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A numerical Haar wavelet-finite difference hybrid method and its convergence for nonlinear hyperbolic partial differential equation

  • Weidong Lei , Muhammad Ahsan , Waqas Khan , Zaheer Uddin EMAIL logo and Masood Ahmad
Published/Copyright: May 11, 2023
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 56 Issue 1

Abstract

In this research work, we proposed a Haar wavelet collocation method (HWCM) for the numerical solution of first- and second-order nonlinear hyperbolic equations. The time derivative in the governing equations is approximated by a finite difference. The nonlinear hyperbolic equation is converted into its full algebraic form once the space derivatives are replaced by the finite Haar series. Convergence analysis is performed both in space and time, where the computational results follow the theoretical statements of convergence. Many test problems with different nonlinear terms are presented to verify the accuracy, capability, and convergence of the proposed method for the first- and second-order nonlinear hyperbolic equations.

Keywords: Haar wavelet; hyperbolic equation; collocation method; single- and double-soliton wave
MSC 2010: 35Exx; 65Mxx; 65Nxx; 00A69; 65T60; 35-XX; 65H05; 35C08

1 Introduction

Most of the scientific and physical phenomena govern the nonlinear hyperbolic partial differential equations (NLHPDEs) and have an important place in atomic physics, aerospace, industry, biology, and engineering problems. To find the exact solution of these kinds of NLHPDEs is very complicated due to the nonlinear term; therefore, implementation of numerical schemes is an alternative option to find their solution. In this article, we consider the first- and second-order NLHPDEs.

The first-order NLHPDEs can also be used in modeling the vibration of structures (like machines, buildings, and beams) and are considered as the foundation for the basic equations of atomic physics (see [1,2] and the references therein). Particular types of NLHPDEs are the Wave and Telegraph equations, which have wide applications in signal analysis for transmission, communication, and broadcasting of electric signals [3], random walk theory [4], and wave circulation or propagation [5].

The second-order NLHPDEs have many applications in mathematical chemistry, mathematical physics, and mathematical biology, such as chemical kinetics, fluid dynamics, solid state physics, quantum field theory, optics, atomic physics, and plasma physics.

Due to the aforementioned applications, different numerical techniques have been developed to solve the important NLHPDEs. These numerical techniques are the finite difference method [6–10], the Galerkin method [1,11–13], the B-spline collocation method [14], the spectral collocation method [15,16], the Chebyshev approach [17], and the wavelet collocation method [2]. Some recent contributions are also reported in [18–20].

Recently, the research has been focused on the Haar wavelet application in the investigation and analysis of different problems in applied sciences. Different algorithms based on weak and strong formulations contain the meshless wavelet method [21], the Daubechies wavelet-based method [22], the wavelet Galerkin method [23], and the wavelet collocation method [24,25]. A thorough introduction of the wavelet schemes for PDEs is given in [26]. Different scientific and engineering phenomena have been represented in the forms of ordinary differential equations, integro-differential equations, and PDEs, which have been solved by Haar wavelets in the references [27–42]. A further development of Haar wavelet is related to the solution of challenging fractional differential and integral equations [43–47]. The further extensions of the Haar wavelet approach are presented to solve linear and nonlinear direct problems [25,48–50] and inverse problems [41,42,51–53]. The latest contribution on Haar wavelets is presented in [54] for identification of software piracy.

Haar wavelets-based algorithms have also been reported for hyperbolic PDEs. In [55], a second-order linear hyperbolic PDE has been solved with the Haar wavelet operational matrix method. In [56], ∂ 3 φ ( s , τ ) ∂ τ ∂ s 2 has been approximated by Haar wavelets to just find the numerical solution in the unit interval [ 0 , 1 ] using some transformation that converted the governing equation into the system of PDEs.

1.1 The governing equations

Mathematically, the first-order NLHPDE can be presented as follows:

(1) ∂ φ ( s , τ ) ∂ τ + ∂ φ ( s , τ ) ∂ s + f ( φ ( s , τ ) ) = g ( s , τ ) , a ≤ s ≤ b , 0 ≤ τ ≤ T ,

with the initial and boundary conditions φ ( s , 0 ) = I ( s ) and φ ( a , τ ) = B ( τ ) .

The second-order NLHPDE can be written as follows:

(2) ∂ 2 φ ( s , τ ) ∂ τ 2 + ∂ 2 φ ( s , τ ) ∂ s 2 + f φ ( s , τ ) , ∂ φ ( s , τ ) ∂ τ = g ( s , τ ) , a ≤ s ≤ b , 0 ≤ τ ≤ T ,

with the initial conditions

φ ( s , 0 ) = I 1 ( s ) , ∂ φ ( s , 0 ) ∂ τ = I 2 ( s )

and boundary conditions

φ ( a , τ ) = B 1 ( τ ) , φ ( b , τ ) = B 2 ( τ ) .

In the above equations, f represents the nonlinear term, g , I , I 1 , I 2 , B , B 1 , and B 2 are known functions, and φ is the only unknown function that is to be determined.

When f φ ( s , τ ) , ∂ φ ( s , τ ) ∂ τ = ∂ φ ( s , τ ) ∂ τ , equation (2) is known as the damped wave equation, whereas when f φ ( s , τ ) , ∂ φ ( s , τ ) ∂ τ = ∂ φ ( s , τ ) ∂ τ + φ ( s , τ ) , equation (2) is known as the telegraph equation. These damped wave and telegraph equations have applications like wave phenomena and electric signal propagation in transmission wires. In fact, this partial differential equation is more appropriate than the ordinary diffusion equation for representing the reaction-diffusion models in many scientific fields, such as biology, where biologists come across these types of equations in the development of pulsating blood flow in arteries and the random motion of bugs along a hedge [57].

If f φ ( s , τ ) , ∂ φ ( s , τ ) ∂ τ = φ ( s , τ ) + φ ( s , τ ) k , where k = 2 or 3, then equation (2) is named Klein-Gordon equation and can be studied in field theory and relativistic quantum mechanics, which have enormous significance for physicists [58] and also described dissemination of dislocations in crystals and the activities of basic particles.

When f φ ( s , τ ) , ∂ φ ( s , τ ) ∂ τ = sin ( φ ( s , τ ) ) , equation (2) is classified as Sine-Gordon equation, which has importance in a range of relevant fields, such as relativistic field theory and differential geometry, and it also appears in other related physics topics, including the movement of a rigid pendulum attached to an expanded cable, the transmission of fluxons in Josephson junctions, the transmission in ferromagnetic materials of waves taking rotary motions in the course of magnetization, laser pulses in two-state medium and dislocations in crystals [59,60].

In this article, the potential of Haar wavelets is investigated on the first- and second-order NLHPDEs. The theoretical convergence is supported by our numerical results in the interval [a, b]. The time and space derivatives are discretized using finite-difference and Haar wavelets, respectively. Due to the discontinuity of the Haar functions, the approximation starts from the highest-order derivatives in the model equation. By further integration of the series, the unknown solution can be obtained. By introducing these approximations in equation (1) or (2), a system of algebraic equations can be easily solved. The details of the proposed methods are given in the subsequent sections.

2 Haar functions

A generalized representation of the Haar functions is defined as follows:

h i ( s ) = 1 for s ∈ [ ζ 1 ( i ) , ζ 2 ( i ) ) , − 1 for s ∈ [ ζ 2 ( i ) , ζ 3 ( i ) ) , 0 elsewhere ,

where

ζ 1 ( i ) = a + ( b − a ) k m , ζ 2 ( i ) = a + ( b − a ) ( k + 0.5 ) m , ζ 3 ( i ) = a + ( b − a ) ( k + 1 ) m .

Here, m = 2 j , j = 0 , 1 , … , represents the level of the wavelet, k = 0 , 1 , … , m − 1 is the translation parameter, and i = m + k + 1 . We note that i ≥ 2 . We define

h 1 ( s ) = 1 for s ∈ [ a , b ] , 0 elsewhere ,

which is also known as the mother wavelet. To keep the derivations simple, we intend to introduce some notations for the following integrals for i = 2 , 3 , 4 , … ,

p i , 1 ( s ) = ∫ a s h i ( s ′ ) d s ′ = 0 for s < ζ 1 ( i ) , s − ζ 1 ( i ) for s ∈ [ ζ 1 ( i ) , ζ 2 ( i ) ) , s − ζ 1 ( i ) − 2 ( s − ζ 2 ( i ) ) for s ∈ [ ζ 2 ( i ) , ζ 3 ( i ) ) , s − ζ 1 ( i ) − 2 ( s − ζ 2 ( i ) ) + ( s − ζ 3 ( i ) ) for s ≥ ζ 3 ( i ) , p i , 2 ( s ) = ∫ a s p i , 1 ( s ′ ) d s ′ = 0 for s < ζ 1 ( i ) , 1 2 ( s − ζ 1 ( i ) ) 2 for s ∈ [ ζ 1 ( i ) , ζ 2 ( i ) ) , 1 2 [ ( s − ζ 1 ( i ) ) 2 − 2 ( s − ζ 2 ( i ) ) 2 ] for s ∈ [ ζ 2 ( i ) , ζ 3 ( i ) ) , 1 2 [ ( s − ζ 1 ( i ) ) 2 − 2 ( s − ζ 2 ( i ) ) 2 + ( s − ζ 3 ( i ) ) 2 ] for s ≥ ζ 3 ( i ) ,

and

(3) C = ∫ a b p i , 1 ( s ′ ) d s ′ = ( b − a ) 2 4 m 2 .

As p i , 1 ( s ) is increasing in the interval [ ζ 1 ( i ) , ζ 2 ( i ) ) and decreasing in the interval [ ζ 2 ( i ) , ζ 3 ( i ) ) , but the maximum value is at ζ 2 ( i ) . Hence,

(4) max s ( p i , 1 ( s ) ) = p i , 1 ( ζ 2 ( i ) ) = b − a 2 m .

It is noteworthy to mention that the following formula has been proved in [61]:

(5) max s ( p i , 2 ( s ) ) = ( b − a ) 2 4 m 2 .

3 Haar approximation

As we consider here the first- and second-order NLHPDEs, we approximate them with two different Haar wavelet collocation methods (HWCMs) in Sections 3.1 and 3.2.

3.1 HWCM for first-order NLHPDE

In this subsection, we consider Haar wavelets as a basic part of our numerical technique for the spatial discretization of equation (1). To construct HWCM for first-order NLHPDE, we start approximating the first-order derivative with Haar functions as follows:

(6) ∂ φ ( s , τ ) ∂ s = ∑ i = 1 ∞ λ i ( τ ) h i ( s ) .

Integrating equation (6) w.r.t s , from a to s , we obtain

φ ( s , τ ) = φ ( a , τ ) + ∑ i = 1 ∞ λ i ( τ ) p i , 1 ( s ) .

If τ n be the current time level and τ n + 1 be the next time level, then equation (1) can be linearized in the following manner:

∂ φ ( s , τ n + 1 ) ∂ τ + ∂ φ ( s , τ n + 1 ) ∂ s + f ( φ ( s , τ n ) ) = g ( s , τ n + 1 ) .

Using the forward difference approximation for time derivative, we obtain

(7) φ ( s , τ n + 1 ) − φ ( s , τ n ) Δ τ + ∂ φ ( s , τ n + 1 ) ∂ s + f ( φ ( s , τ n ) ) = g ( s , τ n + 1 ) − O ( Δ τ ) .

For a Haar wavelet-based numerical solution, we define M = 2 J for some J > 0 and

(8) φ M ( s , τ ) = φ ( a , τ ) + ∑ i = 1 2 M λ i ( τ ) p i , 1 ( s ) .

If j = J , then k = 2 J − 1 and i = m + k + 1 = 2 J + 2 J − 1 + 1 = 2 × 2 J = 2 M (Section 2). Differentiating equation (8) w.r.t s , we obtain

∂ φ M ( s , τ ) ∂ s = ∑ i = 1 2 M λ i ( τ ) h i ( s ) .

The relationship between exact and approximate representations is

φ ( s , τ ) = φ M ( s , τ ) + E M ( s , τ ) , and ∂ φ ( s , τ ) ∂ s = ∂ φ M ( s , τ ) ∂ s + ∂ E M ( s , τ ) ∂ s ,

where E M ( s , τ ) = ∑ i = 2 M + 1 ∞ λ i ( τ ) p i , 1 ( s ) and ∂ E M ( s , τ ) ∂ s = ∑ i = 2 M + 1 ∞ λ i ( τ ) h i ( s ) .

Now the exact form of equation (7) using Haar wavelet is

φ M ( s , τ n + 1 ) − φ M ( s , τ n ) Δ τ + E M ( s , τ n + 1 ) − E M ( s , τ n ) Δ τ + ∂ φ M ( s , τ n + 1 ) ∂ s + ∂ E M ( s , τ n + 1 ) ∂ s + f ( φ M ( s , τ n ) ) + f ( E M ( s , τ n ) ) = g ( s , τ n + 1 ) − O ( Δ τ ) , φ M ( s , τ n + 1 ) − φ M ( s , τ n ) Δ τ + ∂ φ M ( s , τ n + 1 ) ∂ s + f ( φ M ( s , τ n ) ) = g ( s , τ n + 1 ) − O ( Δ τ ) − E M ( s , τ n + 1 ) − E M ( s , τ n ) Δ τ − ∂ E M ( s , τ n + 1 ) ∂ s − f ( E M ( s , τ n ) ) .

Dropping all the error terms and using the collocation points s l = a + ( b − a ) ( l − 0.5 ) 2 M , l = 1 , 2 , … , 2 M , we have

φ M ( s l , τ n + 1 ) − φ M ( s l , τ n ) Δ τ + ∂ φ M ( s l , τ n + 1 ) ∂ s + f ( φ M ( s l , τ n ) ) = g ( s l , τ n + 1 ) .

Now defining

(9) P M , l n ≔ φ ( a , τ ) + ∑ i = 1 2 M λ i M , n p i , 1 ( s l ) , and ∂ P M , l n ∂ s ≔ ∑ i = 1 2 M λ i M , n h i ( s l ) ,

we obtain

(10) P M , l n + 1 − P M , l n Δ τ + ∂ P M , l n + 1 ∂ s + f ( P M , l n ) = g ( s l , τ n + 1 ) .

Putting equation (9) in equation (10), we obtain a system of 2 M equations with 2 M unknowns, which can be easily solved for λ i s, i.e.,

(11) λ i M , n + 1 ( p i , 1 ( s l ) + Δ τ h i ( s l ) ) = P M , l n + Δ τ g ( s l , τ n + 1 ) − Δ τ f ( P M , l n ) − φ ( a , τ ) .

Equation (11) can be written as follows:

(12) [ ℋ ] 2 M × 2 M [ X ] 2 M × 1 = [ B ] 2 M × 1 ,

where ℋ is the Haar coefficient matrix, X = [ λ 1 M , n + 1 λ 2 M , n + 1 … λ 2 M M , n + 1 ] is the unknown Haar wavelet coefficients, and B is a vector that represents the right side of equation (11). Equation (12) can be easily solved for X . By inserting X in equation (9), we obtain the desired numerical solution. We define the following formula to interpolate the solution at any point s :

P M n + 1 ( s ) ≔ φ ( a , τ n + 1 ) + ∑ i = 1 2 M λ i M , n + 1 p i , 1 ( s ) ,

then P M n + 1 ( s ) ≈ φ ( s , τ n + 1 ) .

3.2 HWCM for second-order NLHPDE

To construct HWCM for second-order NLHPDE, we start by approximating the second-order derivative with Haar functions as follows:

(13) ∂ 2 φ ( s , τ ) ∂ s 2 = ∑ i = 1 ∞ λ i ( τ ) h i ( s ) .

Integrating equation (13) w.r.t s , from a to s , we obtain

(14) ∂ φ ( s , τ ) ∂ s = ∂ φ ( a , τ ) ∂ s + ∑ i = 1 ∞ λ i ( τ ) p i , 1 ( s ) .

Integrating equation (14) w.r.t s , from a to b , we obtain

(15) ∂ φ ( a , τ ) ∂ s = φ ( b , τ ) − φ ( a , τ ) b − a − ∑ i = 1 ∞ λ i ( τ ) C b − a .

Eliminating ∂ φ ( a , τ ) ∂ s from equations (14) and (15), we obtain

(16) ∂ φ ( s , τ ) ∂ s = w ¯ o ( τ ) + ∑ i = 1 ∞ λ i ( τ ) h ¯ i ( s ) ,

where w ¯ o ( τ ) = φ ( b , τ ) − φ ( a , τ ) b − a and h ¯ i ( s ) = p i , 1 ( s ) − C b − a . Again, partially integrating equation (16) w.r.t s , from a to s , we obtain

(17) φ ( s , τ ) = w ˜ o ( s , τ ) + ∑ i = 1 ∞ λ i ( τ ) h ˜ i ( s ) , where w ˜ o ( s , τ ) = φ ( a , τ ) + ( s − a ) w ¯ o ( τ ) , and h ˜ i ( s ) = p i , 2 ( s ) − ( s − a ) C b − a .

If τ n be the current time level and τ n + 1 be the next time level, then equation (2) can be linearized in the following manner:

∂ 2 φ ( s , τ n + 1 ) ∂ τ 2 + ∂ 2 φ ( s , τ n + 1 ) ∂ s 2 + f 1 ( φ ( s , τ n ) ) ∂ φ ( s , τ n + 1 ) ∂ τ = g ( s , τ n + 1 ) ,

where f 1 is the linearized form of the function f . Using forward difference approximation for time derivative, we obtain

(18) φ ( s , τ n + 1 ) − 2 φ ( s , τ n ) + φ ( s , τ n − 1 ) Δ τ 2 + ∂ 2 φ ( s , τ n + 1 ) ∂ s 2 + f 1 ( φ ( s , τ n ) ) φ ( s , τ n + 1 ) − φ ( s , τ n ) Δ τ = g ( s , τ n + 1 ) − O ( Δ τ 2 ) − O ( Δ τ ) .

To find the numerical solution based on Haar wavelet, we define

(19) φ M ( s , τ ) = w ˜ o ( s , τ ) + ∑ i = 1 2 M λ i ( τ ) h ˜ i ( s ) .

Differentiating equation (19) w.r.t s , we obtain

∂ φ M ( s , τ ) ∂ s = w ¯ o ( τ ) + ∑ i = 1 2 M λ i ( τ ) h ¯ i ( s ) , ∂ 2 φ M ( s , τ ) ∂ s 2 = ∑ i = 1 2 M λ i ( τ ) h i ( s ) .

The relationship between exact and approximate representations are

φ ( s , τ ) = φ M ( s , τ ) + E M ( s , τ ) , where E M ( s , τ ) = ∑ i = 2 M + 1 ∞ λ i ( τ ) h ˜ i ( s ) , ∂ φ ( s , τ ) ∂ s = ∂ φ M ( s , τ ) ∂ s + ∂ E M ( s , τ ) ∂ s , where ∂ E M ( s , τ ) ∂ s = ∑ i = 2 M + 1 ∞ λ i ( τ ) h ¯ i ( s ) ,

and

∂ 2 φ ( s , τ ) ∂ s 2 = ∂ 2 φ M ( s , τ ) ∂ s 2 + ∂ 2 E M ( s , τ ) ∂ s 2 , where ∂ 2 E M ( s , τ ) ∂ s 2 = ∑ i = 2 M + 1 ∞ λ i ( τ ) h i ( s ) .

Now, the exact form of equation (18) using Haar wavelet is

φ M ( s , τ n + 1 ) − 2 φ M ( s , τ n ) + φ M ( s , τ n − 1 ) Δ τ 2 + ∂ 2 φ M ( s , τ n + 1 ) ∂ s 2 + f 1 ( φ M ( s , τ n ) ) φ M ( s , τ n + 1 ) − φ M ( s , τ n ) Δ τ = g ( s , τ n + 1 ) − E M ( s , τ n + 1 ) − 2 E M ( s , τ n ) + E M ( s , τ n − 1 ) Δ τ 2 − ∂ 2 E M ( s , τ n + 1 ) ∂ s 2 − f 1 ( E M ( s , τ n ) ) E M ( s , τ n + 1 ) − E M ( s , τ n ) Δ τ − O ( Δ τ ) .

Dropping all the error terms and using the collocation points s l = a + ( b − a ) ( l − 0.5 ) 2 M , l = 1 , 2 , … , 2 M , we have

φ M ( s l , τ n + 1 ) − 2 φ M ( s l , τ n ) + φ M ( s l , τ n − 1 ) Δ τ 2 + ∂ 2 φ M ( s l , τ n + 1 ) ∂ s 2 + f 1 ( φ M ( s l , τ n ) ) φ M ( s l , τ n + 1 ) − φ M ( s l , τ n ) Δ τ = g ( s l , τ n + 1 ) .

Now defining

(20) P M , l n ≔ w ˜ o ( s l , τ n ) + ∑ i = 1 2 M λ i M , n h ˜ i ( s l ) , ∂ P M , l n ∂ s ≔ w ¯ o ( τ n ) + ∑ i = 1 2 M λ i M , n h ¯ i ( s l ) , ∂ 2 P M , l n ∂ s 2 ≔ ∑ i = 1 2 M λ i M , n h i ( s l ) ,

we obtain

(21) P M , l n + 1 − 2 P M , l n + P M , l n − 1 Δ τ 2 + ∂ 2 P M , l n + 1 ∂ s 2 + f 1 ( P M , l n ) P M , l n + 1 − P M , l n Δ τ = g ( s l , τ n + 1 ) .

Putting equation (20) in equation (21), we obtain a system of 2 M equations with 2 M unknowns as follows:

(22) [ ℋ ] 2 M × 2 M [ X ] 2 M × 1 = [ B ] 2 M × 1 ,

where ℋ is the Haar coefficient matrix, X = [ λ 1 M , n + 1 λ 2 M , n + 1 … λ 2 M M , n + 1 ] is the unknown Haar wavelet coefficients and B is a vector that represents the right side of equation (21). The equation (22) can be easily solved for X . By using X in equation (20), one can obtain the desired numerical solution. We define the following formula to interpolate the solution at any point s ,

P M n + 1 ( s ) ≔ w ˜ o ( s , τ n + 1 ) + ∑ i = 1 2 M λ i M , n + 1 h ˜ i ( s ) ,

then P M n + 1 ( s ) ≈ φ ( s , τ n + 1 ) .

3.3 Summary of the algorithms

The algorithm for the first- and second-order NLHPDEs is presented as follows:

Input: N = 2 M , M = 2 J , J ∈ N 0 .

Step 1: Compute h i ( x ) , p i , 1 ( x ) and p i , 2 ( x ) .

For n = 1 , … , P , where P is the maximum number of iteration.

Step 2: Construct ℋ and B according to equations (12) or (22).

Step 3: Calculate the unknown Haar wavelet coefficients with the help of

X = ℋ − 1 B .

Step 4: Construct an approximate solution from equations (9) or (20).

Output: If the maximum of absolute error is acceptable, then the for loop will end; otherwise, go to Step 2.

4 Convergence analysis

Let [ 0 , T ] be partitioned into 0 = τ 0 < τ 1 < … τ P = T , where P is a positive integer. In this section, we derive the rate of convergence when φ ( s , τ P ) is approximated by the solution given by the numerical method designed in Section 3.

Theorem 1

Assume that ∂ φ ∂ τ , ∂ φ ∂ s , and ∂ 2 φ ∂ s 2 exist and are bounded in [ a , b ] × [ 0 , T ] . For any M = 2 J , J = 0 , 1 , 2 , … , and p = 0 , 1 , … P , where P is a positive integer, if P M p ( s ) is the Haar wavelet solution and φ ( s , τ p ) is the exact solution, then

max 0 ≤ p ≤ P ‖ φ ( . , τ p ) − P M p ‖ L ∞ ( a , b ) ≤ O 1 M + O ( Δ τ ) , a s J → ∞ a n d P → ∞ ,

where Δ τ = max 0 ≤ p ≤ P − 1 ( τ p + 1 − τ p ) .

Proof

For p = 1 , 2 , … , P , we have

‖ φ ( . , τ p ) − P M p ‖ L ∞ ( a , b ) ≤ ‖ E M ‖ L ∞ ( a , b ) + ‖ φ M ( . , τ p ) − P M p ‖ L ∞ ( a , b ) ,

where ‖ E M ‖ L ∞ ( a , b ) is defined as follows:

‖ E M ‖ L ∞ ( a , b ) ≔ ‖ φ ( . , τ p ) − φ M ( . , τ p ) ‖ L ∞ ( a , b ) = max s ∑ i = 2 M + 1 ∞ λ i p i , 1 ( s ) .

It is shown in [61, equation (18)] that λ i ≤ β ∕ 2 j + 1 . In fact, it can be shown that ∣ λ i ∣ ≤ β ∕ 2 j + 1 (which should also have been required in [61]). Therefore,

‖ E M ‖ L ∞ ( a , b ) ≤ β ∑ i = 2 M + 1 ∞ 1 2 j + 1 max s ∣ p i , 1 ( s ) ∣ .

By using equation (4), we obtain

‖ E M ‖ L ∞ ( a , b ) ≤ β ∑ i = 2 M + 1 ∞ 1 2 j + 1 max s ∣ p i , 1 ( s ) ∣ ≤ β ( b − a ) ∑ i = 2 M + 1 ∞ 1 2 j + 1 2 ≤ β ( b − a ) ∑ j = J + 1 ∞ ∑ k = 0 2 j − 1 1 2 j + 1 2 = β ( b − a ) 2 ∑ j = J + 1 ∞ 1 2 j + 1 ≤ β ( b − a ) 4 1 2 J + 1 = β ( b − a ) 4 M ,

where b − a is the length of the interval, M = 2 J , and b − a ≪ M when J → ∞ , so

‖ E M ‖ L ∞ ( a , b ) ≤ O 1 M .

The second part ‖ φ M ( . , τ p ) − P M p ‖ L ∞ ( a , b ) is the error due to the time iteration where we used forward difference approximation, which is the first-order accurate in time, i.e., ‖ φ M ( . , τ p ) − P M p ‖ ≤ O ( Δ τ ) .□

Theorem 2

Assume that ∂ φ ∂ τ , ∂ 2 φ ∂ τ 2 , ∂ w ∂ s , ∂ 2 w ∂ s 2 , and ∂ 3 w ∂ s 3 exist and are bounded in [ a , b ] × [ 0 , T ] . For any M = 2 J , J = 0 , 1 , 2 , … , and p = 0 , 1 , … P , where P is a positive integer, if P M p ( s ) is the Haar wavelet solution and φ ( s , τ p ) is the exact solution, then

max 0 ≤ p ≤ P ‖ φ ( . , τ p ) − P M p ‖ L ∞ ( a , b ) ≤ O 1 M 2 + O ( Δ τ ) , a s J → ∞ a n d P → ∞ ,

where Δ τ = max 0 ≤ p ≤ P − 1 ( τ p + 1 − τ p ) .

Proof

For p = 1 , 2 , … , P , we have

‖ φ ( . , τ p ) − P M p ‖ L ∞ ( a , b ) ≤ ‖ E M ‖ L ∞ ( a , b ) + ‖ φ M ( . , τ p ) − P M p ‖ L ∞ ( a , b ) ,

where ‖ E M ‖ L ∞ ( a , b ) is defined as follows:

‖ E M ‖ L ∞ ( a , b ) ≔ ‖ φ ( . , τ p ) − φ M ( . , τ p ) ‖ L ∞ ( a , b ) = max s ∑ i = 2 M + 1 ∞ λ i h ˜ i ( s ) .

‖ E M ‖ L ∞ ( a , b ) ≤ β ∑ i = 2 M + 1 ∞ 1 2 j + 1 max s ∣ h ˜ i ( s ) ∣ .

By using successively equation (17), the triangle inequality, equations (3) and (5), we obtain

‖ E M ‖ L ∞ ( a , b ) ≤ β ∑ i = 2 M + 1 ∞ 1 2 j + 1 max s ( p i , 2 ( s ) ) ∣ + max s ∣ ( s − a ) C b − a ≤ 2 β ( b − a ) 2 ∑ i = 2 M + 1 ∞ 1 2 j + 1 3 ≤ 2 β ( b − a ) 2 ∑ j = J + 1 ∞ ∑ k = 0 2 j − 1 1 2 j + 1 3 = β ( b − a ) 2 ∑ j = J + 1 ∞ 1 2 j + 1 2 ≤ β ( b − a ) 2 3 1 2 J + 1 2 = O 1 M 2 .

The second part ‖ φ M ( . , τ p ) − P M p ‖ L ∞ ( a , b ) is the error due to the time iteration where we used a finite difference approximation that is first-order accurate in time, i.e., ‖ φ M ( . , τ p ) − P M p ‖ ≤ O ( Δ τ ) .□

5 Test cases

We implement the HWCM by calculating the results of different numerical test problems. The E ∞ error norm has been used for accuracy measurements, which is defined as follows:

E ∞ = max 1 ≤ l ≤ 2 M ( ∣ φ ( s l , τ P ) − P M , l P ∣ ) .

All results were obtained by “MATLAB R2009b” software on DELL PC Laptop (Intel(R) Core(TM)i3-3110M CPU 2.40 GHz, 4.0 GB RAM). For all computations, we used CPU time having the unit “second.”

Test Problem 1. We consider the following linear case of first-order hyperbolic equation with f ( φ ( s , τ ) ) = φ ( s , τ ) in equation (1):

∂ φ ∂ τ + ∂ φ ∂ s + φ = g ( s , τ ) , a ≤ s ≤ b , 0 ≤ τ ≤ T ,

with the initial and boundary conditions,

φ ( s , 0 ) = cos ( s ) and φ ( a , τ ) = cos ( a + τ ) ,

where

g ( s , τ ) = − 2 sin ( s + τ ) + cos ( s + τ ) .

The exact solution is given in [1,2]

φ ( s , τ ) = cos ( s + τ ) .

Tables 1 and 2 describe the maximum error, the convergence, and the CPU time. The theoretical rate of convergence is 1 (Theorem 1), which is in good agreement with the experimental rate of convergence given in Table 1. The comparison of Haar wavelet-based solution with the exact solution and the absolute error at different T are shown in Figure 1. From the aforementioned figures and tables, it is concluded that the proposed HWCM is efficient and accurate.

Table 1

The numerical results at M = 16 , a = 0 , b = 1 , and T = 1 for Test Problem 1. The theoretical rate of convergence is 1 (Theorem 1)

Δ τ E ∞ Experimental rate of convergence
1/10 0.0575 —
1/20 0.0290 0.9843
1/30 0.0195 0.9878
1/40 0.0146 0.9900
1/50 0.0117 0.9915
1/60 0.0098 0.9926
1/70 0.0084 0.9935
1/80 0.0074 0.9942
1/90 0.0065 0.9949
1/100 0.0059 0.9954
Table 2

The numerical results at a = 0 , b = 1 , and T = 1 for Test Problem 1 with different M

M E ∞ CPU time
( Δ τ = 0.01 )
1 1.579 × 1 0 − 2 0.5020
2 6.605 × 1 0 − 3 0.5312
4 5.791 × 1 0 − 3 0.6467
8 5.851 × 1 0 − 3 0.8029
16 5.890 × 1 0 − 3 1.5616
( Δ τ = 0.001 )
1 1.389 × 1 0 − 2 0.5855
2 4.005 × 1 0 − 3 0.7357
4 1.112 × 1 0 − 3 1.1304
8 6.208 × 1 0 − 4 3.0479
16 5.905 × 1 0 − 4 7.2140
Figure 1 Numerical solutions for Test Problem 1 at different T T with Δ τ = 0.01 \Delta \tau =0.01 .
Figure 1

Numerical solutions for Test Problem 1 at different T with Δ τ = 0.01 .

Test Problem 2. Now considering the following first-order NLHPDE with f ( φ ( s , τ ) ) = sin ( φ ( s , τ ) ) in equation (1):

∂ φ ∂ τ + ∂ φ ∂ s + sin ( φ ) = g ( s , τ ) , a ≤ s ≤ b , 0 ≤ τ ≤ T ,

with the initial and boundary conditions,

φ ( s , 0 ) = sin ( s ) and φ ( a , τ ) = sin ( a − τ ) ,

where

g ( s , τ ) = sin ( sin ( s − τ ) ) .

The exact solution is

φ ( s , τ ) = sin ( s − τ ) .

Tables 3, 4, 5, 6 illustrate the maximum error, the convergence, and the CPU time, where the accuracy of the solution depends on parameters M and Δ τ . The theoretical and experimental rates of convergence are inline, i.e., 1 (Theorem 1 and Table 6). The maximum errors for T ≤ 10 and T > 10 are given in Tables 3 and 4, and the algorithm gives stable results for T ≫ 10 . From these tables, it is concluded that as the resolution M increases and Δ τ decreases, the accuracy of the proposed HWCM also increases.

Table 3

The numerical results at M = 16 , a = 0 , b = 1 , and Δ τ = 0.01 when T ≤ 10 for Test Problem 2

T 1 2 4 6 8 10
E ∞ 6.115 × 1 0 − 3 2.112 × 1 0 − 3 6.610 × 1 0 − 3 4.947 × 1 0 − 3 1.717 × 1 0 − 3 6.808 × 1 0 − 3
CPU time 1.060 1.988 3.982 5.954 7.913 9.852
Table 4

The numerical results at M = 16 , a = 0 , b = 1 , and Δ τ = 0.01 when T > 10 for Test Problem 2

T 20 40 60 80 100
E ∞ 5.624 × 1 0 − 3 3.078 × 1 0 − 3 6.477 × 1 0 − 3 3.727 × 1 0 − 3 4.122 × 1 0 − 3
CPU time 12.550 26.065 38.383 50.015 62.849
Table 5

The numerical results at a = 0 , b = 1 , and T = 1 for Test Problem 2 with different M

M E ∞ CPU time E ∞ CPU time
( Δ τ = 0.01 ) ( Δ τ = 0.001 )
1 1.402 × 1 0 − 2 0.050 1.485 × 1 0 − 2 0.138
2 7.530 × 1 0 − 3 0.093 5.070 × 1 0 − 3 0.252
4 6.410 × 1 0 − 3 0.111 1.455 × 1 0 − 3 0.733
8 6.146 × 1 0 − 3 0.315 7.933 × 1 0 − 4 2.606
16 6.115 × 1 0 − 3 1.060 6.649 × 1 0 − 4 7.084
Table 6

The numerical results at M = 16 , a = 0 , b = 1 , and T = 1 for Test Problem 2. The The theoretical rate of convergence is 1 (Theorem 1)

Δ τ E ∞ Experimental rate of convergence
1/10 0.0562 —
1/20 0.0291 0.9492
1/30 0.0197 0.9621
1/40 0.0149 0.9682
1/50 0.0120 0.9717
1/60 0.0100 0.9739
1/70 0.0086 0.9754
1/80 0.0076 0.9764
1/90 0.0067 0.9771
1/100 0.0061 0.9775

Test Problem 3. We consider the following first-order NLHPDE with f ( φ ( s , τ ) ) = e φ ( s , τ ) in equation (1)

∂ φ ∂ τ + ∂ φ ∂ s + e φ = g ( s , τ ) , a ≤ s ≤ b , 0 ≤ τ ≤ T ,

with the initial and boundary conditions,

φ ( s , 0 ) = s 2 − s and φ ( a , τ ) = a 2 − a + τ ,

where

g ( s , τ ) = 2 s + e ( s 2 − s + τ ) .

The exact solution is

φ ( s , τ ) = s 2 − s + τ .

In Tables 7 and 8 the maximum error, the convergence, and the CPU time are displayed, where the precision of the solution depends on M and Δ τ . The theoretical and experimental rates of convergence are aligned, i.e., 1 (Theorem 1 and Table 8). Figure 2 depicts the space-time graph of approximate and exact solutions. The comparison of numerical and exact solutions at different T are also given in Figure 3. From these tables and figures it is concluded that the proposed HWCM can easily and accurately handle the first-order NLHPDE with various types of nonlinear terms.

Table 7

The numerical results at a = 0 , b = 1 , and T = 1 for Test Problem 3 with different M

M E ∞ CPU time E ∞ CPU time
( Δ τ = 0.01 ) ( Δ τ = 0.001 )
1 4.499 × 1 0 − 2 0.148 4.205 × 1 0 − 2 0.231
2 1.439 × 1 0 − 3 0.284 1.243 × 1 0 − 3 0.435
4 8.217 × 1 0 − 3 0.295 3.522 × 1 0 − 3 0.934
8 7.714 × 1 0 − 3 0.482 9.917 × 1 0 − 4 2.719
16 7.625 × 1 0 − 3 1.136 8.042 × 1 0 − 4 10.030
Table 8

The numerical results at M = 16 , a = 0 , b = 1 , and T = 1 for Test Problem 2. Theoretical rate of convergence is 1 (Theorem 1)

Δ τ E ∞ Experimented rate of convergence
1/10 0.0765 —
1/20 0.0379 1.0103
1/30 0.0253 1.0021
1/40 0.0189 0.9991
1/50 0.0151 0.9974
1/60 0.0126 0.9962
1/70 0.0108 0.9953
1/80 0.0095 0.9946
1/90 0.0084 0.9939
1/100 0.0076 0.9933
Figure 2 The 3D plots for Test Problem 3 at M = 16 M=16 , T = 1 T=1 , and Δ τ = 0.01 \Delta \tau =0.01 .
Figure 2

The 3D plots for Test Problem 3 at M = 16 , T = 1 , and Δ τ = 0.01 .

Figure 3 Comparison of exact and numerical solutions for Test Problem 3 at different T T with M = 16 M=16 and Δ τ = 0.01 \Delta \tau =0.01 .
Figure 3

Comparison of exact and numerical solutions for Test Problem 3 at different T with M = 16 and Δ τ = 0.01 .

Test Problem 4. We consider the following dissipative second-order NLHPDE [3,62]:

∂ 2 φ ∂ τ 2 − ∂ 2 φ ∂ s 2 + 2 φ ∂ φ ∂ τ = ( π 2 − 1 − 2 sin ( π s ) sin ( τ ) ) sin ( π s ) cos ( τ ) , 0 ≤ s ≤ 1 , 0 ≤ τ ≤ T ,

with the initial conditions,

φ ( s , 0 ) = sin ( π s ) and ∂ φ ( s , 0 ) ∂ τ = 0 ,

and the boundary conditions

φ ( 0 , τ ) = 0 = φ ( 1 , τ ) .

The exact solution is

φ ( s , τ ) = sin ( π s ) cos ( τ ) .

The 3D contours of exact and numerical solutions are compared in Figure 4, where the peaks appear by increasing the time T . The absolute errors are calculated for various time T = 1 , 7, 9, 12, 15, and 19 and are presented graphically in Figure 5. In Table 9, we fixed Δ τ = 0.0001 for different values of M to check the spatial convergence of the HWCM, and it has been found that the theoretical and experimental rates of convergence for the space variable are in good agreement, and the CUP times are also shown. In Table 10, we fixed M = 16 and T = 1 for different Δ τ to check the time variable convergence, where the theoretical and experimental rates of convergence for the time variable are also in good agreement. Hence, the HWCM is convergent and efficient.

Figure 4 The 3D contour plots for Test Problem 4 at M = 16 M=16 .
Figure 4

The 3D contour plots for Test Problem 4 at M = 16 .

Figure 5 The absolute error for Test Problem 4 at different T T with M = 16 M=16 and Δ τ = 0.01 \Delta \tau =0.01 .
Figure 5

The absolute error for Test Problem 4 at different T with M = 16 and Δ τ = 0.01 .

Table 9

The numerical results at Δ τ = 0.0001 , a = 0 , b = 1 , and T = 1 for Test Problem 4. The theoretical rate of convergence is 2 (Theorem 1)

M E ∞ Experimental rate of convergence CPU time (second unit)
1 7.5329 × 1 0 − 2 — 2.7428
2 3.0358 × 1 0 − 2 1.3111 5.6209
4 8.3629 × 1 0 − 3 1.8600 11.1650
8 2.0812 × 1 0 − 3 2.0065 24.4861
16 4.5977 × 1 0 − 4 2.1784 41.5712
Table 10

The numerical results at M = 16 , a = 0 , b = 1 , and T = 1 for Test Problem 4. The theoretical rate of convergence is 1 (Theorem 1)

Δ τ E ∞ Experimental rate of convergence CPU time (second unit)
1/10 7.5368 × 1 0 − 2 — 1.3518
1/20 3.9613 × 1 0 − 2 0.9279 1.4755
1/30 2.6765 × 1 0 − 2 0.9669 1.8109
1/40 2.0146 × 1 0 − 2 0.9874 2.3692
1/50 1.6110 × 1 0 − 2 1.0019 2.7356
1/60 1.3391 × 1 0 − 2 1.0137 3.1376
1/70 1.1435 × 1 0 − 2 1.0241 3.6975
1/80 9.9613 × 1 0 − 2 1.0336 4.1004
1/90 8.8100 × 1 0 − 2 1.0427 4.5785
1/100 7.8861 × 1 0 − 2 1.0514 5.0365

Test Problem 5. We consider the following Klein-Gordon equation with two different cases as a challenging problem [63,64]:

(23) ∂ 2 φ ∂ τ 2 − a 1 2 ∂ 2 φ ∂ s 2 + a 1 φ − b 1 φ 3 = 0 , − 10 ≤ s ≤ 10 , 0 ≤ τ ≤ T ,

where a 1 , b 1 ∈ R , and a 1 b 1 ≠ 0 .

Case (i) (Single-soliton wave): Equation (23) represents a single-soliton wave if the initial conditions are

φ ( s , 0 ) = A sech ( B s ) and ∂ φ ( s , 0 ) ∂ τ = c o A B sech ( B s ) tanh ( B s ) ,

and the boundary conditions are

φ ( 0 , τ ) = A sech ( B ( − 10 − c o τ ) ) and φ ( 1 , τ ) = A sech ( B ( 10 − c o τ ) ) .

The exact solution is

φ ( s , τ ) = A sech ( B ( s − c o τ ) ) ,

where A = 2 a 1 b 1 ; B = a 1 a 1 2 − c o 2 ; a 1 , b 1 , a 1 2 − c o 2 > 0 ; c o represents the velocity, and A is the wave amplitude. In our computation, we have considered a 1 = 0.3 , b 1 = 1 and c o = 0.25 .

Case (ii) (Double-soliton wave): Equation (23) represents a double-soliton wave if the initial conditions are

φ ( s , 0 ) = A sech [ μ 1 ( s − x o ) ] + A sech [ μ 2 ( s − x ́ o ) ] and ∂ φ ( s , 0 ) ∂ τ = c 1 μ 1 A sech [ μ 1 ( s − x o ) ] tanh [ μ 1 ( s − x o ) ] + c 2 μ 2 A sech [ μ 2 ( s − x ́ o ) ] tanh [ μ 2 ( s − x ́ o ) ] ,

and the boundary conditions are

φ ( − 10 , τ ) = A sech [ μ 1 ( − 10 − x o ) − c 1 τ ] + A sech [ μ 2 ( − 10 − x ́ o ) − c 2 τ ] and φ ( 10 , τ ) = A sech [ μ 1 ( 10 − x o ) − c 1 τ ] + A sech [ μ 2 ( 10 − x ́ o ) − c 2 τ ] .

The exact solution is

φ ( s , τ ) = A sech [ μ 1 ( s − x o ) − c 1 τ ] + A sech [ μ 2 ( s − x ́ o ) − c 2 τ ] ,

where A = 2 a 1 b 1 is the wave amplitude; μ 1 = a 1 a 1 2 − c 1 2 ; μ 2 = a 1 a 1 2 − c 2 2 ; and a 1 , b 1 , a 1 2 − c 1 2 , a 1 2 − c 2 2 > 0 . In our computation, we have considered a 1 = 0.3 , b 1 = 1 , c 1 = − c 2 = 0.25 , x o = − 2 , and x ́ o = 2 .

In Figure 6, the Haar wavelet based numerical solution is presented at different T and Δ τ by fixing M , where the HWCM has easily captured the single soliton and there does not seem to be any blow-up phenomenon.

Figure 6 The numerical solution for Test Problem 5(i) at M = 512 M=512 .
Figure 6

The numerical solution for Test Problem 5(i) at M = 512 .

In Figure 7, the high resolution-based numerical solution is shown, where the HWCM captured the double soliton and also there does not seem to be any blow-up phenomenon, therefore the HWCM is stable. This example also illustrates the validity and capability of the proposed HWCM.

Figure 7 The numerical solutions for Test Problem 5(ii) at T = 1 T=1 , M = 512 M=512 , and Δ τ = 0.01 \Delta \tau =0.01 .
Figure 7

The numerical solutions for Test Problem 5(ii) at T = 1 , M = 512 , and Δ τ = 0.01 .

Test Problem 6. We consider the following Sine-Gordon equation as a special case of second-order NLHPDE

∂ 2 φ ∂ τ 2 − ∂ 2 φ ∂ s 2 + sin ( φ ) = 0 , − 20 ≤ s ≤ 20 , 0 ≤ τ ≤ T .

The exact solution is given in [63]

φ ( s , τ ) = 4 tan − 1 c o sinh s 1 − c o 2 cosh c o τ 1 − c o 2 .

The initial and boundary conditions can be obtained from the exact solution. For our numerical calculation, we have considered the velocity c o = 0.5 .

This Test Problem is challenging and represents the collision of two Sine-Gordon kink solitons. In Figure 8, the comparison of the exact and numerical solutions for different time T = 0.1 , 5, 10, and 20 are shown and the collision of kink solitons can be clearly observed. The 3D view of the numerical and exact solution up to T = 8 is presented in Figure 9. In Table 11, we fixed Δ τ = 0.0001 for different values of M to check the spatial convergence of the HWCM, and it has been found that the theoretical and experimental rates of convergence for the space variable are in good agreement. The numerical results at M = 16 and T = 1 for different Δ τ are given in Table 12, where the theoretical and experimental rates of convergence for the time variable are also in good agreement in this case.

Figure 8 The comparison of exact and numerical solutions for Test Problem 6 at different T T with M = 16 M=16 and Δ τ = 0.01 \Delta \tau =0.01 .
Figure 8

The comparison of exact and numerical solutions for Test Problem 6 at different T with M = 16 and Δ τ = 0.01 .

Figure 9 The 3D plots for Test Problem 6 at M = 16 M=16 .
Figure 9

The 3D plots for Test Problem 6 at M = 16 .

Table 11

The numerical results at Δ τ = 0.0001 , a = 0 , b = 1 , and T = 1 for Test Problem 6. The theoretical rate of convergence is 2 (Theorem 1)

M E ∞ Experimental rate of convergence CPU time (second unit)
1 4.5990 × 1 0 − 3 — 2.4199
2 2.3806 × 1 0 − 3 0.9499 5.4584
4 6.9063 × 1 0 − 4 1.7853 15.5490
8 2.0234 × 1 0 − 4 1.7710 55.3128
Table 12

The numerical results at M = 16 , a = 0 , b = 1 , and T = 1 for Test Problem 6. The theoretical rate of convergence is 1 (Theorem 1)

Δ τ E ∞ Experimental rate of convergence
1/10 0.0562 —
1/20 0.0261 0.9721
1/30 0.0175 0.9847
1/40 0.0132 0.9893
1/50 0.0105 0.9917
1/60 0.0088 0.9932
1/70 0.0075 0.9942
1/80 0.0066 0.9949
1/90 0.0059 0.9954
1/100 0.0053 0.9958

6 Conclusion

In this work, we have proposed the HWCM for the numerical solution of the first- and second-order NLHPDEs. The E ∞ error norm and the rate of convergence show that the proposed numerical method is accurate and applicable to solve NLHPDEs. Considering the different types of nonlinear equations discussed earlier, we may conclude that the proposed HWCM is practical, efficient, and effective for solving the first- and second-order NlHPDEs numerically. Due to the high potential achievements of the HWCM, the current scheme can be implemented to 2D and coupled NLHPDEs. These topics are the focus of our forthcoming work.

  1. Funding information: The first author would like to acknowledge the financial support from the research grants: 2020B1515120083 by the Guangdong Basic and Applied Basic Research Foundation and JCYJ20210324121402008 by the Shenzhen Science and Technology Innovation Commission. The corresponding author (Zaheer Uddin) is thankful to the CECOS University for research encouragement.

  2. Author contributions: All authors have equally contributed to finalizing the article.

  3. Conflict of interest: The authors declare that there is no conflict of interest.

  4. Data availability statement: Not applicable.

References

[1] E. H. Doha, R. M. Hafez, and Y. H. Youssri, Shifted Jacobi spectral-Galerkin method for solving hyperbolic partial differential equations, Comput. Math. Appl. 78 (2019), no. 3, 889–904. 10.1016/j.camwa.2019.03.011Search in Google Scholar

[2] S. Singh, V. Kumar Patel, and V. K. Singh, Application of wavelet collocation method for hyperbolic partial differential equations via matrices, Appl. Math. Comput. 320 (2018), 407–424. 10.1016/j.amc.2017.09.043Search in Google Scholar

[3] S. N. Jator, F. F. Ngwane, and N. O. Kirby, Functionally fitted block method for solving the general oscillatory second-order initial value problems and hyperbolic partial differential equations, Math. Prob. Eng. 2019 (2019), 1–14. 10.1155/2019/1535430Search in Google Scholar

[4] J. Banasiak and J. R. Mika, Singularly perturbed telegraph equations with applications in the random walk theory, J. Appl. Math. Stoch. Anal. 11 (1998), no. 1, 9–28. 10.1155/S1048953398000021Search in Google Scholar

[5] V. H. Weston and S. He, Wave splitting of the telegraph equation in R3 and its application to inverse scattering, Inv. Prob. 9 (1993), no. 6, 789. 10.1088/0266-5611/9/6/013Search in Google Scholar

[6] L.-B. Liu and H.-W. Liu, Compact difference schemes for solving telegraphic equations with Neumann boundary conditions, Appl. Math. Comput. 219 (2013), no. 19, 10112–10121. 10.1016/j.amc.2013.04.021Search in Google Scholar

[7] H. Moghaderi and M. Dehghan, A multigrid compact finite difference method for solving the one-dimensional nonlinear Sine-Gordon equation, Math. Methods Appl. Sci. 38 (2015), no. 17, 3901–3922. 10.1002/mma.3326Search in Google Scholar

[8] J. Rashidinia and R. Mohammadi, Tension spline solution of nonlinear Sine-Gordon equation, Numer. Alg. 56 (2011), no. 1, 129–142. 10.1007/s11075-010-9377-xSearch in Google Scholar

[9] W. Bao, Y. Feng, and W. Yi, Long time error analysis of finite difference time domain methods for the nonlinear Klein-Gordon equation with weak nonlinearity, Commun. Comput. Phys 26 (2019), 1307–1334. 10.4208/cicp.2019.js60.03Search in Google Scholar

[10] Y. Luo, X. Li, and C. Guo, Fourth-order compact and energy conservative scheme for solving nonlinear Klein-Gordon equation, Numer. Methods Partial Differential Equations 33 (2017), no. 4, 1283–1304. 10.1002/num.22143Search in Google Scholar

[11] A. G. Atta, W. M. Abd-Elhameed, G. M. Moatimid, and Y. H. Youssri, Shifted fifth-kind Chebyshev Galerkin treatment for linear hyperbolic first-order partial differential equations, Appl. Numer. Math. 167 (2021), 237–256. 10.1016/j.apnum.2021.05.010Search in Google Scholar

[12] W. M. Abd-Elhameed, E. H. Doha, Y. H. Youssri, and M. Bassuony, New Tchebyshev-Galerkin operational matrix method for solving linear and nonlinear hyperbolic telegraph type equations, Numer. Methods Partial Differential Equations 32 (2016), no. 6, 1553–1571. 10.1002/num.22074Search in Google Scholar

[13] Ş. Yüzbaşı and M. Karaçayır, A Galerkin-type method to solve one-dimensional telegraph equation using collocation points in initial and boundary conditions, Int. J. Comput. Methods 15 (2018), no. 5, 1850031. 10.1142/S0219876218500317Search in Google Scholar

[14] R. C. Mittal and R. Bhatia, Numerical solution of nonlinear Sine-Gordon equation by modified cubic B-spline collocation method, Int. J. Partial Differential Equations 2014 (2014), 1–8. 10.1155/2014/343497Search in Google Scholar

[15] W. Liu, B. Wu, and J. Sun, Space-time spectral collocation method for the one-dimensional Sine-Gordon equation, Numer. Methods Partial Differential Equations 31 (2015), no. 3, 670–690. 10.1002/num.21910Search in Google Scholar

[16] Y. H. Youssri and R. M. Hafez, Exponential Jacobi spectral method for hyperbolic partial differential equations, Math. Sci. 13 (2019), no. 4, 347–354. 10.1007/s40096-019-00304-wSearch in Google Scholar

[17] A. G. Atta, W. M. Abd-Elhameed, G. M. Moatimid, and Y. H. Youssri, Advanced shifted sixth-kind Chebyshev tau approach for solving linear one-dimensional hyperbolic telegraph type problem, Math. Sci. (2022), 1–15, DOI: https://doi.org/10.1007/s40096-022-00460-6. 10.1007/s40096-022-00460-6Search in Google Scholar

[18] Q. Jin and W. Wang, Analysis of the iteratively regularized Gauss-Newton method under a heuristic rule, Inverse Problems 34 (2018), no. 3, 1–24. 10.1088/1361-6420/aaa0fbSearch in Google Scholar

[19] Q. Jin, On a heuristic stopping rule for the regularization of inverse problems by the augmented Lagrangian method, Numer. Math. 136 (2017), no. 4, 973–992. 10.1007/s00211-016-0860-8Search in Google Scholar

[20] Q. Jin and X. Lu, A fast nonstationary iterative method with convex penalty for inverse problems in Hilbert spaces, Inverse Problems 30 (2014), no. 4, 1–21. 10.1088/0266-5611/30/4/045012Search in Google Scholar

[21] Y. Liu, Y. Liu, and Z. Cen, Daubechies wavelet meshless method for 2-D elastic problems, Tsinghua Sci. Tech. 13 (2008), no. 5, 605–608. 10.1016/S1007-0214(08)70099-3Search in Google Scholar

[22] L. A. Diiiaz, M. T. Martiiin, and V. Vampa, Daubechies wavelet beam and plate finite elements, Finite Elem. Anal. Des. 45 (2009), no. 3, 200–209. 10.1016/j.finel.2008.09.006Search in Google Scholar

[23] G.-W. Jang, Y. Y. Kim, and K. K. Choi, Remesh-free shape optimization using the wavelet-Galerkin method, Int. J. Sold. Struc. 41 (2004), no. 22, 6465–6483. 10.1016/j.ijsolstr.2004.05.010Search in Google Scholar

[24] Siraj-ul-Islam, I. Aziz, and A. S. Al-Fhaid, An improved method based on Haar wavelets for numerical solution of nonlinear integral and integro-differential equations of first and higher orders, J. Comput. Appl. Math. 260 (2014), 449–469. 10.1016/j.cam.2013.10.024Search in Google Scholar

[25] M. Ahsan, M. Bohner, A. Ullah, A. A. Khan, and S. Ahmad, A Haar wavelet multi-resolution collocation method for singularly perturbed differential equations with integral boundary conditions, Math. Comput. Simulation 204 (2023), 166–180. 10.1016/j.matcom.2022.08.004Search in Google Scholar

[26] W. Dahmen, A. J. Kurdila, and P. Oswald (eds.), Multiscale wavelet methods for partial differential equations, Wavelet Analysis and its Applications, vol. 6, Academic Press, Inc., San Diego, CA, 1997. Search in Google Scholar

[27] T. Tran, E. P. Stephan, and P. Mund, Hierarchical basis preconditioners for first kind integral equations, Appl. Anal. 65 (1997), no. 3–4, 353–372. 10.1080/00036819708840566Search in Google Scholar

[28] K. Maleknejad and F. Mirzaee, Using rationalized Haar wavelet for solving linear integral equations, Appl. Math. Comput. 160 (2005), no. 2, 579–587. 10.1016/j.amc.2003.11.036Search in Google Scholar

[29] U. Lepik, Haar wavelet method for nonlinear integro-differential equations, Appl. Math. Comput. 176 (2006), no. 1, 324–333. 10.1016/j.amc.2005.09.021Search in Google Scholar

[30] Siraj-ul-Islam, M. Ahsan, and I. Hussian, A multi-resolution collocation procedure for time-dependent inverse heat problems, Int. J. Therm. Sci. 128 (2018), 160–174. 10.1016/j.ijthermalsci.2018.01.001Search in Google Scholar

[31] M. Ahsan, I. Ahmad, M. Ahmad, and I. Hussian, A numerical Haar wavelet-finite difference hybrid method for linear and non-linear Schrödinger equation, Math. Comput. Simulation 165 (2019), 13–25. 10.1016/j.matcom.2019.02.011Search in Google Scholar

[32] C.-H. Hsiao and W.-J. Wang, Haar wavelet approach to nonlinear stiff systems, Math. Comput. Simulation 57 (2001), no. 6, 347–353. 10.1016/S0378-4754(01)00275-0Search in Google Scholar

[33] F. A. Shah, R. Abass, and J. Iqbal, Numerical solution of singularly perturbed problems using Haar wavelet collocation method, Cogent Math. 3 (2016), Art. 1202504, 13. 10.1080/23311835.2016.1202504Search in Google Scholar

[34] F. A. Shah and R. Abass, An operational Haar wavelet collocation method for solving singularly perturbed boundary-value problems, SeMA J. Bol. Soci. Espan. Matem. Apli. 74 (2017), no. 4, 457–474. 10.1007/s40324-016-0094-9Search in Google Scholar

[35] G. Rana, M. Asif, N. Haider, R. Bilal, M. Ahsan, Q. Al-Mdallal, et al., A modified algorithm based on Haar wavelets for the numerical simulation of interface models, J. Funct. Spaces 2022 (2022), 1–15. 10.1155/2022/1541486Search in Google Scholar

[36] I. Aziz, Siraj-ul-Islam, and F. Khan, A new method based on Haar wavelet for the numerical solution of two-dimensional nonlinear integral equations, J. Comput. Appl. Math. 272 (2014), 70–80. 10.1016/j.cam.2014.04.027Search in Google Scholar

[37] Siraj-ul-Islam, I. Aziz, and F. Haq, A comparative study of numerical integration based on Haar wavelets and hybrid functions, Comput. Math. Appli. 59 (2010), no. 6, 2026–2036. 10.1016/j.camwa.2009.12.005Search in Google Scholar

[38] I. Aziz, Siraj-ul-Islam, and B. Šarler, Wavelets collocation methods for the numerical solution of elliptic BV problems, Appl. Math. Model. 37 (2013), no. 3, 676–694. 10.1016/j.apm.2012.02.046Search in Google Scholar

[39] M. Ahsan, W. Lei, M. Ahmad, M. S. Hussein, and Z. Uddin, A wavelet-based collocation technique to find the discontinuous heat source in inverse heat conduction problems, Phys. Scr. 97 (2022), no. 12, 125208. 10.1088/1402-4896/ac9dc6Search in Google Scholar

[40] Siraj-ul-Islam, I. Aziz, and M. Ahmad, Numerical solution of two-dimensional elliptic PDEs with nonlocal boundary conditions, Comput. Math. Appl. 69 (2015), no. 3, 180–205. 10.1016/j.camwa.2014.12.003Search in Google Scholar

[41] M. Ahsan, Siraj-ul-Islam, and I. Hussain, Haar wavelets multi-resolution collocation analysis of unsteady inverse heat problems, Inverse Probl. Sci. Engrg. 27 (2019), no. 11, 1498–1520. 10.1080/17415977.2018.1481405Search in Google Scholar

[42] M. Ahsan, M. Ahmad, W. Khan, E. E. Mahmoud, and A.-H. Abdel-Aty, Meshless analysis of nonlocal boundary value problems in anisotropic and inhomogeneous media, Mathematics 8 (2020), no. 11, 2045. 10.3390/math8112045Search in Google Scholar

[43] J. Majak, B. Shvartsman, M. Pohlak, K. Karjust, M. Eerme, and E. Tungel, Solution of fractional order differential equation by the Haar wavelet method. Numerical convergence analysis for most commonly used approach, AIP Conference Proceedings, vol. 1738, AIP Publishing LLC, 2016, p. 480110. 10.1063/1.4952346Search in Google Scholar

[44] Y. Li and W. Zhao, Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. Math. Comput. 216 (2010), no. 8, 2276–2285. 10.1016/j.amc.2010.03.063Search in Google Scholar

[45] M. Yi and J. Huang, Wavelet operational matrix method for solving fractional differential equations with variable coefficients, Appl. Math. Comput. 230 (2014), 383–394. 10.1016/j.amc.2013.06.102Search in Google Scholar

[46] U. Saeed and M. ur Rehman, Haar wavelet Picard method for fractional nonlinear partial differential equations, Appl. Math. Comput. 264 (2015), 310–322. 10.1016/j.amc.2015.04.096Search in Google Scholar

[47] L. Wang, Y. Ma, and Z. Meng, Haar wavelet method for solving fractional partial differential equations numerically, Appl. Math. Comput. 227 (2014), 66–76. 10.1016/j.amc.2013.11.004Search in Google Scholar

[48] X. Liu, M. Ahsan, M. Ahmad, I. Hussain, M. M. Alqarni, and E. E. Mahoud, Haar wavelets multi-resolution collocation procedures for two-dimensional nonlinear Schrödinger equation, Alex. Eng. J. 60 (2021), no. 3, 3057–3071. 10.1016/j.aej.2021.01.033Search in Google Scholar

[49] X. Liu, M. Ahsan, M. Ahmad, M. Nisar, X. Liu, I. Ahmad, et al., Applications of Haar wavelet-finite difference hybrid method and its convergence for hyperbolic nonlinear Schrödinger equation with energy and mass conversion, Energies 14 (2021), no. 23, 1–17, Art. ID 7831. 10.3390/en14237831Search in Google Scholar

[50] M. Ahsan, T. Tran, Siraj-ul-Islam, and I. Hussain, A multiresolution collocation method and its convergence for Burgers’ type equations, Math. Methods Appl. Sci. (2022), 1–24, DOI: https://doi.org/10.1002/mma.8764. 10.1002/mma.8764Search in Google Scholar

[51] M. Ahsan, S. Lin, M. Ahmad, M. Nisar, I. Ahmad, H. Ahmed, et al., A Haar wavelet-based scheme for finding the control parameter in nonlinear inverse heat conduction equation, Open Phys. 19 (2021), no. 1, 722–734. 10.1515/phys-2021-0080Search in Google Scholar

[52] M. Ahsan, I. Hussain, and M. Ahmad, A finite-difference and Haar wavelets hybrid collocation technique for non-linear inverse Cauchy problems, Appl. Math. Sci. Eng. 30 (2022), no. 1, 121–140. 10.1080/17415977.2022.2026350Search in Google Scholar

[53] M. Ahsan, K. Shams-ul-Haq, X. Liu, S. Ahmad, and M. Nisar, A Haar wavelets based approximation for nonlinear inverse problems influenced by unknown heat source, Math. Methods Appl. Sci. 46 (2023), no. 2, 2475–2487. 10.1002/mma.8655Search in Google Scholar

[54] S. Nazir, S. Shahzad, R. Wirza, R. Amin, M. Ahsan, N. Mukhtar, et al., Birthmark based identification of software piracy using Haar wavelet, Math. Comput. Simulation 166 (2019), 144–154. 10.1016/j.matcom.2019.04.010Search in Google Scholar

[55] S. M. Aznam and M. S. Chowdhury, Generalized Haar wavelet operational matrix method for solving hyperbolic heat conduction in thin surface layers, Results Phys. 11 (2018), 243–252. 10.1016/j.rinp.2018.08.021Search in Google Scholar

[56] S. Pandit, R. Jiwari, K. Bedi, and M. E. Koksal, Haar wavelets operational matrix based algorithm for computational modelling of hyperbolic type wave equations, Eng. Comput. 34 (2017), no. 8, 2793–2814. 10.1108/EC-10-2016-0364Search in Google Scholar

[57] R. K. Mohanty, New unconditionally stable difference schemes for the solution of multi-dimensional telegraphic equations, Int. J. Comput. Math. 86 (2009), no. 12, 2061–2071. 10.1080/00207160801965271Search in Google Scholar

[58] W. Greiner, Relativistic Quantum Mechanics, vol. 2, Springer, Berlin, 2000. 10.1007/978-3-662-04275-5Search in Google Scholar

[59] A. Scott, Nonlinear Science: Emergence and Dynamics of Coherent Structures, Oxford University Press, Oxford, 2003. 10.1093/oso/9780198528524.001.0001Search in Google Scholar

[60] T. Dauxois and M. Peyrard, Physics of Solitons, Cambridge University Press, Cambridge, 2006. Search in Google Scholar

[61] J. Majak, B. S. Shvartsman, M. Kirs, M. Pohlak, and H. Herranen, Convergence theorem for the Haar wavelet based discretization method, Compos. Struct. 126 (2015), 227–232. 10.1016/j.compstruct.2015.02.050Search in Google Scholar

[62] R. Jiwari, Lagrange interpolation and modified cubic B-spline differential quadrature methods for solving hyperbolic partial differential equations with Dirichlet and Neumann boundary conditions, Comput. Phys. Commun. 193 (2015), 55–65. 10.1016/j.cpc.2015.03.021Search in Google Scholar

[63] A. Mohebbi and M. Dehghan, High-order solution of one-dimensional Sine-Gordon equation using compact finite difference and DIRKN methods, Math. Comput. Modell. 51 (2010), no. 5–6, 537–549. 10.1016/j.mcm.2009.11.015Search in Google Scholar

[64] Y. Duan, L. Kong, and M. Guo, Numerical simulation of a class of nonlinear wave equations by lattice Boltzmann method, Commun. Math. Stat. 5 (2017), no. 1, 13–35. 10.1007/s40304-016-0098-xSearch in Google Scholar
Received: 2021-07-08
Revised: 2022-11-08
Accepted: 2023-01-20
Published Online: 2023-05-11

© 2023 the author(s), published by De Gruyter

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

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  3. Duality for convolution on subclasses of analytic functions and weighted integral operators
  4. Existence of a solution to an infinite system of weighted fractional integral equations of a function with respect to another function via a measure of noncompactness
  5. On the existence of nonnegative radial solutions for Dirichlet exterior problems on the Heisenberg group
  6. Hyers-Ulam stability of isometries on bounded domains-II
  7. Asymptotic study of Leray solution of 3D-Navier-Stokes equations with exponential damping
  8. Semi-Hyers-Ulam-Rassias stability for an integro-differential equation of order 𝓃
  9. Jordan triple (α,β)-higher ∗-derivations on semiprime rings
  10. The asymptotic behaviors of solutions for higher-order (m1, m2)-coupled Kirchhoff models with nonlinear strong damping
  11. Approximation of the image of the Lp ball under Hilbert-Schmidt integral operator
  12. Best proximity points in ℱ-metric spaces with applications
  13. Approximation spaces inspired by subset rough neighborhoods with applications
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  16. Fekete-Szegö functional for a class of non-Bazilevic functions related to quasi-subordination
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https://doi.org/10.1515/dema-2022-0203
Keywords for this article
Haar wavelet; hyperbolic equation; collocation method; single- and double-soliton wave
Creative Commons
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Articles in the same Issue

  1. Regular Articles
  2. A novel class of bipolar soft separation axioms concerning crisp points
  3. Duality for convolution on subclasses of analytic functions and weighted integral operators
  4. Existence of a solution to an infinite system of weighted fractional integral equations of a function with respect to another function via a measure of noncompactness
  5. On the existence of nonnegative radial solutions for Dirichlet exterior problems on the Heisenberg group
  6. Hyers-Ulam stability of isometries on bounded domains-II
  7. Asymptotic study of Leray solution of 3D-Navier-Stokes equations with exponential damping
  8. Semi-Hyers-Ulam-Rassias stability for an integro-differential equation of order 𝓃
  9. Jordan triple (α,β)-higher ∗-derivations on semiprime rings
  10. The asymptotic behaviors of solutions for higher-order (m1, m2)-coupled Kirchhoff models with nonlinear strong damping
  11. Approximation of the image of the Lp ball under Hilbert-Schmidt integral operator
  12. Best proximity points in ℱ-metric spaces with applications
  13. Approximation spaces inspired by subset rough neighborhoods with applications
  14. A numerical Haar wavelet-finite difference hybrid method and its convergence for nonlinear hyperbolic partial differential equation
  15. A novel conservative numerical approximation scheme for the Rosenau-Kawahara equation
  16. Fekete-Szegö functional for a class of non-Bazilevic functions related to quasi-subordination
  17. On local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) -preinvexity involving local fractional integral operators with Mittag-Leffler kernel
  18. On some geometric results for generalized k-Bessel functions
  19. Convergence analysis of M-iteration for 𝒢-nonexpansive mappings with directed graphs applicable in image deblurring and signal recovering problems
  20. Some results of homogeneous expansions for a class of biholomorphic mappings defined on a Reinhardt domain in ℂn
  21. Graded weakly 1-absorbing primary ideals
  22. The existence and uniqueness of solutions to a functional equation arising in psychological learning theory
  23. Some aspects of the n-ary orthogonal and b(αn,βn)-best approximations of b(αn,βn)-hypermetric spaces over Banach algebras
  24. Numerical solution of a malignant invasion model using some finite difference methods
  25. Increasing property and logarithmic convexity of functions involving Dirichlet lambda function
  26. Feature fusion-based text information mining method for natural scenes
  27. Global optimum solutions for a system of (k, ψ)-Hilfer fractional differential equations: Best proximity point approach
  28. The study of solutions for several systems of PDDEs with two complex variables
  29. Regularity criteria via horizontal component of velocity for the Boussinesq equations in anisotropic Lorentz spaces
  30. Generalized Stević-Sharma operators from the minimal Möbius invariant space into Bloch-type spaces
  31. On initial value problem for elliptic equation on the plane under Caputo derivative
  32. A dimension expanded preconditioning technique for block two-by-two linear equations
  33. Asymptotic behavior of Fréchet functional equation and some characterizations of inner product spaces
  34. Small perturbations of critical nonlocal equations with variable exponents
  35. Dynamical property of hyperspace on uniform space
  36. Some notes on graded weakly 1-absorbing primary ideals
  37. On the problem of detecting source points acting on a fluid
  38. Integral transforms involving a generalized k-Bessel function
  39. Ruled real hypersurfaces in the complex hyperbolic quadric
  40. On the monotonic properties and oscillatory behavior of solutions of neutral differential equations
  41. Approximate multi-variable bi-Jensen-type mappings
  42. Mixed-type SP-iteration for asymptotically nonexpansive mappings in hyperbolic spaces
  43. On the equation fn + (f″)m ≡ 1
  44. Results on the modified degenerate Laplace-type integral associated with applications involving fractional kinetic equations
  45. Characterizations of entire solutions for the system of Fermat-type binomial and trinomial shift equations in ℂn#
  46. Commentary
  47. On I. Meghea and C. S. Stamin review article “Remarks on some variants of minimal point theorem and Ekeland variational principle with applications,” Demonstratio Mathematica 2022; 55: 354–379
  48. Special Issue on Fixed Point Theory and Applications to Various Differential/Integral Equations - Part II
  49. On Cauchy problem for pseudo-parabolic equation with Caputo-Fabrizio operator
  50. Fixed-point results for convex orbital operators
  51. Asymptotic stability of equilibria for difference equations via fixed points of enriched Prešić operators
  52. Asymptotic behavior of resolvents of equilibrium problems on complete geodesic spaces
  53. A system of additive functional equations in complex Banach algebras
  54. New inertial forward–backward algorithm for convex minimization with applications
  55. Uniqueness of solutions for a ψ-Hilfer fractional integral boundary value problem with the p-Laplacian operator
  56. Analysis of Cauchy problem with fractal-fractional differential operators
  57. Common best proximity points for a pair of mappings with certain dominating property
  58. Investigation of hybrid fractional q-integro-difference equations supplemented with nonlocal q-integral boundary conditions
  59. The structure of fuzzy fractals generated by an orbital fuzzy iterated function system
  60. On the structure of self-affine Jordan arcs in ℝ2
  61. Solvability for a system of Hadamard-type hybrid fractional differential inclusions
  62. Three solutions for discrete anisotropic Kirchhoff-type problems
  63. On split generalized equilibrium problem with multiple output sets and common fixed points problem
  64. Special Issue on Computational and Numerical Methods for Special Functions - Part II
  65. Sandwich-type results regarding Riemann-Liouville fractional integral of q-hypergeometric function
  66. Certain aspects of Nörlund ℐ-statistical convergence of sequences in neutrosophic normed spaces
  67. On completeness of weak eigenfunctions for multi-interval Sturm-Liouville equations with boundary-interface conditions
  68. Some identities on generalized harmonic numbers and generalized harmonic functions
  69. Study of degenerate derangement polynomials by λ-umbral calculus
  70. Normal ordering associated with λ-Stirling numbers in λ-shift algebra
  71. Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operators
  72. Compositions of positive integers with 2s and 3s
  73. Kinematic-geometry of a line trajectory and the invariants of the axodes
  74. Hahn Laplace transform and its applications
  75. Discrete complementary exponential and sine integral functions
  76. Special Issue on Recent Methods in Approximation Theory - Part II
  77. On the order of approximation by modified summation-integral-type operators based on two parameters
  78. Bernstein-type operators on elliptic domain and their interpolation properties
  79. A class of strongly convergent subgradient extragradient methods for solving quasimonotone variational inequalities
  80. Special Issue on Recent Advances in Fractional Calculus and Nonlinear Fractional Evaluation Equations - Part II
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  82. On some conformable boundary value problems in the setting of a new generalized conformable fractional derivative
  83. A certain class of fractional difference equations with damping: Oscillatory properties
  84. Weighted Hermite-Hadamard inequalities for r-times differentiable preinvex functions for k-fractional integrals
  85. Special Issue on Recent Advances for Computational and Mathematical Methods in Scientific Problems - Part II
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  87. Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves
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  91. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part I
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  93. Pell-Lucas polynomials for numerical treatment of the nonlinear fractional-order Duffing equation
  94. Impacts of Brownian motion and fractional derivative on the solutions of the stochastic fractional Davey-Stewartson equations
  95. Some results on fractional Hahn difference boundary value problems
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  97. Special Issue on Development of Fuzzy Sets and Their Extensions - Part I
  98. The cross-border e-commerce platform selection based on the probabilistic dual hesitant fuzzy generalized dice similarity measures
  99. Comparison of fuzzy and crisp decision matrices: An evaluation on PROBID and sPROBID multi-criteria decision-making methods
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