Gegenbauer spectral tau algorithm for solving fractional telegraph equation with convergence analysis

https://doi.org/10.1007/s12043-021-02113-0

Gegenbauer spectral tau algorithm for solving fractional telegraph equation with convergence analysis

HODA F AHMED , M R A MOUBARAK and W A HASHEM ^∗ Mathematics Department, Faculty of Science, Minia University, Minia, Egypt

∗Corresponding author. E-mail: wabdelraheem@yahoo.com, waleed.hashim@mu.edu.eg MS received 23 July 2020; revised 8 January 2021; accepted 12 January 2021

Abstract. In this article, a novel shifted Gegenbauer operational matrix (SGOM) of fractional derivative in the Caputo sense is derived. Based on this operational matrix, an accurate and effective numerical algorithm is proposed.

The SGOM of fractional derivative in conjunction with the tau method are used for solving the constant and variable coefficients space–time fractional telegraph equations (FTE) with various types of boundary conditions, namely, Neumann, Dirichlet and Robin conditions. The convergence analysis of the proposed method is established in L²_ωα. Finally, miscellaneous test examples are given and compared with other methods to clarify the accuracy and efficiency of the presented algorithm.

Keywords. Telegraph equation; Caputo fractional derivative; shifted Gegenbauer tau method; operational matrices; convergence analysis.

PACS Nos 02.60.−x; 02.30.Jr; 02.30.Mv

1. Introduction

The applications of fractional calculus in physics and engineering were not related to its appearance in math- ematics. It was delayed for many years because of the widespread controversy about the multiple non- equivalent definitions of the fractional derivatives besides the absence of geometrical interpretation of the frac- tional derivatives due to their non-local properties [1,2].

However, it was proved in the last 10 years that the frac- tional calculus is a precious tool for modelling many phenomena and interdisciplinary applications. Frac- tional partial differential equations where the derivatives have more degrees of fluency are quite flexible and relevant for precisely modelling some phenomena in science and engineering more than the classical type;

for instance, anomalous diffusion processes [3], seep- age flow in porous media [4], the seismic analysis of mechanical models in the presence of viscoelastic dampers [5] and other scientific branches [6–8].

The telegraph model arose after noticing reflection of wave motion of electromagnetic type on the wire throughout the improvement of transmission line joining the voltage and current waves. The mathematical model of telegraph equation can be precisely represented by

fractional partial differential equation where the depen- dent variable denotes the voltage and the independent variables denote position and time. Special choice of the coefficients and the fractional derivative domain of the fractional telegraph equation convert it into fractional non-homogeneous Klein–Gordon equation or fractional reaction dispersion equation or fractional diffusion-wave equation with damping or fractional wave equation.

Recently, a number of numerical approaches have been applied to solve the FTEs. Jiang and Lin [9] repro- duced kernal theorem with Caputo fractional deriva- tives to find exact solution for the temporal FTE with Robin boundary condition. Zhao and Li [10] pre- sented Galerkin finite element method in the Riemann–

Liouville sense and finite difference method in the Caputo sense for solving the spatial and temporal FTE respectively. Mohebbi et al [11] proposed meshless method depending on the approximation of the solution by radial basis functions with the spectral collocation method to solve the one- and two-dimensional time FTE. Bhrawyet al[12] solved the two-sided time FTE using operational integration matrices based on shifted Chebyshev tau method with Riesz fractional derivative.

Akram et al [13] used Caputo’s fractional derivative

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combined with the extended cubic B-splines method as a basis for a finite difference scheme to solve time FTE.

Spectral methods such as Galerkin, collocation and tau methods are a class of the most elegant tools because of their ability to reach accurate numerical solutions with fewer degrees of freedom to the classical and frac- tional differential equations. We must know that the utilised basis functions are the main factor in obtain- ing the precise approximation characteristics of spectral methods. Using orthogonal polynomials as basis func- tions gives preferable results, such as, trigonometric functions for approximating the periodic problems, Gegenbauer, Jacobi, Chebyshev and Legendre functions for approximating non-periodic problems. The spectral numerical procedure first expresses approximate solu- tion as an expansion of certain orthogonal polynomials, then the problem is transformed into an algebraic system of equations. The coefficients are selected in such a way to make the difference between the exact and approxi- mate solutions as accurate as possible. In recent years, spectral methods, together with the operational matri- ces of fractional derivatives and integrals, were applied for finding numerical solutions of fractional differential equations of various kinds; for instance, see [14–17].

The present work focusses on the Gegenbauer polyno- mials (GPs) due to their fundamental properties. They are eigenfunctions of some differential operators. They achieve rapid rates of convergence for small range of spectral expansion terms. GPs are successfully applied to solve different kinds of fractional differential equa- tions [18,19], fractional variational problems and opti- mal control problems [20]. In this respect, El-Kalaawy et al[21] derived fractional integral operational matrix of the fractional Gegenbauer functions, then applied it together with the Rayleigh–Ritz method for solv- ing fractional variational and optimal control problems.

Ahmed et al [22] presented a novel spectral colloca- tion method together with the SGPs to provide accurate numerical solutions for the one- and two-dimensional time- and space-fractional coupled Burgers’ equations.

The main objective of this paper is to present a new spectral algorithm depending on SGPs and operational matrix of fractional derivative in Caputo sense combined with the spectral tau method to discretise the second- order one-dimensional space–time fractional telegraph equation with space variable coefficients. Hence, a linear system of algebraic equations is generated which greatly simplifies the problem. Moreover, the convergence of the proposed method is analysed. The proposed method is an easily implementable tool for the numerical solu- tion of the variable coefficients telegraph equation. It is free from round-off errors. Also, it has an exponential convergence rate in both spatial and temporal discreti- sation according to the numerical result. It is worth

pointing out that, the spectral approximation based on Chebyshev functions and Legendre functions are special cases of the present approximation.

The structure of this article is arranged as follows:

In §2, we present short notes on some fractional cal- culus concepts, Gegenbauer orthogonal polynomials and their properties in approximate functions of one and two independent variables. In §3, the operational matrix of fractional derivative of the shifted Gegenbauer polynomials is derived. In §4, a numerical algorithm based on shifted Gegenbauer tau method for solving constant and variable coefficient one-dimensional frac- tional telegraph equation with several kinds of boundary conditions is constructed. In §5, we investigate the convergence analysis of the proposed method. In §6, numerical examples are tested and compared with other methods to demonstrate the accuracy of the proposed algorithms. Conclusion is given in §7.

2. Preliminaries and basic definitions

In this section, we mention some of the preliminary definitions and properties of the fractional integration, fractional differentiation and shifted Gegenbauer poly- nomials which are required in the following sections.

2.1 Fractional integration and differentiation DEFINITION 2.1

The integral of fractional order v > 0 according to Riemann–Liouville has the form

I^vf(x)= 1 (v)

_x

0 (x−ξ)^v−1f(ξ)dξ, v >0, x>0. (1) DEFINITION 2.2

The derivative of fractional order v > 0 according to Caputo has the form

D^vf(x)=I^m−vD^mf(x)

= 1 (m−v)

_x

0

(x−ξ)^m−v−1 d^m

dξ^m f(ξ)dξ,

m−1< v ≤m, x >0, (2)

wherem is the ceiling function ofv. The operator D^v satisfies the following equation:

D^vx^α =

⎧⎨

⎩

(α+1)

(α+1−v)x^α−v α≥v,

0 α < v. (3)

DEFINITION 2.3

The two-parameter Mittag–Leffler function is defined as

E_α,β(t)= ∞ k=0

t^k

(αk+β), α, β ∈R⁺,t ∈C (4) which is a generalisation of the one-parameter Mittag–

Leffler function(β =1)and the latter is a generalisation of the exponential function e^t.

2.2 Shifted Gegenbauer polynomials

The Gegenbauer polynomials (GPs) with parameter α > −¹₂ and degree i ∈ Z⁺ are classical orthogonal polynomials on the finite interval[−1,1]which has the formC_i^(α)(x)and satisfy the following recurrence rela- tion:

(i+1)C_i^(α)₊₁(x)

=2(i+α)xC_i^(α)(x)−(i+2α−1)C_i^(α)₋₁(x), i>0, (5) where the first two terms areC₀^(α)(x)=1 andC₁^(α)(x)= 2αx. Moreover, GPs satisfy the following symmetry and inequality relations:

C_i^(α)(x)=(−1)ⁱC_i^(α)(−x), i ≥0,

|C_i^(α)(x)| ≤C_i^(α)(1),

i≥0, α >0, |x| ≤1.

Doha [18] standardised the GPs,C_i^(α)(x)such that C_i^(α)(1)=1, i=0,1,2, . . . .

Under this standardisation, the GPs are defined as C_i^(α)(x)= i!

α+¹₂

i+α+¹₂P ^α−

1 2,α−¹₂

i (x),

i =0,1,2, . . . ,

where P_i^(α,β)(x)is the Jacobi polynomial. The orthog- onality relation of the GPs with respect to the weight functionω^α(x)=(1−x²)^α−12 is defined as

C_i^(α)(x),C^(α)_j (x) = ₁

−1

C_i^(α)(x)C^(α)_j (x)ω^α(x)dx

= h^(α)_i δi,j, where

h^(α)_i = 2^2α−1i!

α+¹₂2

(i+α)(i+2α) .

It is worth mentioning that GPs can be identically the Legendre polynomial and Chebyshev polynomial. More specifically, we have

Ti(x)=C_i⁽⁰⁾(x), Li(x)=C

1 2

i (x), Ui(x)=(i+1)C_i⁽¹⁾(x), i≥0,

where Ti(x) is the Chebyshev polynomial of the first kind of degreei, Li(x)is the Legendre polynomial of degreei andU_i(x)is the Chebyshev polynomial of the second kind of degreei.

To use GPs in the interval[0,L], a change of variable x → ^2x_L −1 has been performed on GPs,C_i^(α)(x)turn- ing it into the so-called shifted Gegenbauer polynomials (SGPs)C^(α)_L,i(x).

The SGPs have the orthogonality properties C^(α)_L,i(x),C^(α)_L,j(x) =

L 0

C^(α)_L,i(x)C^(α)_L,j(x)ω^α_L(x)dx

=h^(α)_L,iδi,j, (6) where

C^(α)_L,i(x)=C_i^(α) 2x

L −1

, ω^αL(x)=

L x−x²_α−¹₂

(7) h^(α)_L,i =

L 2

2α

h^(α)_i . (8)

The analytic form of SGPs is given by C^(α)_L,i(x)=

i k=0

k,ix^k, 0<x <L, (9) where

k,i =(−1)^i−k i! α+¹₂

(i+k+2α)

k+α+¹₂

(i+2α)(i−k)!k!L^k.

• In terms of the first (N + 1) SGPs, we can approximate the square integrable function of the one independent variableu(x), as follows:

u(x) N

i=0

a_iC^(α)_L,i(x)A^TφL,N(x), (10) where the coefficientsa_i are calculated from

a_i= 1 h^(α)_L,i

_L

0

u(x)C_L^(α)_,i(x)ω^(α)_L (x)dx, i=0,1, . . . ,N. (11) The last term of eq. (10) represents the matrix form of the approximatedu(x), with

A^T ≡[a0,a1, . . . ,aN], (12)

φL,N(x)≡

C_L^(α)_,0(x),C^(α)_L,1(x), . . . ,C^(α)_L,N(x)T

. (13)

• The function of two independent variables u(x,t) where(x,t)∈(0,L] ×(0, τ]is approximated by using double SGPs as

u_N,M(x,t) M

i=0

N j=0

b_i,jC_τ,^(α)_i(t)C^(α)_L,j(x)

φ_τ,M^T (t)BφL,N(x), (14) where B = [bi,j] with i = 0,1, . . . ,M, j = 0,1, . . . ,Nis an(M+1)×(N+1)matrix. The matrix coefficientbi,j can be determined from

bi,j = 1 h^(α)_τ,ih^(α)_L,j

_τ

0

_L

0 [u(x,t)C_τ,^(α)_i(t) C_L^(α)_,j(x)ω^(α)_τ (t)ω^(α)_L (x)dxdt],

i=0,1, . . . ,M, j =0,1, . . . ,N. (15) 3. SGOM of fractional differentiation

The main objective of this section is inferring the SGOM of fractional Caputo derivative. For this purpose, firstly, some lemmas are deduced to get the SGOM.

Lemma3.1. The Caputo fractional derivative of order v >0o f SGPs has the form

D^vC_L^(α)_,j(x)

=

⎧⎨

⎩ j

k=vk,j (k+1)

(k+1−v)x^k−v j ≥ v,

0 j<v.

Proof. Using the linearity of the fractional derivative, we get

D^vC_L^(α)_,j(x)= j k=0

k,jD^vx^k.

Now, by using eq. (3) the lemma is proved easily.

Lemma3.2. Forv >0, the following relation is satis- fied:

_L

0

D^vC_L^(α)_,i(x)C^(α)_L,j(x)ω^αL(x)dx

= i k=v

j f=0

k,if,jL^f+k−v+2α

×(k+1)

f +k−v+α+¹₂

α+¹₂ (k+1−v) (f +k−v+2α+1) .

Proof. According to the fractional derivative of the SGPs as in Lemma 3.1, with eqs (7) and (9) we can write

_L

0

D^vC_L^(α)_,i(x)C^(α)_L,j(x)ω^α_L(x)dx

= i k=v

j f=0

k,if,j (k+1) (k+1−v)

× _L

0

x^{f+k−v+α−12}(L−x)^α−12 dx

= i k=v

j f=0

k,if,jL^f+k−v+2α (k+1) (k+1−v)

× ₁

0

x L

f+k−v+α−¹₂

1− x L

_α−¹₂ d x

L .

The last integral part represents the beta function B(f +k−v+α+ ¹₂, α+ ¹₂), and using the relation between beta and gamma functions completes the proof.

The fractional differentiation of the shifted Gegen- bauer vectorφL,N(x)of order(v >0)can be expressed by

D^vφL,N(x)D^(v)_L φL,N(x), (16) whereD^(v)_L denotes the operational matrix of differen- tiation of φL,N(x) which is the main objective to be computed in this section.

Theorem 3.3. Let φL,N(x)be the shifted Gegenbauer vector and by applying the Caputo fractional derivative of order (v >0)we get

D^vφL,N(x)D^(v)_L φL,N(x),

whereD^(v)_L is the(N+1)square matrix called opera- tional matrix of fractional differentiation of the form

D^(v)_L =

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

0 0 . . . 0

... ... ... ...

0 0 . . . 0

(v,0, α, v) (v,1, α, v) . . . (v,N, α, v)

... ... ... ...

(i,0, α, v) (i,1, α, v) . . . (i,N, α, v)

... ... ... ...

(N,0, α, v) (N,1, α, v) . . . (N,N, α, v)

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠ .

Here D^(v)_L =

(i, j, α, v)i≥ v,

0 i <v,

where

(i, j, α, v)= i k=v

k,i

k!L^k (k+1−v)

× j

f=0

f,j

2(j+α)(j+2α)(f +k−v+α+¹₂)L^f−v

j!(α+¹₂)(f +k−v+2α+1) . (17)

Proof. Applying eq. (16) and the orthogonality property in eq. (6),we can write

D^(v)_L = D^vφL,N(x), φ^T_L,N(x)ϒ⁻¹

for whichD^vφL,N(x), φ^T_L,N(x)andϒ⁻¹are twoN+ 1 square matrices defined by

D^vφL,N(x), φ_L^T_,N(x)

= _L

0

D^vC^(α)_L,i(x)C_L^(α)_,j(x)ω^α_L(x)dx N

i,j=0

ϒ⁻¹= δi,j

h^(α)_L,i N

i,j=0

,

where δi,j is the Kronecker delta function. Then, by using Lemma 3.2 we get the desired result.

4. Space–time FTE

In this section, the shifted Gegenbauer operational matrix of fractional differentiation with the tau method

are utilised to solve the space–time FTE with various boundary conditions.

4.1 Operational matrices multiplication product We present the following operational matrices product for completely using the spectral tau method on the pro- posed 1D fractional telegraph equation in the subsequent subsection.

Consider the functions(x)which is smooth and con- tinuous. Therefore,s(x)can be approximated by SGPs as

sN(x)=S^TφL,N(x),

whereS^T = [s0,s1, . . . ,sN]and the elementssk;k = 0,1, . . . ,N of the matrixS^T are calculated from sk = 1

h^(α)_{L,k L}

0

s(x)C^(α)_L,k(x)ω^(α)_L (x)dx,

k =0,1, . . . ,N. (18)

Now, we are looking for a matrixWsuch that

φL,N(x)φ^T_L,N(x)S=W^TφL,N(x) (19) which can be written in the form

N k=0

s_kC_L^(α)_,k(x)C_L^(α)_,j(x)= N k=0

wk jC_L^(α)_,k(x), j =0,1, . . . ,N.

Multiplying both sides of the previous relation by C_L^(α)_,i(x)ω^α_L(x),i = 0,1, . . . ,N and integrating on the interval[0,L], we get

N k=0

s_{k L}

0

C^(α)_L,k(x)C^(α)_L,j(x)C^(α)_L,i(x)ω^α_L(x)dx

= N k=0

wi j

_L

0

C_L^(α)_,k(x)C_L^(α)_,i(x)w^α_L(x)dx.

Now, by applying the orthogonality property on the right-hand side, we get

W= 1

h^(α)_L,i N k=0

sk

_L

0

C_L^(α)_,k(x)C_L^(α)_,j(x)C^(α)_L,i(x)w^αL(x)dx ^N

i,j=0

(20)

which defines the desired(N +1)square matrixW.

4.2 The proposed discretisation methodology

In this subsection, we consider the general 1D space–

time FTE of the following form:

∂^ηu(x,t)

∂t^η +p(x)∂^vu(x,t)

∂t^v +q(x)u(x,t)

=r(x)∂^θu(x,t)

∂x^θ + f(x,t),

0<x<L,0<t≤τ (21) with initial conditions

u(x,0)=h0(x), ∂u(x,0)

∂t =h1(x), 0≤x ≤L (22) and Dirichlet boundary conditions

u(0,t)=g0(t), u(L,t)=g1(t), 0≤t ≤τ (23)

where 1 < η ≤ 2, 0 < v ≤ 1, 1 < θ ≤ 2, p(x) >0, q(x)≥0 ,r(x) >0.

The known and unknown functions f(x,t),p(x), q(x),r(x)andu(x,t)are approximated by finite terms of SGPs and written in matrix forms as follows:

uN,M(x,t)=φ_τ,^TM(t)BφL,N(x), fN,M(x,t)=φ_τ,^T_M(t)FφL,N(x), pN(x)=P^TφL,N(x),

qN(x)=Q^TφL,N(x),

rN(x)=R^TφL,N(x), (24)

whereBis the unknown(M+1)×(N+1)coefficient matrix to be determined,Fis known as(M+1)×(N+1) coefficient matrix whose elements fi,j can be deter- mined as in eq. (15),P^T,Q^T andR^T are(N+1)row coefficient matrices which can be calculated as in eq.

(11).

Now by employing eqs (16) and (24) in eq. (21), we get

φ_τ,^T_M(t)(D^(η)_τ )^TBφL,N(x)

+(P^TφL,N(x))(φ_τ,M^T (t)(D^(v)_τ )^TBφL,N(x))

+

Q^TφL,N(x)

φ_τ,M^T (t)BφL,N(x)

=

R^TφL,N(x)

φ_τ,^T_M(t)BD^(θ)_L φL,N(x) +φ_τ,^TM(t)FφL,N(x)

which can be written as φ_τ,^T_M(t)(D^(η)_τ )^TBφL,N(x)

+φ_τ,M^T (t)(D^(v)_τ )^TB(φL,N(x)φ^T_L,N(x)P) +φ_τ,^TM(t)B(φL,N(x)φL^T,N(x)Q)

=φ_τ,^TM(t)BD^(θ)_L

φL,N(x)φL^T,N(x)R +φ_τ,^T_M(t)FφL,N(x).

Applying eq. (19) yields φ_τ,^TM(t)(D^(η)_τ )^TBφL,N(x)

+φ_τ,^T_M(t)(D^(v)_τ )^TBW^T₁φL,N(x) +φ_τ,^TM(t)BW^T₂φL,N(x)

=φ_τ,^TM(t)BD^(θ)_L W₃^TφL,N(x)+φ_τ,^TM(t)FφL,N(x), where W1,W2 and W3 are calculated from eq. (20).

Then the residual has the form

Residual=φ_τ,^TM(t)((D^(η)_τ )^TB+(D^(v)_τ )^TBW^T₁ +BW₂^T −BD^(θ)_L W^T₃ −F)φL,N(x)

=φ_τ,^TM(t)HφL,N(x) (25) with

H=(D^(η)_τ )^TB+(D^(v)_τ )^TBW^T₁+BW^T₂−BD^(θ)_L W₃^T−F, (26) where His an(M+1)×(N +1) matrix whose ele- ments are linear algebraic equations of the unknown coefficients;ai j withi=0, . . . ,M, j =0, . . . ,N.

By applying eqs (14) and (16), the initial conditions;

eq. (22) takes the form φ_τ,M^T (0)BφL,N(x)=h0(x), φ_τ,^TM(0)

D⁽_τ¹⁾T

BφL,N(x)=h1(x). (27) Employing eq. (14), the Dirichlet boundary condition eq. (23), can be expressed as

φ_τ,M^T (t)BφL,N(0)=g0(t), φ_τ,M^T (t)BφL,N(L)=g1(t).

(28) 4.2.1 Robin boundary condition. If the boundary con- ditions of the FTE is defined as a linear combination of the dependent variable value and its normal derivative which have the form

μ1u(0,t)+μ2u_x(0,t)=g₀(t),

σ1u(L,t)+σ2ux(L,t)=g1(t). (29)

Then in terms of eqs (14) and (16), we get φ_τ,^T_M(t)(μ1B+μ2BD⁽_L¹⁾)φL,N(0)=g₀(t),

φ_τ,^TM(t)(σ1B+σ2BD⁽_L¹⁾)φL,N(L)=g1(t). (30) 4.2.2 Neumann boundary condition. If the boundary conditions of the FTE is defined as a value of dependent variable normal derivatives, which have the form ux(0,t)=g0(t), ux(L,t)=g1(t) (31) then in terms of eq. (16), we get

φ_τ,^T_M(t)BD⁽_L¹⁾φL,N(0)=g0(t),

φ_τ,^TM(t)BD⁽_L¹⁾φL,N(L)=g1(t). (32) The outline of the SG tau algorithm for solving the 1D-FTE represented by eqs (21)–(23) is described in the sequel.

Input: 1D-FTE; eqs (21)–(23)

(p(x),q(x),r(x),h0(x),h1(x),g0(t),g1(t),f(x,t)) Step1. Choose M,N, α, v, β,L, τ.

Step2. ComputeHi j =0;i=0, . . . ,M−2, j= 0, . . . ,N−2 by using eq. (26).

Step3. Collocate every initial condition of eq. (27) at(N−1)roots ofC^(α)_L,N−1(x).

Step4. Collocate every boundary condition of eq.

(23) at(M+1)roots ofC_τ,^(α)_M+1(t).

Step5. Join the(M+1)×(N+1)linear algebraic equations resulting from Steps 2–4 and solve forbi j;i =0, . . . ,M, j=0, . . . ,N. Output. The approximate solutionu_N,M(x,t)is com-

puted from eq. (14).

5. Convergence analysis

The aim of this section is to analyse the convergence of the shifted Gegenbauer tau method for the numer- ical solution of FTEs based on the SGPs by using the operator theory. The spectral rate of convergence for the proposed method is established in theL²_ωα-norm. Here, we shall confine ourselves to the FTE of the following form:

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

∂^ηu(x,t)

∂t^η +p(x)∂^vu(x,t)

∂t^v +q(x)u(x,t)=r(x)∂^θu(x,t)

∂x^θ + f(x,t), u(x,0)=0, ∂u(x,0)

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

(33)

where 1 < η ≤ 2,0 < α ≤ 1,1 < θ ≤ 2. The more complicated boundary conditions for convergence will be presented in each individual case in the exam- ples presented in the next section. The error function of the tau approximation is defined as eN,M(x,t) = u_N,M(x,t)−u(x,t), where u_N,M(x,t) is the shifted Gegenbauer tau approximation solution of eq. (33) and u(x,t)is the exact solution of eq. (33). Throughout this section, we will denote byC_i;i =0,1,2, . . .the posi- tive constants independent ofN,Mbut will depend on m,n, v, ηandθ.

The set of SGPs forms a completeL²_ωα()orthogo- nal system, where=(0,1),ωα is the shifted weight function and L²_ωα() functions u : → R with u_L²_ωα() <∞. We define

u²_L2

ωα() = u,u_L²_ωα =

u²(x)ωα(x)dx.

H^m_ωα indicates the Sobolev space of all functionsu(x) on such thatu(x)and all its weak derivatives up to

ordermare inL²_ωα(). The norm and the semi-norm of H^m_ωα are defined by

u²_H^m

ωα() = m k=0

∂^k

∂t^ku(x) ²

L²_ωα(),

|u|²_Hm:M ωα () =

N k=min(m,N+1)

∂^k

∂t^ku(x) ²

L²_ωα().

LetPN()be the space of all polynomials of degree up toN. DenoteN as the orthogonal projection oper- atorN :L²_ωα()→PN()such that

Nu−u, v =0, ∀v∈PN.

According to [23], the estimate of the truncation error of SGPs forλ≥0,μ≤λandu∈H^λ_ωα():

Nu−u_H^μ_ωα()≤C N^σ(μ,λ)u_Hλ

ωα(), (34)

where σ(μ, λ)=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

2μ−λ−1

2, μ >1, 3

2μ−λ, 0≤μ≤1, μ−λ, μ <0.

Now, we consider a two-dimensional domain, say =t ×x and consider

PN,M =span

C^(α)_L,0(x),C_L^(α)_,1(x), ...,C^(α)_L,N(x),C^(α)_L,0(t),C^(α)_L,1(t), ...,C_L^(α)_,M(t) . The Hilbert spaceH^a_ω^,_α^b()of measurable functionsu:→Ris defined as follows:

H^a_ω^,_α^b()=H^b_ωα(t;H^a_ωα(x))=

u ∈L²_ωα()∂^i+ju

∂xⁱ∂t^j ∈L²_ωα(),0≤i≤a,0≤ j ≤b endowed with the norm

u²_Ha,b ωα =

a i=0

b j=0

∂^i+ju

∂xⁱ∂t^{j 2}

L²_ωα(),

whereH^a_ω^,_α⁰=L²_ωα(t;H^a_ωα(x))andH⁰_ω^,_α^b =H^b_ωα(t; L²_ωα(x)), endowed with the norms

u²_Ha,0 ωα =

1 0

u(·,t)²_H^a_ωα(x)dt, u²_H0,b

ωα = b

j=0

∂^ju

∂t^{j 2}

L²_ωα().

Theorem 5.1. Consider the orthogonal projection N,M : L²_ωα() → PN,M, i.e., for any function u ∈L²_ωα()

N,M(u(x,t))=uN,M(x,t).

Then, for alla,b≥0,we have u−N,Mu

L²_ωα()≤C1M^−au_H^a,0 ωα()

+C₂N^−bu_H0,b ωα()

for alluin which the norms on the right-hand side are finite.

Proof. LetN andM be one-dimensional orthogonal projections. Then,

N,Mu =N ◦(Mu).

Using inequality (34) leads to u−N,Mu

L²_ωα()

≤ u−Nu_L²_ωα()+ N ◦(u−Mu)_L²_ωα()

≤ u−Nu_L²_ωα()+C₂(u−Mu)_L²_ωα()

≤C1M^−au_Ha,0

ωα()+C2N^−bu_H0,b

ωα(). (35)

This completes the proof.

Now, we state and prove the convergence theorem anal- ysis of the shifted Gegenbauer tau method which is the main result of this section.

Theorem 5.2. Letu(x,t)be the exact solution of FTE eq.(33)andN,M(u(x,t))=u_N,M(x,t)be the spec- tral Gegenbauer tau approximation eq.(14)tou(x,t) defined by eq.(25). Assume thatu∈L²_ωα(). Then, for sufficiently smooth functions p(x),q(x)andr(x)in eq.

(33)and for sufficiently large N andMwe get uN,M(x,t)−u(x,t)

L²_ωα() →0.