Explicit Chebyshev Petrov–Galerkin scheme for time-fractional fourth-order uniform Euler–Bernoulli pinned–pinned beam equation

Mohamed Moustafa; Youssri Hassan Youssri; Ahmed Gamal Atta

doi:10.1515/nleng-2022-0308

Article Open Access

Explicit Chebyshev Petrov–Galerkin scheme for time-fractional fourth-order uniform Euler–Bernoulli pinned–pinned beam equation

Mohamed Moustafa , Youssri Hassan Youssri and Ahmed Gamal Atta

Published/Copyright: August 17, 2023

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From the journal Nonlinear Engineering Volume 12 Issue 1

Abstract

In this research, a compact combination of Chebyshev polynomials is created and used as a spatial basis for the time fractional fourth-order Euler–Bernoulli pinned–pinned beam. The method is based on applying the Petrov–Galerkin procedure to discretize the differential problem into a system of linear algebraic equations with unknown expansion coefficients. Using the efficient Gaussian elimination procedure, we solve the obtained system of equations with matrices of a particular pattern. The L ∞ and L 2 norms estimate the error bound. Three numerical examples were exhibited to verify the theoretical analysis and efficiency of the newly developed algorithm.

Keywords: time-fractional fourth-order partial differential equations; Chebyshev polynomials; Petrov–Galerkin method; error analysis

MSC 2010: 65M70; 42C05; 65G99; 35R11

1 Introduction

Research in many different fields has focused heavily on fractional differential equations. This fact demonstrates how fractional derivatives can describe various phenomena in several areas, including turbulence, wave propagation, signal processing, porous media, and anomalous diffusion [1,2]. Additional applications of fractional and ordinary differential problems can be found in refs [3–9]. To generate the fractional diffusion equation, a time fractional derivative term of order s in interval (0, 1) can be used to replace the time derivative term in the classical diffusion equation. Much work has been done on the time fractional fourth-order sub-diffusion equation, including theoretical study and numerical computation. As we all know, exact solutions are never handy for real-world use, and most fractional differential equations cannot find the exact solutions [10].

Transversely vibrating beams with external forcing functions are described by the beam models. They have received extensive literary study [11]. A flexible straight beam’s undamped transverse vibrations with no support contribution are specifically taken into account by the Euler–Bernoulli beam model [12]. Although it is one of the most straightforward models, it offers reasonable solutions to many engineering challenges. Numerous algorithms have been developed to solve the Euler–Bernoulli problem, such as finite difference methods [13], alternating direction implicit methods [14], alternating group explicit iterative methods [15], Adomian decomposition methods [16], and Legendre–Laguerre Galerkin method [17].

As far as we know, the spectral method is the most potent numerical/semi-analytic method for partial differential equations due to its high-order accuracy whenever it works appropriately. Standard spectral/weighted residual methods have been extensively investigated in solving stochastic partial differential equations [18], fractional Lane–Emden equation [19], linear hyperbolic first-order partial differential equations [20], one- and two-dimensional heat equations [21], second-order diffusion equation [22], and other recent spectral methods for different types of partial differential equations [23–28].

Finding approximate solutions using spectral methods requires selecting appropriate basis functions. Among these functions, which have proven their efficiency in finding these solutions, are the Chebyshev polynomials of the first kind because of their valuable properties and trigonometric representation. An interested reader of Chebyshev polynomials can refer to [29]. Chebyshev polynomials have recently been used to handle many differential problems via spectral methods, see, for instance, [30,31].

The main objective of this work is to find an explicit Chebyshev–Petrov–Galerkin solution for a time-fractional fourth-order uniform Euler–Bernoulli pinned–pinned beam equation. The article is outlined as follows: Section 2 is devoted to the essential properties of fractional calculus and Chebyshev polynomials; Section 3 is the main section of the spectral scheme for handling the underlying fractional partial differential equation; Section 4 is devoted to finding a truncation error estimate; in Section 5, we study some numerical examples with comparisons; and some concluding remarks are reported in Section 6.

2 Preliminaries and essential relations

In this section, the essential definition of the Caputo fractional derivative operator and some relevant properties of the shifted first-kind Chebyshev polynomials (S1KCPs) are reported, which will subsequently be of important use.

2.1 The fractional derivative in the Caputo sense

Definition 1

[1] The Caputo fractional derivative of order s is defined as follows:

(1) D t s χ ( t ) = 1 Γ ( m − s ) ∫ 0 t ( t − z ) m − s − 1 χ ( m ) ( z ) d z , s > 0 , t > 0 ,

where m − 1 ⩽ s < m , m ∈ N .

The following properties are satisfied by the operator D t s for m − 1 ⩽ s < m , m ∈ N ,

(2) D t s b = 0 , ( b is a constant )

(3) D t s t m = 0 , if m ∈ N 0 and m < ⌈ s ⌉ , Γ ( m + 1 ) Γ ( m − s + 1 ) t m − s , if m ∈ N 0 and m ≥ ⌈ s ⌉ ,

where N = { 1 , 2 , 3 , … } , N 0 = { 0 } ∪ N , and the notation ⌈ s ⌉ denotes the ceiling function.

2.2 An account on the S1KCPs

The S1KCPs on the interval [ 0 , 1 ] are a sequence of polynomials { T m ∗ ( x ) = T m ( 2 x − 1 ) : m = 0 , 1 , 2 , … } , which can be defined as follows [31,32]:

(4) T m ∗ ( x ) = m ∑ k = 0 m ( − 1 ) m − k 2 2 k ( m + k − 1 ) ! ( m − k ) ! ( 2 k ) ! x k , m > 0 ,

and satisfy the following orthogonality relation with respect to the weight function w ˆ ( x ) = 1 x ( 1 − x ) [31,32]:

(5) ∫ 0 1 w ˆ ( x ) T m ∗ ( x ) T n ∗ ( x ) d x = h m , n ,

where

(6) h m , n = π , if m = n = 0 , π 2 , if m = n > 0 , 0 , if m ≠ n .

Moreover, the inversion formula is defined as follows [31,32]:

(7) x j = 2 1 − 2 j ( 2 j ) ! ∑ p = 0 j ε p ( j − p ) ! ( j + p ) ! T p ∗ ( x ) , j ≥ 0 ,

where

(8) ε m = 1 2 , if m = 0 , 1 , if m > 0 .

3 Petrov–Galerkin approach for time-fractional fourth-order partial differential equations (TF4PDEs)

In this section, we consider the following TF4PDE [33]:

(9) D t α u ( x , t ) + β ∂ 4 u ( x , t ) ∂ x 4 = f ( x , t ) , 0 < α ≤ 1 ,

subject to the following initial condition:

(10) u ( x , 0 ) = g ( x ) , 0 < x ≤ 1 ,

and boundary conditions:

(11) u ( 0 , t ) = u ( 1 , t ) = u x x ( 0 , t ) = u x x ( 1 , t ) = 0 , 0 < t ≤ 1 ,

where β is arbitrary positive constant and f ( x , t ) is the source term.

3.1 Trial functions

Consider the following basis functions:

(12) ζ i ∗ ( x ) = T i + 4 ∗ ( x ) − 2 ( i + 2 ) ( 2 i 2 + 8 i + 15 ) ( i + 1 ) ( 2 i 2 + 4 i + 3 ) T i + 2 ∗ ( x ) + ( i + 3 ) ( 2 i 2 + 12 i + 19 ) ( i + 1 ) ( 2 i 2 + 4 i + 3 ) T i ∗ ( x ) ,

along with T i ∗ ( x ) defined in Eq. (4).

Corollary 1

[34] For every positive integer q, the qth derivative of T j ∗ ( x ) can be expressed in terms of their original polynomials as follows:

(13) D q T j ∗ ( x ) = ∑ p = 0 ( j + p + q ) even j − q λ j , p , q T p ∗ ( x ) ,

where

(14) λ j , p , q = j 2 2 q ε p ( q ) 1 2 ( j − p − q ) 1 2 ( j − p − q ) ! 1 2 ( j + p + q ) 1 − q ,

and ε p is defined in Eq. (8).

3.2 Petrov–Galerkin solution for TF4PDE

Now, one may set

(15) Λ M = span { ζ i ∗ ( x ) T j ∗ ( t ) : i , j = 0 , 1 , … , M } , Δ M = { u ∈ Λ M : u ( 0 , t ) = u ( 1 , t ) = u x x ( 0 , t ) = u x x ( 1 , t ) = 0 } ,

then any function u M ( x , t ) ∈ Δ M may be written as follows:

(16) u M ( x , t ) = ∑ i = 0 M ∑ j = 0 M c i j ζ i ∗ ( x ) T j ∗ ( t ) = ζ C T ∗ T ,

(17) ζ = [ ζ 0 ∗ ( x ) , ζ 1 ∗ ( x ) , … , ζ M ∗ ( x ) ] , T ∗ = [ T 0 ∗ ( t ) , T 1 ∗ ( t ) , … , T M ∗ ( t ) ] ,

and C ˆ = ( c i j ) 0 ≤ j ≤ M is the matrix of unknowns and has the order ( M + 1 ) 2 .

The residual R ( x , t ) of Eq. (9) can be written as follows:

(18) R ( x , t ) = D t α u M ( x , t ) + β ∂ 4 u M ( x , t ) ∂ x 4 − f ( x , t ) .

One of the most important advantages of using Petrov–Galerkin method is its flexibility in choosing test functions, so let us choose the following test functions:

(19) ψ r , s ( x , t ) = T r ∗ ( x ) T s ∗ ( t ) .

Hence, the application of Petrov–Galerkin method is used to find u M ( x , t ) ∈ Δ M such that

(20) ( R ( x , t ) , ψ r , s ( x , t ) ) ω ( x , t ) = 0 , 0 ≤ r ≤ M , 0 ≤ s ≤ M − 1 ,

where ω ( x , t ) = w ˆ ( x ) w ˆ ( t ) .

Suppose that

F ˆ = ( f r s ) ( M + 1 ) × M , f r s = ( f ( x , t ) , T r ∗ ( x ) T s ∗ ( t ) ) ω ( x , t ) , A ˆ = ( a i r ) ( M + 1 ) × ( M + 1 ) , a i r = ( ζ i ∗ ( x ) , T r ∗ ( x ) ) w ˆ ( x ) , B ˆ = ( b s j ) M × ( M + 1 ) , b s j = ( T s ∗ ( t ) , D t α T j ∗ ( t ) ) w ˆ ( t ) , H ˆ = ( h ¯ i r ) ( M + 1 ) × ( M + 1 ) , h ¯ i r = d 4 ζ i ∗ ( x ) d x 4 , T r ∗ ( x ) w ˆ ( x ) , K ˆ = ( k s j ) M × ( M + 1 ) , k s j = ( T s ∗ ( t ) , T j ∗ ( t ) ) w ˆ ( t ) ,

where

(21) ( u ( y , t ) , v ( y , t ) ) ω ( y , t ) = ∫ 0 1 ∫ 0 1 u ( y , t ) v ( y , t ) ω ( y , t ) d y d t , ( u ( y ) , v ( y ) ) w ˆ ( y ) = ∫ 0 1 u ( y ) v ( y ) w ˆ ( y ) d y .

Then, Eq. (20) can be rewritten alternatively in matrix form as follows:

(22) A ˆ C ˆ B ˆ T − β H ˆ C ˆ K ˆ T + F ˆ .

Moreover, the initial condition (10) implies that

(23) ∑ i = 0 M ∑ j = 0 M c i j a i r ( − 1 ) j = g ¯ i r , r : 0 , … , M ,

where g ¯ r = ( g ( x ) , T r ∗ ( x ) ) w ˆ ( x ) .

Now, Eqs (22) and (23) constitute a system of algebraic equations of order ( M + 1 ) 2 that may be solved using Gauss elimination procedure.

Theorem 1

The elements of matrices K ˆ , A ˆ , H ˆ , and B ˆ in system (22) are given as follows:

(24) k s j = h s , j , a i r = h i + 4 , r − 2 ( i + 2 ) ( 2 i 2 + 8 i + 15 ) ( i + 1 ) ( 2 i 2 + 4 i + 3 ) h i + 2 , r + ( i + 3 ) ( 2 i 2 + 12 i + 19 ) ( i + 1 ) ( 2 i 2 + 4 i + 3 ) h i , r , h ¯ i r = 128 π ( i + 1 ) ( i + 2 ) ( i + 3 ) ( i + 4 ) , i f i − r = 0 , 256 π i ( i + 2 ) ( 2 i 4 + 18 i 3 + 43 i 2 + 63 i + 54 ) 2 i 2 + 4 i + 3 , i f i − r = 2 , 384 π ( i − 1 ) i ( 2 i 5 + 24 i 4 + 81 i 3 + 218 i 2 + 489 i + 474 ) ( i + 1 ) ( 2 i 2 + 4 i + 3 ) , i f i − r = 4 , 0 , o t h e r w i s e , b s j = ( − 1 ) j + 1 j 2 π Γ 3 2 − α F 3 4 1 , 1 − j , j + 1 , 3 2 − α 3 2 , − s − α + 2 , s − α + 2 1 .

Proof

The elements of matrices K ˆ and A ˆ can be easily obtained after using the orthogonality relation of T i ∗ ( x ) defined in Eq. (5) and the definition of ζ i ( x ) .

To find the elements of matrix H ˆ .

The elements h ¯ i r can be written as follows:

(25) h ¯ i r = d 4 ζ i ∗ ( x ) d x 4 , T r ∗ ( x ) w ˆ ( x )

with the aid of Corollary 1 after putting q = 4 , the last equation becomes

(26) h ¯ i r = ∑ p = 0 ( i + p + 8 ) even i λ i + 4 , p , 4 T p ∗ ( x ) − 2 ( i + 2 ) ( 2 i 2 + 8 i + 15 ) ( i + 1 ) ( 2 i 2 + 4 i + 3 ) ∑ p = 0 ( i + p + 6 ) even i − 2 λ i + 2 , p , 4 T p ∗ ( x ) + ( i + 3 ) ( 2 i 2 + 12 i + 19 ) ( i + 1 ) ( 2 i 2 + 4 i + 3 ) ∑ p = 0 ( i + p + 4 ) even i − 4 λ i , p , 4 T p ∗ ( x ) , T r ∗ ( x ) w ˆ ( x ) .

Now, if we simplify the right side of the previous relation using the orthogonality relation of T i ∗ ( x ) , we obtain the following result:

(27) h ¯ i r = 128 π ( i + 1 ) ( i + 2 ) ( i + 3 ) ( i + 4 ) , if i − r = 0 , 256 π i ( i + 2 ) ( 2 i 4 + 18 i 3 + 43 i 2 + 63 i + 54 ) 2 i 2 + 4 i + 3 , if i − r = 2 , 384 π ( i − 1 ) i ( 2 i 5 + 24 i 4 + 81 i 3 + 218 i 2 + 489 i + 474 ) ( i + 1 ) ( 2 i 2 + 4 i + 3 ) , if i − r = 4 , 0 , otherwise .

At the end, to find the elements of matrix B ˆ .

The elements b s j can be written as follows:

(28) b s j = ( T s ∗ ( t ) , D t α T j ∗ ( t ) ) w ˆ ( t ) ,

which can be written after using the fractional derivative (3) and Eq. (4) as follows:

(29) b s j = ∫ 0 1 [ D t α T j ∗ ( t ) ] T s ∗ ( t ) w ˆ ( t ) d t = j s ∑ k = 0 j ∑ n = 0 s ( − 1 ) j − k 2 2 k ( j + k − 1 ) ! Γ ( k + 1 ) ( − 1 ) s − n 2 2 n ( s + n − 1 ) ! ( j − k ) ! ( 2 k ) ! Γ ( k − α + 1 ) ( s − n ) ! ( 2 n ) ! ∫ 0 1 t k + n − α t ( 1 − t ) d t .

Integrating and simplifying the right-hand side of the last equation lead to the following result:

(30) b s j = ( − 1 ) j + 1 j 2 π Γ 3 2 − α F 3 4 1 , 1 − j , j + 1 , 3 2 − α 3 2 , − s − α + 2 , s − α + 2 1 .

Remark 1

Algorithm 1 shows all the steps required to obtain the numerical solution of Eq. (9) governed by the conditions (10) and (11).

Algorithm 1 Coding algorithm for the proposed technique
Input α , β , g ( x ) and f ( x , t ) .
Step 1. Assume an approximate solution u M ( x , t ) as in (16).
Step 2. Apply Petrov–Galerkin method to obtain the system in (22) and (23).
Step 3. Use Theorem 1 to get the elements of matrices K ˆ , A ˆ , H ˆ , and B ˆ in system (22).
Step 4. Use NDsolve command to solve the system in (22) and (23) to get c i j .
Output u M ( x , t )

4 Error analysis

In this section, we study the error analysis of the numerical solution u M ( x , t ) to the exact solution u ( x , t ) of Eq. (9), according to the following two cases:

Error analysis in L ∞ -norm.
Error analysis in L 2 -norm.

4.1 Error analysis in L ∞ -norm

Let u M ( x , t ) ∈ Δ M be the best approximation of u ( x , t ) ; then, the definition of the best approximation enables us to write the following inequality:

(31) ‖ u ( x , t ) − u M ( x , t ) ‖ ∞ ≤ ‖ u ( x , t ) − u ˆ M ( x , t ) ‖ ∞ , ∀ u ˆ M ( x , t ) ∈ Δ M .

Moreover, the previous inequality is also true if u ˆ M denotes the interpolating polynomial for u ( x , t ) at points ( x i , t j ) , where x i are the roots of ζ i ( x ) and t j are the roots of T j ∗ ( t ) .

Now, if we take similar steps as in refs [35,36], we obtain

(32) u ( x , t ) − u ˆ M ( x , t ) = ∂ M + 1 u ( η , t ) ∂ x M + 1 ( M + 1 ) ! ∏ i = 0 M ( x − x i ) + ∂ M + 1 u ( x , μ ) ∂ t M + 1 ( M + 1 ) ! ∏ j = 0 M ( t − t j ) − ∂ 2 M + 2 u ( η ˆ , μ ˆ ) ∂ x M + 1 ∂ t M + 1 ( ( M + 1 ) ! ) 2 ∏ i = 0 M ( x − x i ) ∏ j = 0 M ( t − t j ) .

where η , η ˆ , μ , μ ˆ ∈ [ 0 , 1 ] . Now,

(33) ‖ u ( x , t ) − u ˆ M ( x , t ) ‖ ∞ ≤ max ( x , t ) ∈ Ω ∂ M + 1 u ( η , t ) ∂ x M + 1 ∥ ∏ i = 0 M ( x − x i ) ∥ ∞ ( M + 1 ) ! + max ( x , t ) ∈ Ω ∂ M + 1 u ( x , μ ) ∂ t M + 1 ∥ ∏ j = 0 M ( t − t j ) ∥ ∞ ( M + 1 ) ! − max ( x , t ) ∈ Ω ∂ 2 M + 2 u ( η ˆ , μ ˆ ) ∂ x M + 1 ∂ t M + 1 × ∥ ∏ i = 0 M ( x − x i ) ∥ ∞ ∥ ∏ j = 0 M ( t − t j ) ∥ ∞ ( ( M + 1 ) ! ) 2 .

Since u is a smooth function on Ω = [ 0 , 1 ] 2 , there exist three constants g 1 , g 2 and g 3 , such that

(34) max ( x , t ) ∈ Ω ∂ M + 1 u ( x , t ) ∂ x M + 1 ≤ g 1 , max ( x , t ) ∈ Ω ∂ M + 1 u ( x , μ ) ∂ t M + 1 ≤ g 2 , max ( x , t ) ∈ Ω ∂ 2 M + 2 u ( η ˆ , μ ˆ ) ∂ x M + 1 ∂ t M + 1 ≤ g 3 .

To minimize the factor ∥ ∏ i = 0 M ( x − x i ) ∥ ∞ , let us use the one-to-one mapping x = 1 2 ( z + 1 ) between the intervals [ − 1 , 1 ] and [ 0 , 1 ] to deduce that

(35) min x i ∈ [ 0 , 1 ] max x ∈ [ 0 , 1 ] ∏ i = 0 M ( x − x i ) = min z i ∈ [ − 1 , 1 ] max z ∈ [ − 1 , 1 ] ∏ i = 0 M 1 2 ( z − z i ) = 1 2 M + 1 min z i ∈ [ − 1 , 1 ] max z ∈ [ − 1 , 1 ] ∏ i = 0 M ( z − z i ) = 1 2 M + 1 min z i ∈ [ − 1 , 1 ] max z ∈ [ − 1 , 1 ] ζ M − 3 ( z ) ζ M ¯ ,

where ζ M ¯ = 2 M is the leading coefficient of ζ M − 3 ( z ) = T M + 1 ( z ) − 2 ( M − 1 ) M − 2 T M − 1 ( z ) + M M − 2 T M − 3 ( z ) and z i are the roots of ζ M − 3 ( z ) .

Similarly, the factor ∥ ∏ j = 0 M ( t − t j ) ∥ ∞ , can be minimized by using the one-to-one mapping t = 1 2 ( t ¯ + 1 ) between the intervals [ − 1 , 1 ] and [ 0 , 1 ] to deduce that

(36) min t j ∈ [ 0 , 1 ] max t ∈ [ 0 , 1 ] ∏ j = 0 M ( t − t j ) = 1 2 M + 1 min t ¯ j ∈ [ − 1 , 1 ] max t ¯ ∈ [ − 1 , 1 ] T M + 1 ( t ¯ ) T ˆ M ,

where T ˆ M = 2 M is the leading coefficient of T M + 1 ( t ¯ ) and t ¯ j are the roots of T M + 1 ( t ¯ ) .

It is known that

(37) ϒ M = max z ∈ [ − 1 , 1 ] ∣ ζ M − 3 ( z ) ∣ = 4 ( M − 1 ) M − 2 ,

and

(38) max t ¯ ∈ [ − 1 , 1 ] ∣ T M + 1 ( t ¯ ) ∣ = ∣ T M + 1 ( 1 ) ∣ = 1 .

Hence, inequality (34), along with Eqs (35)–(38), helps us obtain the following desired result:

(39) ‖ u ( x , t ) − u M ( x , t ) ‖ ∞ ≤ g 1 ( 1 2 ) M + 1 ϒ M ζ M ¯ ( M + 1 ) ! + g 2 ( 1 2 ) M + 1 T ˆ M ( M + 1 ) ! + g 3 ( 1 4 ) M + 1 ϒ M ζ M ¯ T ˆ M ( ( M + 1 ) ! ) 2 ,

which represents an upper bound of the absolute error (AE).

4.2 Error analysis in L 2 -norm

To study the error analysis in L 2 -norm, we follow the study by Sadri and Aminikhah [37].

Theorem 2

Assume that ∂ i + j u ( x , t ) ∂ x i ∂ t j ∈ C ( Ω ) , i , j = 0 , 1 , 2 , … , M + 1 , and u M ( x , t ) is the approximate solution obtained from the method belonging to Δ M and

(40) ℒ M = sup ( x , t ) ∈ Ω ∂ 2 ( M + 1 ) u ( x , t ) ∂ x M + 1 ∂ t M + 1 ,

where Ω = ( 0 , 1 ) × ( 0 , 1 ) . Then, the following estimation holds:

(41) ‖ u ( x , t ) − u M ( x , t ) ‖ 2 ≲ ℒ M M 3 4 ( Γ ( M + 2 ) ) 2 ,

where q ˆ ≲ q ¯ means that there exists a generic constant n such that q ˆ ≤ n q ¯ .

Proof

Assume that

(42) v M ( x , t ) = ∑ i = 0 M ∑ j = 0 M − i ∂ i + j u ( x , t ) ∂ x i ∂ t j ( 0 , 0 ) x i t j i ! j !

is the Taylor expansion of u ( x , t ) about the point ( 0 , 0 ) , and

(43) u ( x , t ) − v M ( x , t ) = x M + 1 t M + 1 ∂ 2 ( M + 1 ) u ( n ¯ , n ˆ ) ( Γ ( M + 2 ) ) 2 ∂ x M + 1 ∂ t M + 1 , ( n ¯ , n ˆ ) ∈ Ω .

Since u M ( x , t ) is the best approximate solution of u ( x , t ) , then according to the concept of the best approximation, we obtain

(44) ‖ u ( x , t ) − u M ( x , t ) ‖ 2 2 ≤ ‖ u ( x , t ) − v M ( x , t ) ‖ 2 2 = ∫ 0 1 ∫ 0 1 ℒ M 2 x 2 ( M + 1 ) t 2 ( M + 1 ) ( Γ ( M + 2 ) ) 4 t − 1 2 ( 1 − t ) − 1 2 d x d t = ℒ M 2 π ( Γ ( M + 2 ) ) 4 Γ 2 M + 5 2 Γ ( 2 M + 4 ) .

According to the following inequality [38],

(45) Γ ( z + a ) Γ ( z + b ) ≤ o z a , b z a − b ,

where z ≥ 1 , z + a > 1 , and z + b > 1 , a and b , are any constants and

(46) o z a , b = exp a − b 2 ( z + b − 1 ) + 1 12 ( z + a − 1 ) + ( a − b ) 2 z = 1 + O ( z − 1 ) .

The following estimation may be obtained

(47) ‖ u ( x , t ) − u M ( x , t ) ‖ 2 ≲ ℒ M M 3 4 ( Γ ( M + 2 ) ) 2 .

This completes the proof of this theorem.

Theorem 3

Suppose that u ( x , t ) , u M ( x , t ) , and ∂ i + j u ( x , t ) ∂ x i ∂ t j satisfy the condition of Theorem 2 and

(48) T M , m = sup ( x , t ) ∈ Ω ∂ 2 M − m + 2 u ( x , t ) ∂ x M − m + 1 ∂ t M + 1 , m = 1 , 2 , 3 , 4 .

Then, the following estimation holds:

(49) ∂ m ( u ( x , t ) − u M ( x , t ) ) ∂ x m 2 ≲ T M , m ( 2 ( M − m ) + 3 ) 1 2 M 1 4 Γ ( M + 2 ) Γ ( M − m + 2 ) .

Proof

Assume that ∂ m v M ( x , t ) ∂ x m is the Taylor expansion of ∂ m u ( x , t ) ∂ x m about the point ( 0 , 0 ) , then the residual between ∂ m u ( x , t ) ∂ x m and ∂ m v M ( x , t ) ∂ x m can be written as follows:

(50) ∂ m ( u ( x , t ) − v M ( x , t ) ) ∂ x m = x M − m + 1 t M + 1 ∂ 2 M − m + 2 u ( n 1 ¯ , n 2 ˆ ) Γ ( M + 2 ) Γ ( M − m + 2 ) ∂ x M − m + 1 ∂ t M + 1 , ( n 1 ¯ , n 2 ˆ ) ∈ Ω .

Since ∂ m u M ( x , t ) ∂ x m is the best approximate solution of ∂ m u ( x , t ) ∂ x m , then according to the definition of the best approximation, we obtain

(51) ∂ m ( u ( x , t ) − u M ( x , t ) ) ∂ x m 2 ≤ ∂ m ( u ( x , t ) − v M ( x , t ) ) ∂ x m 2 .

Now, by following the same steps as in Theorem 2, we obtain the desired outcome.

Theorem 4

Suppose that D t α u ( x , t ) ∈ C ( Ω ) satisfy the conditions of Theorem 2 and set

(52) P M , α = sup ( x , t ) ∈ Ω ∂ 2 M − α + 2 u ( x , t ) ∂ x M + 1 ∂ t M − α + 1 , 0 < α ≤ 1 .

Then, the following estimation holds:

(53) ‖ D t α ( u ( x , t ) − u M ( x , t ) ) ‖ 2 ≲ P M , α ( 2 M + 3 ) 1 2 ( M − α ) 1 4 Γ ( M + 2 ) Γ ( M − α + 2 ) .

Proof

This theorem’s proof is simple to obtain after utilizing the Caputo operator’s characteristics (3) and replicating identical procedures from Theorems 2 and 3.

Theorem 5

‖ R ( x , t ) ‖ 2 will be small enough for the big enough values of M .

Proof

Eqs (9) and (18) enable us to write R ( x , t ) as

(54) R ( x , t ) = D t α u M ( x , t ) + β ∂ 4 u M ( x , t ) ∂ x 4 − f ( x , t ) = D t α ( u M ( x , t ) − u ( x , t ) ) + β ∂ 4 ( u M ( x , t ) − u ( x , t ) ) ∂ x 4 .

Taking L 2 -norm and using Theorems 3 and 4, we obtain

(55) ‖ R ( x , t ) ‖ 2 ≲ P M , α ( 2 M + 3 ) 1 2 ( M − α ) 1 4 Γ ( M + 2 ) Γ ( M − α + 2 ) + β T M , 4 ( 2 M − 5 ) 1 2 M 1 4 Γ ( M + 2 ) Γ ( M − 2 ) .

Finally, it is evident from Eq. (55) that ‖ R ( x , t ) ‖ 2 will be small enough for the big enough values of M . This completes the theorem’s proof.

5 Illustrative examples

Test Problem 1

[33,39,40] Consider the TF4PDE of the form

(56) D t α u ( x , t ) + β ∂ 4 u ( x , t ) ∂ x 4 = f ( x , t ) , 0 < α ≤ 1 ,

subject to the following initial and boundary conditions:

(57) u ( x , 0 ) = sin ( π x ) , 0 < x ≤ 1 , u ( 0 , t ) = u ( 1 , t ) = u x x ( 0 , t ) = u x x ( 1 , t ) = 0 , 0 < t ≤ 1 ,

where f ( x , t ) = 1 Γ ( 2 − α ) t 1 − α + β π 4 ( t + 1 ) sin ( π x ) and u ( x , t ) = ( t + 1 ) sin ( π x ) is the exact solution of this problem.

Table 1 shows a comparison of maximum absolute error (MAE) between our method and methods followed in refs [33,39,40] at α = 0.5 . Table 2 reports the MAE at different values of M when α = 0.5 . Figure 1 shows the AE (left) and approximate solution (right) at α = 0.4 and M = 12 . This figure proves that the approximate solution is quite near to the precise one. Table 3 presents the MAE at different values of t when α = 0.1 and M = 12 for 0 < x < 1 . This table indicates the advantage of our method for obtaining the MAE at small values of M .

Table 1

Comparison of the MAE for Test problem 1

At Δ t = 0.00001 , m = 40 , n = 500			At M = 12
Method in ref. [33]	Method in ref. [40]	Method in ref. [39]	Our method
4.6930 × 1 0 ‒ 8	1.4312 × 1 0 ‒ 4	2.3241 × 1 0 ‒ 4	1.17473 × 1 0 ‒ 11

Table 2

MAE for Test problem 1

M	2	4	6	8	10	12
Error	1.0927 × 1 0 ‒ 2	2.5938 × 1 0 ‒ 4	3.77548 × 1 0 ‒ 6	3.83646 × 1 0 ‒ 8	3.44999 × 1 0 ‒ 10	1.17473 × 1 0 ‒ 11

Figure 1

The AE (left) and approximate solution (right) for Test problem 1.

Table 3

MAE for Test problem 1

( x , t )	( x , 0.1 )	( x , 0.3 )	( x , 0.5 )	( x , 0.7 )	( x , 0.9 )
Error	7.57172 × 1 0 ‒ 13	1.22746 × 1 0 ‒ 12	1.53166 × 1 0 ‒ 12	1.79967 × 1 0 ‒ 12	2.11764 × 1 0 ‒ 12

Test Problem 2

[33] Consider the TF4PDE of the form

(58) D t α u ( x , t ) + β ∂ 4 u ( x , t ) ∂ x 4 = f ( x , t ) , 0 < α ≤ 1 ,

subject to the following initial and boundary conditions:

(59) u ( x , 0 ) = 0 , 0 < x ≤ 1 , u ( 0 , t ) = u ( 1 , t ) = u x x ( 0 , t ) = u x x ( 1 , t ) = 0 , 0 < t ≤ 1 ,

where f ( x , t ) = 2 Γ ( 3 − α ) t 2 − α + 16 β π 4 t 2 sin ( 2 π x ) and u ( x , t ) = t 2 sin ( 2 π x ) is the exact solution of this problem.

Table 4 presents a comparison of MAE between our method and the method followed by Roul and Goura [33] at different values of α . Figure 2 illustrates the AE at different values of t when α = 0.9 and M = 12 . Figure 3 shows the AE (left) and approximate solution (right) at α = 0.1 and M = 12 . It can be seen that the approximate solution is quite near to the precise one.

Table 4

Comparison of the MAE for Test problem 2

α	Method in ref. [33] at m = 200 , n = 160	Our method at M = 14
0.4	7.3899 × 1 0 ‒ 8	3.55836 × 1 0 ‒ 9
0.5	1.4865 × 1 0 ‒ 7	6.53638 × 1 0 ‒ 9
0.7	5.7410 × 1 0 ‒ 7	4.26957 × 1 0 ‒ 9

Figure 2

The AE for Test problem 2.

Figure 3

The AE (left) and approximate solution (right) for Test problem 2.

Test Problem 3

Consider the TF4PDE of the form

(60) D t α u ( x , t ) + β ∂ 4 u ( x , t ) ∂ x 4 = f ( x , t ) , 0 < α ≤ 1 ,

subject to the following initial and boundary conditions:

(61) u ( x , 0 ) = sin ( π x ) , 0 < x ≤ 1 , u ( 0 , t ) = u ( 1 , t ) = u x x ( 0 , t ) = u x x ( 1 , t ) = 0 , 0 < t ≤ 1 ,

where f ( x , t ) is chosen such that u ( x , t ) = e α t sin ( π x ) is the exact solution of this problem.

Table 5 reports the MAE at different values of M when α = 0.9 . This table indicates the advantage of our method for obtaining the MAE at small values of M . Figure 4 shows the AE (left) and approximate solution (right) at α = 0.1 and M = 12 . Table 6 presents the MAE at different values of t when α = 0.5 and M = 12 for 0 < x < 1 . It can be seen that the approximate solution is quite near the precise one.

Table 5

MAE for Test problem 3

M	2	4	6	8	10	12
Error	6.36454 × 1 0 ‒ 2	1.00154 × 1 0 ‒ 3	5.92147 × 1 0 ‒ 6	5.00238 × 1 0 ‒ 8	3.69535 × 1 0 ‒ 10	2.0759 × 1 0 ‒ 12

Figure 4

The AE (left) and approximate solution (right) for Test problem 3.

Table 6

MAE for Test problem 3

( x , t )	( x , 0.1 )	( x , 0.3 )	( x , 0.5 )	( x , 0.7 )	( x , 0.9 )
Error	1.31561 × 1 0 ‒ 12	1.45817 × 1 0 ‒ 12	1.61648 × 1 0 ‒ 12	1.78835 × 1 0 ‒ 12	1.15707 × 1 0 ‒ 12

6 Concluding remarks

Herein, the time fractional fourth-order Euler–Bernoulli pinned–pinned beam was studied via a compact combination of Chebyshev polynomials as its spatial basis. The approach is based on discretizing the differential problem into a set of linear algebraic equations with unknown expansion coefficients using the Petrov–Galerkin algorithm. We solve the resulting system of equations with matrices that follow a specific pattern using the effective Gaussian elimination approach. The error bound is estimated. The theoretical analysis and effectiveness of the recently created algorithm are supported by numerical evidence. We plan to generalize the offered algorithm to more general differential problems in the near future. As an expected future work, we aim to employ the developed theoretical results in this article along with suitable spectral methods to treat more advanced problems, for instance, ref. [41]. All codes were written and debugged by Mathematica 11 on HP Z420 Workstation, Processor: Intel (R) Xeon(R) CPU E5-1620 – 3.6 GHz, 16 GB Ram DDR3, and 512 GB storage.

Acknowledgments

The authors would like to thank the anonymous reviewers for carefully reading the article and providing constructive and valuable comments that have improved the article’s present form.

Funding information: The authors received no financial support for the research.
Author contributions: All authors contributed equally to the work.
Conflict of interest: The authors declare that they have no conflict of interest.
Data availability statement: The authors did not use any scientific data during this research.

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Received: 2023-06-05

Revised: 2023-07-05

Accepted: 2023-07-22

Published Online: 2023-08-17

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

Articles in the same Issue

https://doi.org/10.1515/nleng-2022-0308

Keywords for this article

time-fractional fourth-order partial differential equations; Chebyshev polynomials; Petrov–Galerkin method; error analysis

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