , k = 0 , 1 , 2 , 3 ,

then the LEE in the temporal direction satisfies

(31) ‖ e r ‖ ∞ = O ( Δ τ 3 ) .

Proof

We now rewrite Relation (28) in a new formulation as:

(32) w r − 1 ( x ) = w r ( x ) − Δ τ 2 [ ε ( w r − 1 ″ ( x ) + w r ″ ( x ) ) + μ ( B r − 1 ( x ) w r ′ ( x ) + B r ( x ) w r − 1 ′ ( x ) ) − ( A r − 1 ( x ) w r ( x ) + A r ( x ) w r − 1 ( x ) ) + h r ( x ) + h r − 1 ( x ) ] .

Let w ( x , τ ) be the smooth solution of the original equation (1). By using the Taylor formula with integral remainder, we have

(33) w ( x , τ r − 1 ) = w ( x , τ r ) − Δ τ w τ ( x , τ r ) + Δ τ 2 2 w τ τ ( x , τ r ) + 1 2 ∫ τ r τ r − 1 ( τ r − 1 − ξ ) 2 ∂ 3 w ∂ τ 3 ( x , ξ ) d ξ .

We now utilize expression (23) with w ( x , τ r ) instead of w r ( x ) to obtain

(34) w τ ( x , τ r ) = ε ∂ 2 w ( x , τ r ) ∂ x 2 + μ B r ( x ) ∂ w ( x , τ r ) ∂ x − A r ( x ) w ( x , τ r ) + h r ( x ) .

For the second-order derivative w τ τ ( x , τ r ) , we use (25) except that on the right-hand side, we utilize the backward difference formula rather than forward one to obtain

(35) Δ τ w τ τ ( x , τ r ) = ε ∂ 2 w ( x , τ r ) ∂ x 2 − ∂ 2 w ( x , τ r − 1 ) ∂ x 2 + μ ( B r ( x ) − B r − 1 ( x ) ) ∂ w ( x , τ r ) ∂ x + μ B r ( x ) ∂ w ( x , τ r ) ∂ x − ∂ w ( x , τ r − 1 ) ∂ x − ( A r − A r − 1 ) w ( x , τ r ) − A r ( x ) ( w ( x , τ r ) − w ( x , τ r − 1 ) ) + h r ( x ) − h r − 1 ( x ) .

The next task is to substitute two forgoing Relations (34) and (35) into (33). Some simplifications give us

(36) w ( x , τ r − 1 ) = w ( x , τ r ) − ε Δ τ 2 ( w x x ( x , τ r ) + w x x ( x , τ r − 1 ) ) − μ Δ τ 2 ( B r − 1 w x ( x , τ r ) + B r w x ( x , τ r − 1 ) ) + Δ τ 2 ( A r − 1 w ( x , τ r ) + A r w ( x , τ r − 1 ) ) − Δ τ 2 ( h ( x , τ r ) + h ( x , τ r − 1 ) ) + O ( Δ τ 3 ) .

We now subtract Relation (36) from (32). Consequently, we find that the LTE satisfies in the following Dirichlet BVPs:

(37) ℒ ε [ e r ] ( x ) = O ( Δ τ 3 ) , e r ( 0 ) = 0 , e r ( L ) = 0 .

The boundedness of inverse operator given in (30) helps us to show that

‖ e r ‖ ∞ ⩽ ‖ ℒ ε − 1 ‖ ∞ O ( Δ τ 3 ) .

Now, the desired result is proved.□

Let us combine the consistency and the stability presented semi-discretized BVP (28)–(29). In the following result, we derive an upper bound for the global error estimate (GEE) defined by:

E r ( x ) ≔ ψ ( x , τ r ) − Ψ r ( x ) , r ⩽ T Δ τ .

It should be stressed that all LEE up to time level contributes to the GEE at time instant τ r .

Lemma 3.4

(GEE) Under the assumption of Lemma 3.3, the GEE in the time direction satisfies

(38) ‖ E r ‖ ∞ = O ( Δ τ 2 ) , ∀ r Δ τ ⩽ T .

Proof

After summing the LEE given via Relation (31) in Lemma 3.3 over s = 1 , 2 , … , r , one obtains

‖ E r ‖ ∞ = ∑ s = 1 r e s ∞ ( r Δ τ ⩽ T ) ⩽ ‖ e 1 ‖ ∞ + ‖ e 2 ‖ ∞ + … + ‖ e r ‖ ∞ ⩽ r ( C 1 Δ τ 3 ) = C 1 ( r Δ τ ) Δ τ 2 ⩽ C 1 T Δ τ 2 = O ( Δ τ 2 ) ,

where the constant C 1 is not dependent on Δ τ .□

The previous Lemma indicates that the presented time discretized approach is uniformly convergent with order 2.

Remark 3.5

Similar lemmas and results can be proved for the semi-discretized approach (22) via the Rothe method. In this case, the LEE and GEE satisfy ‖ e r ‖ ∞ = O ( Δ τ 2 ) and ‖ E r ‖ ∞ = O ( Δ τ ) for all r Δ τ ⩽ T , respectively. Thus, the order of (uniform) convergence is one as expected.

4 Two hybrid approaches

In the previous parts, the Rothe and Taylor approaches are used to discretize the parabolic CDR equation (1) with respect to time. These two discretized time equations are given by Relations (22) as well as (26)–(27). Our goal is here to solve these initial BVPs with respect to the spatial variable x .

In both methods and at each time step τ r , the solutions w r ( x ) are approximated as a combination of SDPs (19). Supposedly, we have W r − 1 , M ( x ) as the Delannoy approximation to w r − 1 ( x ) at time level τ r − 1 . Note that for r = 1 , we utilize the initial condition g ( x ) for w 0 ( x ) . Proceeding in the next time level τ r , we take the approximate solution W r , M ( x ) for r = 1 , 2 , … , R as:

(39) W r , M ( x ) = ∑ m = 0 M p m r D m ⋆ ( x ) , x ∈ Ω x .

In both Rothe-Delannoy and Taylor-Delannoy techniques presented in the following, we seek the unknowns p m r , m = 0 , 1 , … , M . By introducing

P M r = [ p 0 r p 1 r … p M r ] t , D M ( x ) = [ D 0 ⋆ ( x ) D 1 ⋆ ( x ) … D M ⋆ ( x ) ] ,

we may express W r , M ( x ) as:

(40) W r , M ( x ) = D M ( x ) P M r .

It is also not a difficult task to show that the vector of SDFs, i.e., D M ( x ) can be written as:

(41) D M ( x ) = χ M ( x ) K M ,

where χ M ( x ) as the vector consisting of powers of ( x − L ) is given by:

χ M ( x ) = [ 1 x − L ( x − L ) 2 … ( x − L ) M ] ,

and K M as a lower-triangular matrix is defined by:

K M = 0 0 0 0 1 0 1 0 2 0 2 0 … M − 1 0 M − 1 0 M 0 M 0 0 L − 1 1 1 2 1 L − 1 2 1 3 1 … L − 1 M − 1 1 M 1 L − 1 M 1 M + 1 1 0 0 L − 2 2 2 4 2 … L − 2 M − 1 2 M + 1 2 L − 2 M 2 M + 2 2 ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ 0 0 0 … L 1 − M M − 1 M − 1 2 M − 2 M − 1 L 1 − M M M − 1 2 M − 1 M − 1 0 0 0 … 0 L − M M M 2 M M ( M + 1 ) × ( M + 1 ) .

It can be clearly seen that

det ( K M ) = ∏ j = 0 M L − j j j 2 j j = L − M ( M + 1 ) ∕ 2 ∏ j = 1 M 2 j j ,

which is non-zero. Therefore, K M is in an invertible matrix. With the aid of Relation (41), we have

(42) W r , M ( x ) = D M ( x ) P M r = χ M ( x ) K M P M r .

One can also prove that the derivatives of χ M ( x ) can be written in terms of itself. To be more precise, we have

(43) d ν d x ν χ M ( x ) = χ M ( x ) ( E M ) ν , E M = 0 1 0 … 0 0 0 2 … 0 ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 0 ⋱ M 0 0 0 … 0 ( M + 1 ) × ( M + 1 ) ,

for ν = 1 , 2 . Indeed, E M is called the differentiation matrix. A set of collocation points on Ω x must be given to obtain the numerical solutions of BVPs (22) as well as (26)–(27). Thus, a uniform partitioning of Ω x in the following form is considered

(44) x ξ = L ξ M , ξ = 0 , 1 , … , M .

In Relation (42), we differentiate twice with regard to the variable x . Utilizing Relation (43) to obtain

(45) W r , M ( 1 ) ( x ) = χ M ( x ) E M K M P M r , W r , M ( 2 ) ( x ) = χ M ( x ) ( E M ) 2 K M P M r .

The following theorem states the ways that the approximate solutions W r , M ( x ) , d d x W r , M ( x ) , and d 2 d x 2 W r , M ( x ) can be written in the matrix forms. We proceed by defining the following vectors:

(46) X M = χ M ( x 0 ) χ M ( x 1 ) ⋮ χ M ( x M ) , W r = W r , M ( x 0 ) W r , M ( x 1 ) ⋮ W r , M ( x M ) , W r ( 1 ) = W r , M ( 1 ) ( x 0 ) W r , M ( 1 ) ( x 1 ) ⋮ W r , M ( 1 ) ( x M ) , W r ( 2 ) = W r , M ( 2 ) ( x 0 ) W r , M ( 2 ) ( x 1 ) ⋮ W r , M ( 2 ) ( x M ) .

Theorem 4.1

For r = 1 , 2 , … , R , the matrix representations of d ν d x ν W r , M ( x ) at the set of collocation points (44) are given by:

(47) W r = X M K M P M r ,

(48) W r ( 1 ) = X M E M K M P M r ,

(49) W r ( 2 ) = X M ( E M ) 2 K M P M r .

Proof

The proof is straightforward by placing the points (44) into Relations (40)–(45) followed by utilizing the defined vectors (46).□

4.1 Rothe-Delannoy method

We now turn to Relation (22) and write it in a compact form by letting its corresponding coefficients as:

σ 2 , r ( x ) ≔ − ε Δ τ , σ 1 , r ( x ) ≔ − μ Δ τ B r ( x ) , σ 0 , r ( x ) ≔ 1 + Δ τ A r ( x ) , H r ( x ) ≔ Δ τ h r ( x ) + w r − 1 ( x ) .

Afterward, we substitute the set of collocation points (44) into the resulting equations to arrive at:

(50) σ 2 , r ( x ξ ) w r ″ ( x ξ ) + σ 1 , r ( x ξ ) w r ′ ( x ξ ) + σ 0 , r ( x ξ ) w r ( x ξ ) = H r ( x ξ ) ,

for r = 1 , 2 , … , R . In the matrix notation, we express the latter equations compactly as:

(51) Σ 2 , r W r ( 2 ) + Σ 1 , r W r ( 1 ) + Σ 0 , r W r = H r , r = 1 , 2 , … , R ,

where the vector H r and coefficient matrices Σ j , r , j = 0 , 1 , 2 are given by:

H r = H r ( x 0 ) H r ( x 1 ) ⋮ H r ( x M ) ( M + 1 ) × 1 , Σ j , r = σ j , r ( x 0 ) 0 … 0 0 σ j , r ( x 1 ) … 0 ⋮ ⋮ ⋱ ⋮ 0 0 … σ j , r ( x M ) ( M + 1 ) × ( M + 1 ) .

Now, for r = 1 , 2 , … , R , we obtain the fundamental matrix equation by placing Relation (47) into the former equation (51) as:

(52) Y r P M r = H r , [ Y r ; H r ] ,

where

Y r ≔ { Σ 2 , r X M ( E M ) 2 + Σ 1 , r X M E M + Σ 0 , r X M } K M .

4.2 Taylor-Delannoy method

Similar to the latter Rothe-Delannoy approach, we consider Relation (26) and inserting the set of collocation points (44) into it to obtain

(53) σ 2 , r − 1 ( x ξ ) w r ″ ( x ξ ) + σ 1 , r − 1 ( x ξ ) w r ′ ( x ξ ) + σ 0 , r − 1 ( x ξ ) w r ( x ξ ) = H r − 1 ( x ξ ) ,

for r = 1 , 2 , … , R . We next introduce the vector H r − 1 and the coefficient matrices Σ j , r − 1 , j = 0 , 1 , 2 as:

H r − 1 = H r − 1 ( x 0 ) H r − 1 ( x 1 ) ⋮ H r − 1 ( x M ) , Σ j , r − 1 = σ j , r − 1 ( x 0 ) 0 … 0 0 σ j , r − 1 ( x 1 ) … 0 ⋮ ⋮ ⋱ ⋮ 0 0 … σ j , r − 1 ( x M ) .

Therefore, we are now able to rewrite the system of equations in a compact representation form as:

(54) Σ 2 , r − 1 W r ( 2 ) + Σ 1 , r − 1 W r ( 1 ) + Σ 0 , r − 1 W r = H r − 1 , r = 1 , 2 , … , R .

By setting

Y r − 1 ≔ { Σ 2 , r − 1 X M ( E M ) 2 + Σ 1 , r − 1 X M E M + Σ 0 , r − 1 X M } K M ,

and putting Relations (47) into the corresponding equation (54), we arrive at related fundamental matrix equation:

(55) Y r − 1 P M r = H r − 1 , [ Y r − 1 ; H r − 1 ] .

4.3 Matrix expressions of boundary conditions

The fundamental matrix equations in both Rothe and Taylor-Delannoy techniques are obtained in the last parts. The implementations of boundary conditions have so far been left undone. We need to incorporate the related boundary conditions (22) and (27) into it in both approaches. We consider the relation (40) and let x → 0 , L in order to implement the boundary conditions. Thus, we obtain

Y ^ r 0 P M r = a 0 r , Y ^ r 0 ≔ χ M ( 0 ) K M , Y ^ r L P M r = a L r , Y ^ r L ≔ χ M ( L ) K M .

Or equivalently, we obtain two rows [ Y ^ r 0 ; a 0 r ] and [ Y ^ r L ; a L r ] in each time step τ r . We now replace two rows of the fundamental matrix equations [ Y r ; H r ] and [ Y r − 1 ; H r − 1 ] in (52) and (55), respectively, by the two above new rows. We finally obtain the modified matrix equations denoted by [ Y ^ r ; H ^ r ] and [ Y ^ r − 1 ; H ^ r − 1 ] . In each case, by solving the modified equation, we obtain the Delannoy coefficients p 0 r , p 1 r , … , p M r in each time level.

4.4 Convergence analysis of the hybrid schemes

Let us investigate the convergence analysis of two proposed hybrid techniques, namely, the Talyor/Rothe-Delannoy approaches. In this respect, we define the error function:

(56) E r , M ( x ) ≔ w ( x , τ r ) − W r , M ( x ) , x ∈ Ω x .

The next theorem establishes that the method is uniformly convergent with respect to both space and time variables.

Theorem 4.2

Suppose that w ( x , τ ) ∈ C ( 2 , 3 ) ( Ω ) is the exact solution of (1). Also, suppose that the solution w r ( x ) of (26) or (28) satisfies the hypotheses of Theorem 2.6. Then, the following estimate for the (global) error E r , M ( x ) holds

‖ E r , M ‖ ∞ = O Δ τ 2 + 1 M , r Δ τ ⩽ T .

Proof

By applying the triangle inequality, we have

‖ E r , M ‖ ∞ ⩽ ‖ w ( x , τ r ) − w r ( x ) ‖ ∞ + ‖ w r ( x ) − W r , M ( x ) ‖ ∞ .

We use the estimate (38) proved in Lemma 3.4 for the first term in above. For the second term, it is sufficient to use the estimate (20) given in Theorem 2.7 with w r ( x ) instead of f ( x ) .□

In view of Remark (3.5) and under condition w ( x , τ ) ∈ C ( 2 , 2 ) ( Ω ) is the exact solution of (1), we can similarly prove that the Rothe-Delannoy collocation technique is uniformly convergent of order ‖ E r , M ‖ ∞ = O Δ τ + 1 M for all r Δ τ ⩽ T .

5 Graphical and implementation results

In this part, a set of numerical simulations is performed to demonstrate the accuracy of the proposed Rothe/Taylor-Delannoy collocation matrix techniques and support the theoretical findings. For validation of the obtained results, a comparison will be drawn between the numerical outcomes of some available computational procedures. The experiments were conducted on an ASUS laptop computer with 16 GB RAM and with 2.2 GHz Intel Core i7 CPU. We used Matlab software version 2021a.

Test example 5.1

As an academic test case, we first set the related coefficients and non-homogeneous term in (1) as given by [7]:

B ( x , τ ) = − ( 1 + e x ) , A ( x , τ ) = 1 + x 4 ∣ x ∣ , h ( x , τ ) = 10 ( x 2 − x 4 ) e τ 2 .

The initial and boundary conditions are given for L = 1 as:

g ( x ) = x 3 ( 1 − x 3 ) , a 0 ( τ ) = 0 , a L ( τ ) = 0 .

Test example 5.2

In the second test problem, we take the following coefficients as [7–9]:

B ( x , τ ) = − ( τ 2 + 1 + x − x 2 ) , A ( x , τ ) = 1 + 5 x τ , h ( x , τ ) = ( e τ − 1 ) x ( 1 − x ) .

For L = 1 , we have the next initial and boundary conditions:

g ( x ) = 0 , a 0 ( τ ) = 0 , a L ( τ ) = 0 .

No exact solutions are given for these two test cases. Thus, we consider the residual error functions (REFs) defined at time level τ r related to the Rothe and Taylor approaches given by:

(57) R r , M R o t ( x ) ≔ ∣ σ 2 , r ( x ) W r , M ( 2 ) ( x ) + σ 1 , r ( x ) W r , M ( 1 ) ( x ) + σ 0 , r ( x ) W r , M ( x ) − H r ( x ) ∣ , R r , M T a y ( x ) ≔ ∣ σ 2 , r − 1 ( x ) W r , M ( 2 ) ( x ) + σ 1 , r − 1 ( x ) W r , M ( 1 ) ( x ) + σ 0 , r − 1 ( x ) W r , M ( x ) − H r − 1 ( x ) ∣ , x ∈ Ω x ,

for r = 1 , … , R . Note that the coefficients σ i , s ( x ) , i = 0 , 1 , 2 , and s = r , r − 1 are defined in (50) and (26), respectively. Furthermore, the norms of REFs in the L ∞ and L 2 at the final time step τ R are

R ∞ ≡ R ∞ ( R , M ) ≔ max x ∈ Ω x R R , M Rot ∕ Tay ( x ) , R 2 ≡ R 2 ( R , M ) ≔ ‖ R R , M Rot ∕ Tay ( x ) ‖ 2 .

The following formulae are further utilized to calculate the obtained order of convergence (OC) related to both numerical techniques (by fixing R or M ) given by:

(58) OC τ d ≔ log 2 R d ( R , M ) R d ( 2 R , M ) , OC x d ≔ log 2 R d ( R , M ) R d ( R , 2 M ) , d = 2 , ∞ .

5.1 Rothe-Delannoy approach

We first consider the parameters ε , μ = 0.1 . Using M = 10 and Δ τ = 0.1 , the approximate solutions are obtained in the following forms at the time step τ = T for Examples 5.1 and 5.2, respectively,

W 10 , 10 ( x ) = − 3.00485 ( x − 1 ) x 10 − 20.3634 ( x − 1 ) 9 − 61.4892 ( x − 1 ) 8 − 105.775 ( x − 1 ) 7 − 108.697 ( x − 1 ) 6 − 57.5927 ( x − 1 ) 5 − 5.86718 ( x − 1 ) 4 − 4.79809 ( x − 1 ) 3 − 20.5385 ( x − 1 ) 2 − 11.0679 ( x − 1 ) , W 10 , 10 ( x ) = − 0.183164 ( x − 1 ) x 10 − 0.956985 ( x − 1 ) 9 − 2.24824 ( x − 1 ) 8 − 3.34094 ( x − 1 ) 7 − 3.33467 ( x − 1 ) 6 − 2.38496 ( x − 1 ) 5 − 1.32992 ( x − 1 ) 4 − 0.409884 ( x − 1 ) 3 − 0.255003 ( x − 1 ) 2 − 0.258238 ( x − 1 ) .

The profile of approximate solutions in the whole domain ( x , τ ) ∈ [ 0 , 1 ] × [ 0 , 1 ] is visualized in Figures 1 and 2. The related REFs obtained via relation (57) are also depicted along with these plots. We emphasize that we can obtain more accuracy by increasing M or by decreasing Δ τ . To confirm this fact, we take M = 20 and Δ τ = 0.01 in the next experiment. Figure 3 shows the achieved REFs obtained via these aforesaid parameters related to both numerical examples.

Figure 1 Approximate solution obtained via the Rothe-Delannoy approach (left) and the REFs (right) in Example 5.1 with ε , μ = 1 0 ‒ 1 \varepsilon ,\mu =1{0}^{&#x2012;1} , Δ τ = 0.1 \Delta \tau =0.1 , T = 1 T=1 , and M = 10 M=10 .
Figure 1

Approximate solution obtained via the Rothe-Delannoy approach (left) and the REFs (right) in Example 5.1 with ε , μ = 1 0 ‒ 1 , Δ τ = 0.1 , T = 1 , and M = 10 .

Figure 2 Approximate solution obtained via the Rothe-Delannoy approach (left) and the REFs (right) in Example 5.2 with ε , μ = 1 0 ‒ 1 \varepsilon ,\mu =1{0}^{&#x2012;1} , Δ τ = 0.1 \Delta \tau =0.1 , T = 1 T=1 , and M = 10 M=10 .
Figure 2

Approximate solution obtained via the Rothe-Delannoy approach (left) and the REFs (right) in Example 5.2 with ε , μ = 1 0 ‒ 1 , Δ τ = 0.1 , T = 1 , and M = 10 .

Figure 3 Visualization of REFs in the Rothe-Delannoy approach at different time instants τ r {\tau }_{r} , r = 1 , 2 , … , 100 r=1,2,\ldots ,100 in Example 5.1 (left) and Example 5.2 (right) for Δ τ = 0.01 , M = 20 \Delta \tau =0.01,M=20 , ε , μ = 1 0 ‒ 1 , \varepsilon ,\mu =1{0}^{&#x2012;1}, and T = 1 T=1 .
Figure 3

Visualization of REFs in the Rothe-Delannoy approach at different time instants τ r , r = 1 , 2 , … , 100 in Example 5.1 (left) and Example 5.2 (right) for Δ τ = 0.01 , M = 20 , ε , μ = 1 0 ‒ 1 , and T = 1 .

It can be realized that the theoretical OC of the Rothe-Delannoy collocation approach is O ( Δ τ ) in time and has an exponential behavior as proved in Theorem 2.6. To justify these facts, we calculate the OC via (58) in the next results using ε , μ = 1 0 − 2 . The results using fixed M = 4 and different values of Δ τ = 2 − s , s = 1 , 2 , 3 , 4 are tabulated in Tables 1 and 2 for Examples 5.1 and 5.2, respectively. Similarly, the outcomes by fixing Δ τ = 1 ∕ 16 and utilizing various M = 2 s , s = 1 , 2 , 3 , 4 are reported in Tables 1 and 2. As one expected, the first-order accuracy with respect to time and a high-order accuracy in space are observable from the results shown in this table.

Table 1

R 2 and R ∞ error norms the achieved OC in the Rothe-Delannoy approach in Example 5.1 with a fixed M = 4 , Δ τ = 1 ∕ 32 , ε , μ = 1 0 ‒ 2 , T = 1 , and various Δ τ , M

M = 4 Δ τ = 1 ∕ 16
Δ τ R 2 OC τ 2 R ∞ OC τ ∞ M R 2 OCx 2 R ∞ OCx ∞
1 2 4.1642 × 1 0 ‒ 2 — 2.0863 × 1 0 ‒ 1 — 2 6.6509 × 1 0 ‒ 2 — 1.7112 × 1 0 ‒ 1 —
1 4 1.9564 × 1 0 ‒ 2 1.0898 9.9282 × 1 0 ‒ 2 1.0713 4 4.6445 × 1 0 ‒ 3 3.8400 2.3750 × 1 0 ‒ 2 2.8490
1 8 9.4531 × 1 0 ‒ 3 1.0494 4.8228 × 1 0 ‒ 2 1.0417 8 3.6127 × 1 0 ‒ 4 3.6844 7.3733 × 1 0 ‒ 3 1.6875
1 16 4.6445 × 1 0 ‒ 3 1.0253 2.3750 × 1 0 ‒ 2 1.0220 16 4.8071 × 1 0 ‒ 6 6.2318 2.1273 × 1 0 ‒ 4 5.1152
Table 2

R 2 and R ∞ error norms the achieved OC in the Rothe-Delannoy approach in Example 5.2 with a fixed M = 4 , Δ τ = 1 ∕ 32 , ε , μ = 1 0 ‒ 2 , T = 1 , and various Δ τ , M

M = 4 Δ τ = 1 ∕ 16
Δ τ R 2 OC τ 2 R ∞ OC τ ∞ M R 2 OCx 2 R ∞ OCx ∞
1 2 1.5722 × 1 0 ‒ 3 — 1.7828 × 1 0 ‒ 2 — 2 2.0157 × 1 0 ‒ 3 — 5.0315 × 1 0 ‒ 3 —
1 4 7.1666 × 1 0 ‒ 4 1.1334 7.9780 × 1 0 ‒ 3 1.1601 4 1.6405 × 1 0 ‒ 4 3.6190 1.7824 × 1 0 ‒ 3 1.4972
1 8 3.3865 × 1 0 ‒ 4 1.0815 3.7150 × 1 0 ‒ 3 1.1027 8 4.3763 × 1 0 ‒ 5 1.9064 8.5134 × 1 0 ‒ 4 1.0660
1 16 1.6405 × 1 0 ‒ 4 1.0456 1.7824 × 1 0 ‒ 3 1.0595 16 3.7332 × 1 0 ‒ 7 6.8732 1.6209 × 1 0 ‒ 5 5.7148

A comparison with respect to L ∞ error norm has been done in Table 3 to validate our results obtained via the Rothe-Delannoy technique. In this respect, we use ( Δ τ , M ) = ( 1 32 , 16 ) and diverse values of ( ε , μ ) are utilized. Also, the outcomes of an existing numerical method, i.e., the moving mesh-adaptive algorithm (MMAA) [7], are further mentioned in Table 3 with parameters ( Δ τ , M ) = ( 1 32 , 128 ) , ( 1 256 , 1024 ) . In [7], M denotes the number of spatial grid points. Clearly, our algorithm with a small number of M and a higher value of Δ τ is more accurate than MMAA. Similar results using a smaller set of ( ε , μ ) but for ( Δ τ , M ) = ( 1 16 , 10 ) are presented in Table 4 related to Example 5.2. Here, we compared our results with the method developed in [8] besides the MMAA with parameters ( Δ τ , M ) = ( 1 128 , 512 ) . In this approach, the Richardson extrapolation procedure is used for the time stepping, while a combined monotone finite difference operator is designed for the space variable using a piecewise uniform Shishkin mesh. It is evident the proposed numerical technique produces higher accurate results against the methods developed in [7] and [8].

Table 3

Comparisons of error norms in Rothe-Delannoy approach in Example 5.1 with M = 16 , Δ τ = 1 ∕ 32 , and various ε , μ

Present: ( Δ τ , M ) = ( 1 32 , 16 ) MMAA [7] (Maximum errors)
( ε , μ ) R ∞ R 2 ( Δ τ , M ) = ( 1 32 , 128 ) ( Δ τ , M ) = ( 1 256 , 1024 )
( 1 0 ‒ 2 , 1 0 ‒ 4 ) 1.3431 × 1 0 ‒ 4 3.1523 × 1 0 ‒ 6 1.1005 × 1 0 ‒ 2 1.3370 × 1 0 ‒ 3
( 1 0 ‒ 3 , 1 0 ‒ 4 ) 4.2173 × 1 0 ‒ 4 1.1160 × 1 0 ‒ 5 1.1976 × 1 0 ‒ 2 1.4690 × 1 0 ‒ 3
( 1 0 ‒ 4 , 1 0 ‒ 4 ) 6.3133 × 1 0 ‒ 5 1.0522 × 1 0 ‒ 6 1.1985 × 1 0 ‒ 2 1.4780 × 1 0 ‒ 3
( 1 0 ‒ 5 , 1 0 ‒ 4 ) 1.6381 × 1 0 ‒ 4 3.5362 × 1 0 ‒ 6 1.2011 × 1 0 ‒ 2 1.4850 × 1 0 ‒ 3
( 1 0 ‒ 5 , 1 0 ‒ 6 ) 8.9921 × 1 0 ‒ 6 2.4677 × 1 0 ‒ 7 1.1856 × 1 0 ‒ 2 1.4840 × 1 0 ‒ 3
( 1 0 ‒ 6 , 1 0 ‒ 6 ) 6.6773 × 1 0 ‒ 7 2.0028 × 1 0 ‒ 8 1.1922 × 1 0 ‒ 2 1.4790 × 1 0 ‒ 3
( 1 0 ‒ 7 , 1 0 ‒ 6 ) 1.6409 × 1 0 ‒ 6 3.6331 × 1 0 ‒ 8 1.2279 × 1 0 ‒ 2 1.4720 × 1 0 ‒ 3
Table 4

Comparisons of error norms in Rothe-Delannoy approach in Example 5.2 with M = 10 , Δ τ = 1 ∕ 16 , and various ε , μ

Present: ( Δ τ , M ) = ( 1 16 , 10 ) ( Δ τ , M ) = ( 1 128 , 512 )
( ε , μ ) R ∞ R 2 MMAA [7] Richardson [8]
( 1 0 ‒ 6 , 1 0 ‒ 7 ) 1.5753 × 1 0 ‒ 7 5.8572 × 1 0 ‒ 9 1.2824 × 1 0 ‒ 4 6.6241 × 1 0 ‒ 7
( 1 0 ‒ 7 , 1 0 ‒ 7 ) 2.2132 × 1 0 ‒ 8 7.0771 × 1 0 ‒ 10 1.2853 × 1 0 ‒ 4 6.6241 × 1 0 ‒ 7
( 1 0 ‒ 8 , 1 0 ‒ 7 ) 8.5904 × 1 0 ‒ 9 2.6963 × 1 0 ‒ 10 1.2781 × 1 0 ‒ 4 6.6241 × 1 0 ‒ 7
( 1 0 ‒ 9 , 1 0 ‒ 7 ) 7.2362 × 1 0 ‒ 9 2.4376 × 1 0 ‒ 10 1.2804 × 1 0 ‒ 4 6.6241 × 1 0 ‒ 7

Finally, in the first part and in the Rothe-Delannoy technique, we utilize two small values of ε = 1 0 − 8 and μ = 1 0 − 6 . In this case, we use Δ τ = 1 ∕ 32 and M = 14 . The graphical representations of the approximate solution together with the related REFs are shown in Figures 4 and 5 associated with Examples 5.1 and 5.2, respectively.

Figure 4 Visualization of approximate solution (right) and the REFs (left) in the Rothe-Delannoy approach at different time instants τ r {\tau }_{r} , r = 1 , 2 , … , 32 r=1,2,\ldots ,32 in Example 5.1 for Δ τ = 1 ∕ 32 , M = 14 \Delta \tau =1/32,M=14 , ε = 1 0 ‒ 8 , μ = 1 0 ‒ 6 , \varepsilon =1{0}^{&#x2012;8},\mu =1{0}^{&#x2012;6}, and T = 1 T=1 .
Figure 4

Visualization of approximate solution (right) and the REFs (left) in the Rothe-Delannoy approach at different time instants τ r , r = 1 , 2 , … , 32 in Example 5.1 for Δ τ = 1 ∕ 32 , M = 14 , ε = 1 0 ‒ 8 , μ = 1 0 ‒ 6 , and T = 1 .

Figure 5 Visualization of approximate solution (right) and the REFs (left) in the Rothe-Delannoy approach at different time instants τ r {\tau }_{r} , r = 1 , 2 , … , 32 r=1,2,\ldots ,32 in Example 5.2 for Δ τ = 1 ∕ 32 , M = 14 \Delta \tau =1/32,M=14 , ε = 1 0 ‒ 8 , μ = 1 0 ‒ 6 , \varepsilon =1{0}^{&#x2012;8},\mu =1{0}^{&#x2012;6}, and T = 1 T=1 .
Figure 5

Visualization of approximate solution (right) and the REFs (left) in the Rothe-Delannoy approach at different time instants τ r , r = 1 , 2 , … , 32 in Example 5.2 for Δ τ = 1 ∕ 32 , M = 14 , ε = 1 0 ‒ 8 , μ = 1 0 ‒ 6 , and T = 1 .

5.2 Taylor-Delannoy approach

Analog to the Rothe-Delannoy method, we first take the parameters ε , μ = 0.1 . The approximate solutions for Examples 5.1 and 5.2 at time step τ = T utilizing M = 10 and Δ τ = 0.1 take the following forms, respectively,

W 10 , 10 ( x ) = − 2.83681 ( x − 1 ) x 10 − 19.2256 ( x − 1 ) x 9 − 58.179 ( x − 1 ) 8 − 100.26 ( x − 1 ) 7 − 103.057 ( x − 1 ) 6 − 54.2433 ( x − 1 ) 5 − 4.97596 ( x − 1 ) 4 − 4.56666 ( x − 1 ) 3 − 20.0702 ( x − 1.0 ) 2 − 10.8142 ( x − 1 ) , W 10 , 10 ( x ) = 0.185724 ( x − 1 ) 1 0 + 0.978579 ( x − 1 ) 9 + 2.29962 ( x − 1 ) 8 + 3.39476 ( x − 1 ) 7 + 3.36605 ( x − 1 ) 6 + 2.39168 ( x − 1 ) 5 + 1.31676 ( x − 1 ) 4 + 0.398006 ( x − 1 ) 3 + 0.251874 ( x − 1 ) 2 + 0.257017 ( x − 1 ) .

The graphical representation of approximate solutions in the whole domain ( x , τ ) ∈ [ 0 , 1 ] × [ 0 , 1 ] is shown in Figure 6. To obtain a higher accuracy, we increase the number of basis M or decrease Δ τ . For Example 5.1 and in Table 5, we first fix M = 20 and vary Δ τ to see how the OC τ d in time are behaved. Here, we utilize ε , μ = 1 0 − 2 in the computations. Hence, we fix Δ τ = 1 ∕ 16 and vary M = 2 , 4 , 8 , 16 . The results of OCx d are reported in the second part of Table 5 for d = 2 , ∞ . Clearly, the second-order accuracy in τ and a higher-order accuracy in x are obtainable in the Taylor-Delannoy series method. Similar results for the second Example 5.2 are presented in Table 6 but for different fixed values of M = 10 and Δ τ = 1 ∕ 320

Figure 6 Approximate solutions obtained via the Taylor-Delannoy approach for Example 5.1 (left) and Example 5.2 (right) with ε , μ = 1 0 ‒ 1 \varepsilon ,\mu =1{0}^{&#x2012;1} , Δ τ = 0.1 \Delta \tau =0.1 , T = 1 T=1 , and M = 10 M=10 .
Figure 6

Approximate solutions obtained via the Taylor-Delannoy approach for Example 5.1 (left) and Example 5.2 (right) with ε , μ = 1 0 ‒ 1 , Δ τ = 0.1 , T = 1 , and M = 10 .

Table 5

R 2 and R ∞ error norms the achieved OC in the Taylor-Delannoy approach in Example 5.1 with a fixed M = 20 , Δ τ = 1 ∕ 16 , ε , μ = 1 0 ‒ 2 , T = 1 , and various Δ τ , M

M = 20 Δ τ = 1 ∕ 16
Δ τ R 2 OC τ 2 R ∞ OC τ ∞ M R 2 OCx 2 R ∞ OCx ∞
1 2 7.6855 × 1 0 ‒ 6 — 4.0077 × 1 0 ‒ 4 — 2 1.2547 × 1 0 ‒ 1 — 3.2306 × 1 0 ‒ 1 —
1 4 9.3964 × 1 0 ‒ 6 ‒ 0.2900 4.0761 × 1 0 ‒ 4 ‒ 0.0244 4 8.6965 × 1 0 ‒ 3 3.8507 4.4586 × 1 0 ‒ 2 2.8571
1 8 2.8447 × 1 0 ‒ 6 1.7238 1.4570 × 1 0 ‒ 4 1.4842 8 6.9113 × 1 0 ‒ 4 3.6534 1.4085 × 1 0 ‒ 2 1.6624
1 16 6.8003 × 1 0 ‒ 7 2.0646 3.6922 × 1 0 ‒ 5 1.9804 16 9.9786 × 1 0 ‒ 6 6.1140 4.3118 × 1 0 ‒ 4 5.0297
Table 6

R 2 and R ∞ error norms the achieved OC in the Taylor-Delannoy approach in Example 5.2 with a fixed M = 10 , Δ τ = 1 ∕ 320 , ε , μ = 1 0 ‒ 2 , T = 1 , and various Δ τ , M

M = 10 Δ τ = 1 ∕ 320
Δ τ R 2 OC τ 2 R ∞ OC τ ∞ M R 2 OCx 2 R ∞ OCx ∞
1 2 1.3091 × 1 0 ‒ 2 — 6.4420 × 1 0 ‒ 2 — 2 1.9939 × 1 0 ‒ 4 — 4.9898 × 1 0 ‒ 4 —
1 4 2.9828 × 1 0 ‒ 3 2.1339 1.4596 × 1 0 ‒ 2 2.1419 4 1.5828 × 1 0 ‒ 5 3.6550 1.6955 × 1 0 ‒ 4 1.5573
1 8 7.3089 × 1 0 ‒ 4 2.0289 3.5652 × 1 0 ‒ 3 2.0336 8 4.3493 × 1 0 ‒ 6 1.8636 8.4488 × 1 0 ‒ 5 1.0049
1 16 1.8338 × 1 0 ‒ 4 1.9949 8.8603 × 1 0 ‒ 4 2.0085 16 3.6534 × 1 0 ‒ 7 3.5735 2.2103 × 1 0 ‒ 6 5.2564

Similar to the Rothe-Delannoy method, some comparisons are made between our numerical results and the outcomes of the MMAA [7] in Table 7. In these experiments, we have used M = 16 , Δ τ = 1 ∕ 32 , and various small values of ε and μ are considered. In contrast, the values M = 256 , 512 as the number of spatial grid points and Δ τ = 1 ∕ 64 , 1 ∕ 128 were used in the MMAA procedure. Needless to say, our results using a larger time step (and with a smaller number of bases) are more accurate in comparison with the outcomes of MMAA.

Table 7

Comparisons of error norms in Taylor-Delannoy approach in Example 5.1 with M = 16 , Δ τ = 1 ∕ 32 , and various ε , μ

Present: ( Δ τ , M ) = ( 1 32 , 16 ) MMAA [7] (Maximum errors)
( ε , μ ) R ∞ R 2 ( Δ τ , M ) = ( 1 64 , 256 ) ( Δ τ , M ) = ( 1 128 , 512 )
( 1 0 ‒ 5 , 1 0 ‒ 6 ) 1.7491 × 1 0 ‒ 5 4.7820 × 1 0 ‒ 7 5.886 × 1 0 ‒ 3 2.974 × 1 0 ‒ 3
( 1 0 ‒ 6 , 1 0 ‒ 6 ) 1.2824 × 1 0 ‒ 6 3.8829 × 1 0 ‒ 8 5.909 × 1 0 ‒ 3 2.961 × 1 0 ‒ 3
( 1 0 ‒ 7 , 1 0 ‒ 6 ) 3.1737 × 1 0 ‒ 6 7.0724 × 1 0 ‒ 8 5.906 × 1 0 ‒ 3 2.961 × 1 0 ‒ 3
( 1 0 ‒ 8 , 1 0 ‒ 6 ) 3.3630 × 1 0 ‒ 6 7.4950 × 1 0 ‒ 8 6.008 × 1 0 ‒ 3 2.981 × 1 0 ‒ 3

Ultimately, we consider two small values of ( ε , μ ) = ( 1 0 − 8 , 1 0 − 6 ) . For Δ τ = 1 ∕ 32 and M = 12 , the approximate solutions at the final time τ = T is given by:

W 1 , 32 ( x ) = 28.2824 ( x − 1 ) 12 + 176.975 ( x − 1 ) 11 + 471.382 ( x − 1 ) 10 + 668.581 ( x − 1 ) 9 + 479.111 ( x − 1 ) 8 + 43.4217 ( x − 1.0 ) 7 − 196.896 ( x − 1 ) 6 − 109.513 ( x − 1 ) 5 + 35.4617 ( x − 1 ) 4 + 35.5468 ( x − 1 ) 3 − 17.2626 ( x − 1 ) 2 − 14.932 ( x − 1 ) ,

for the first Example 5.1 and

W 1 , 32 ( x ) = − 0.00318951 ( x − 1 ) 12 − 0.0164746 ( x − 1 ) 11 − 0.036348 ( x − 1 ) 10 − 0.0455024 ( x − 1 ) 9 − 0.0320158 ( x − 1 ) 8 − 0.0179865 ( x − 1.0 ) 7 + 0.00832472 ( x − 1 ) 6 − 0.0221956 ( x − 1 ) 5 + 0.0447422 ( x − 1 ) 4 − 0.069024 ( x − 1 ) 3 + 0.0900682 ( x − 1 ) 2 + 0.242765 ( x − 1 ) .

for the second Example 5.2. The snapshots of approximate solutions at all time steps are shown in Figures 7 and 8. The pictorial representations of the related REFs for these two examples further are visualized in Figures 7 and 8 (right panels).

Figure 7 Visualization of approximate solution (left) and the REFs (right) in the Taylor-Delannoy approach at different time instants τ r {\tau }_{r} , r = 1 , 2 , … , 32 r=1,2,\ldots ,32 in Example 5.1 for Δ τ = 1 ∕ 32 , M = 12 \Delta \tau =1/32,M=12 , ε = 1 0 ‒ 8 , μ = 1 0 ‒ 6 , \varepsilon =1{0}^{&#x2012;8},\mu =1{0}^{&#x2012;6}, and T = 1 T=1 .
Figure 7

Visualization of approximate solution (left) and the REFs (right) in the Taylor-Delannoy approach at different time instants τ r , r = 1 , 2 , … , 32 in Example 5.1 for Δ τ = 1 ∕ 32 , M = 12 , ε = 1 0 ‒ 8 , μ = 1 0 ‒ 6 , and T = 1 .

Figure 8 Visualization of approximate solution (left) and the REFs (right) in the Taylor-Delannoy approach at different time instants τ r {\tau }_{r} , r = 1 , 2 , … , 32 r=1,2,\ldots ,32 in Example 5.2 for Δ τ = 1 ∕ 32 , M = 12 \Delta \tau =1/32,M=12 , ε = 1 0 ‒ 8 , μ = 1 0 ‒ 6 , \varepsilon =1{0}^{&#x2012;8},\mu =1{0}^{&#x2012;6}, and T = 1 T=1 .
Figure 8

Visualization of approximate solution (left) and the REFs (right) in the Taylor-Delannoy approach at different time instants τ r , r = 1 , 2 , … , 32 in Example 5.2 for Δ τ = 1 ∕ 32 , M = 12 , ε = 1 0 ‒ 8 , μ = 1 0 ‒ 6 , and T = 1 .

6 Conclusion

Two accurate hybrid matrix collocation techniques based on the (novel) Delannoy functions of graph are developed to find an approximate solution to a class of singularly perturbed CDR problems with variable coefficients and two small parameters. For the time-marching algorithms, the first-order Rothe method and the second-order Taylor series formula are utilized to discretize the underlying CDR model problems. We then apply the Delannoy collocation technique to the discretized time equations at each time level. The proposed schemes have been shown to be uniformly convergent of order O ( Δ τ s + 1 ∕ M 1 2 ) , s = 1 , 2 . Numerical simulations relied on two academic test cases are presented to demonstrate the first- and second-order accuracy of the proposed hybrid approaches. The presented results show that two new hybrid procedures have better accuracy in comparison with the current numerical methods.

Acknowledgements

Khursheed J. Ansari extends his appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through a large group Research Project under Grant Number RGP2/371/44.

  1. Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  2. Conflict of interest: The authors state no conflict of interest.

  3. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during this study.

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Received: 2023-01-18
Revised: 2023-11-06
Accepted: 2024-01-08
Published Online: 2024-03-18

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
https://doi.org/10.1515/dema-2023-0144
Schlagwörter für diesen Artikel
collocation points; convergent analysis; Delannoy functions; stability; Taylor expansion; Rothe method; singularly perturbed problem
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Artikel in diesem Heft

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
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