# New Fifth-Kind Chebyshev Collocation Scheme for First-Order Hyperbolic and Second-Order Convection-Diffusion Partial Differential Equations

Authors: Mahmoud M. Mokhtar; M. H. El Dewaik
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The fifth type of Chebyshev polynomials was used in tandem with the spectral tau method to achieve a semianalytical solution for the partial differential equation of the hyperbolic first order. For this purpose, the problem was diminished to the solution of a set of algebraic equations in unspecified expansion coefficients. The convergence and error analysis of the proposed expansion were studied in-depth. Numerical trials have confirmed the applicability and the accuracy.


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## Body

## 1. Introduction

The first-order partial differential equations (PDEs) modelled various real-life and physical problems. Hyperbolic PDEs characterize the time-dependent physical systems and may be worn to model many phenomena including wave and advection transportation of the material. Advection equations form a special category of conservative hyperbolic first-order PDEs which deliver a given property at a fixed rate through a method. In advection equations, the space and time derivatives of the conserved quantityux,t are proportional to each other, and the interested readers are referred to [1–4], for further implementations on hyperbolic PDE.Spectral methods play a very important role in numerical analysis, especially in the field of numerical solution of ordinary, partial, and differential equations. The key advantage of spectral methods is that differential problems can be quickly transformed into solutions of linear or nonlinear algebraic equations. For more information on spectral methods, please see [5–13].Chebyshev polynomials are very important in many mathematical divisions, especially in numerical analysis. The key idea of Chebyshev polynomials is that they form the foundation for the extension of the differential and integral equation solution. Four well-known Chebyshev polynomial groups are used in the literature. The author in [14] presented the extended Sturm–Liouville differential problem in the fascinating PhD thesis of Masjed-Jamei, and he introduced a basic category of orthogonal-symmetric polynomials. That class has four criteria. Some basic properties are also included, such as a seminal differential equation of order two and a generic relationship containing a three-term recursive relation. For more essential formulae about these polynomials, the reader is referred to the work of [14]. The advantage of this class is that two new types of Chebyshev polynomials, fifth and sixth, are inherited. Such polynomials were used only once in literature by seminal work in the numerical solution of fractional differential equations, and Abd-Elhameed and Youssri [15–20] first used the fifth-type Chebyshev polynomials to handle ODEs and PDEs.


### 1.1. Mathematical Preliminaries

This section presents the properties of the basic class of symmetric orthogonal polynomials (BCSOP) that formed in [14]. The key concept to develop this class of polynomial is focused on the use of an extended Sturm–Liouville differential problem. More precisely, in [14], the author assumed that y=ϕiz is a sequence of symmetric functions that satisfies the following differential equation of second order:(1)A1zϕi″z+A2zϕi′z+μiA3z+A4z+1−−1i2A5zϕiz=0,where Aiz,1≤i≤5 are independent functions, and μi are constants. In [14], it has been shown that Aiz,i=1,3,4,5 is even, and A2z is odd. You may obtain the desired symmetric category of orthogonal polynomials if Aiz,1≤i≤5, and μi are chosen as follows:(2)A1z=z2rz2+s,A2z=zmz2+n,A3z=z2,A4z=0,A5z=−n,μi=−ii−1r+m,where the m,n,r,s parameters are real numbers.By using the above relations, we have the following differential equation:(3)z2s+rz2ϕi″z+zn+mz2ϕi′z−ii−1r+mz2+1+−1i+1n2ϕiz=0.The solution of (3) is the generalized polynomials Gim,n,r,sz which have the explicit form:(4)Gim,n,r,sz=∑k=0i2i2k∏j=0i2−k−11Γi,j,m,n,r,sz2i2−k,where(5)Γi,j,m,n,r,s=2j+−1i+1+2s+n−1i+1+2j+2i/2r+m.Moreover, the author introduced orthogonal symmetric polynomials in [14], denoted by G¯im,n,r,sz defined as(6)G¯im,n,r,sz=∏j=0i2−1Γi,j,m,n,r,sGim,n,r,sz.The polynomialsG¯im,n,r,sz satisfy the following recurrence relation:(7)G¯i+1m,n,r,st=zG¯im,n,r,sz+Ai,m,n,r,sG¯i−1m,n,r,sz,i≥0,with the initials:(8)G¯0m,n,r,sz=1,G¯1m,n,r,sz=z.And(9)Ai,m,n,r,s=rsi2+m−2rs−−1irni+12m−2rn1−−1i2ri+m−r2ri+m−3r.Many properties ofG¯im,n,r,st may be found in [14].There are many specific categories of important orthogonal polynomials ofG¯i,j,m,n,r,sz. The four different types of Chebyshev polynomials could be formed through the expressions:(10)T¯iz=G¯i−1,0,−1,1z,U¯iz=G¯i−3,0,−1,1z,V¯iz=2iG¯2i−3,2,−1,11+z2,W¯iz=2iG¯2i−3,2,−1,11−z2,and T¯iz,U¯iz,V¯iz,W¯iz are the first, second, third, and fourth kinds of Chebyshev polynomials. All these polynomials can be obtained as specific special cases of G¯im,n,r,sz. The two types of orthogonal polynomials in [14], especially, Chebyshev’s fifth- and sixth-kind polynomials, may also be defined, respectively, as(11)X¯iz=G¯i−3,2,−1,1z,Y¯iz=G¯i−5,2,−1,1z.We focus our study on the Chebyshev fifth kind and their shifted polynomials. The property of orthogonality ofX¯iz is(12)∫−11z2X¯jzX¯iz1−z2dz=−1i∏L=1iAL,−3,2,−1,1π2,ifi=j,0,otherwise,where Ai,m,n,r,s is defined in (9).Alternatively, the orthogonality formula above is written as(13)∫−11z2X¯izX¯jz1−z2dz=hi,ifi=j,0,ifi≠j.And(14)hi=π22i+1,ieven,πi+2i22i+1,iodd.It is more reasonable to normalize fifth-type Chebyshev polynomials. For this specific purpose, we set(15)Xiz=1hiX¯iz.Accordingly,Xiz are orthonormal on I=−1,1:(16)∫−11z21−z2XizXjzdz=1,ifi=j,0,ifi≠j.


### 1.2. Fifth Type of Shifted Orthonormal Chebyshev Polynomials (5SOCP)

The 5SOCPCiz can be defined on I∗=0,1 by(17)Ciz=1hiX¯i2z−1.Also,hi is defined in (14).From (16), it is easy to note that Ciz,i≥0 are orthonormal on I∗. Directly, we have(18)∫01w∗zCizCjzdx=1,ifi=j,0,otherwise.Andw∗z=1−2z2/z−z2. The following results are needed in the sequel.Theorem 1.
The polynomialsCiz in (9) are connected with Ti∗z by the following formula:(19)Ciz=∑j=0igi,jTj∗z,where(20)gi,j=22π−1i−j2δj,i,jeven,ji,i,jodd,0otherwise.
And(21)δj=12,ifj=0,1,ifj>0.Proof.
See [15]Theorem 2.
The polynomialsCiz in (9) are connected with Ti∗z by two formulae as follows:(22)C2iz=22π∑r=0i−1i+rδrT2r∗z,where δr is defined in (12).
And(23)C2i+1z=22π2i+32i+1∑r=0i−1i+r2r+1T2r+1∗z.The next corollary shows the above-intended purpose.Corollary 1.
Chebyshev polynomials of the fifth type have the following trigonometric representations:(24)X2icosϕ=2πcos1+2icosϕ.
And(25)X2i+1cosϕ=2π4i2+8i+33+2icos2+2iϕcosϕ−cos3+2iϕcos2ϕ.Proof.
See [15]The following connection theorem is needed in the sequel. The following two theorems are important.Theorem 3.
The analytical formCiz is specifically given as(26)Ciz=∑r=0iϱr,izr,where(27)ϱr,i=22r+32ϕ2r!2∑j=r+12i2−1i2+j−rjδj2j+r−1!2j−r!,ieven,1ii+2∑j=r2i−12−1i+12+j−r2j+122j+r!2j−r+1!,iodd.Theorem 4.
The reflection relation (17) of the analytical relation may be stated as(28)zm=∑ℓ=0mqm,ℓCℓz,where(29)qm,ℓ=π2−2m−122m!2m2+m+1+ℓ2m−ℓ!ℓ+m+2!,ℓeven,ℓ2+ℓm−2+ℓ!m+2+ℓ!+2ℓ+1m+ℓ!m−ℓ!,ℓodd.Proof.
See [15]The derivative formula is also needed.Corollary 2.
These two identities hold for all nonnegative integerq:(30)DqCiz|t=1=αi,q,DqCiz|t=0=−1i+qαi,q,where(31)αi,q=∑j=0iα¯i,j,q.
And(32)α¯i,j,q=22−1i−j2Γq+j−q+j+1qΓjΓq+12δj,ifi,jeven,jii2+i,i,jodd,0,otherwise,where jq is the pochhammer symbol.Proof.
See [15]Theorem 5.
(See [20]). Let j≥1, the following relation holds(33)DCjξ=∑k=0j−1Mk,jCkξ.
AndMk,j are given by(34)Mk,j=22−j+kj+1k+1k+2,jevenkodd,j2+2j+2k+2j,joddj−k+12even,j2+2j−2k−2j,joddj−k−12even,0,otherwise.(35)∂u∂t=∑j=1N∑i=0N∑k=0j−1aijMKijCkxCjt,∂u∂X=∑j=0N∑i=1N∑k=0I−1aijMKjiCixCKt.Theorem 6.
(See [20]). For every j≥2, the following relation holds(36)D2Cjξ=∑k=0j+kevenj−2M¯k,jCkξ,where(37)M¯k,j=π22−jj!k+1!2j−32+3j+kj−k2−1!12−j+k+7+j−32Γ12−j+k+2+j2×_4F3−j2−k2,−j2+k2+1,3−j2−12,−j+12−12−j,3−j2−32,12−j+12|1.


## 1.1. Mathematical Preliminaries

This section presents the properties of the basic class of symmetric orthogonal polynomials (BCSOP) that formed in [14]. The key concept to develop this class of polynomial is focused on the use of an extended Sturm–Liouville differential problem. More precisely, in [14], the author assumed that y=ϕiz is a sequence of symmetric functions that satisfies the following differential equation of second order:(1)A1zϕi″z+A2zϕi′z+μiA3z+A4z+1−−1i2A5zϕiz=0,where Aiz,1≤i≤5 are independent functions, and μi are constants. In [14], it has been shown that Aiz,i=1,3,4,5 is even, and A2z is odd. You may obtain the desired symmetric category of orthogonal polynomials if Aiz,1≤i≤5, and μi are chosen as follows:(2)A1z=z2rz2+s,A2z=zmz2+n,A3z=z2,A4z=0,A5z=−n,μi=−ii−1r+m,where the m,n,r,s parameters are real numbers.By using the above relations, we have the following differential equation:(3)z2s+rz2ϕi″z+zn+mz2ϕi′z−ii−1r+mz2+1+−1i+1n2ϕiz=0.The solution of (3) is the generalized polynomials Gim,n,r,sz which have the explicit form:(4)Gim,n,r,sz=∑k=0i2i2k∏j=0i2−k−11Γi,j,m,n,r,sz2i2−k,where(5)Γi,j,m,n,r,s=2j+−1i+1+2s+n−1i+1+2j+2i/2r+m.Moreover, the author introduced orthogonal symmetric polynomials in [14], denoted by G¯im,n,r,sz defined as(6)G¯im,n,r,sz=∏j=0i2−1Γi,j,m,n,r,sGim,n,r,sz.The polynomialsG¯im,n,r,sz satisfy the following recurrence relation:(7)G¯i+1m,n,r,st=zG¯im,n,r,sz+Ai,m,n,r,sG¯i−1m,n,r,sz,i≥0,with the initials:(8)G¯0m,n,r,sz=1,G¯1m,n,r,sz=z.And(9)Ai,m,n,r,s=rsi2+m−2rs−−1irni+12m−2rn1−−1i2ri+m−r2ri+m−3r.Many properties ofG¯im,n,r,st may be found in [14].There are many specific categories of important orthogonal polynomials ofG¯i,j,m,n,r,sz. The four different types of Chebyshev polynomials could be formed through the expressions:(10)T¯iz=G¯i−1,0,−1,1z,U¯iz=G¯i−3,0,−1,1z,V¯iz=2iG¯2i−3,2,−1,11+z2,W¯iz=2iG¯2i−3,2,−1,11−z2,and T¯iz,U¯iz,V¯iz,W¯iz are the first, second, third, and fourth kinds of Chebyshev polynomials. All these polynomials can be obtained as specific special cases of G¯im,n,r,sz. The two types of orthogonal polynomials in [14], especially, Chebyshev’s fifth- and sixth-kind polynomials, may also be defined, respectively, as(11)X¯iz=G¯i−3,2,−1,1z,Y¯iz=G¯i−5,2,−1,1z.We focus our study on the Chebyshev fifth kind and their shifted polynomials. The property of orthogonality ofX¯iz is(12)∫−11z2X¯jzX¯iz1−z2dz=−1i∏L=1iAL,−3,2,−1,1π2,ifi=j,0,otherwise,where Ai,m,n,r,s is defined in (9).Alternatively, the orthogonality formula above is written as(13)∫−11z2X¯izX¯jz1−z2dz=hi,ifi=j,0,ifi≠j.And(14)hi=π22i+1,ieven,πi+2i22i+1,iodd.It is more reasonable to normalize fifth-type Chebyshev polynomials. For this specific purpose, we set(15)Xiz=1hiX¯iz.Accordingly,Xiz are orthonormal on I=−1,1:(16)∫−11z21−z2XizXjzdz=1,ifi=j,0,ifi≠j.


## 1.2. Fifth Type of Shifted Orthonormal Chebyshev Polynomials (5SOCP)

The 5SOCPCiz can be defined on I∗=0,1 by(17)Ciz=1hiX¯i2z−1.Also,hi is defined in (14).From (16), it is easy to note that Ciz,i≥0 are orthonormal on I∗. Directly, we have(18)∫01w∗zCizCjzdx=1,ifi=j,0,otherwise.Andw∗z=1−2z2/z−z2. The following results are needed in the sequel.Theorem 1.
The polynomialsCiz in (9) are connected with Ti∗z by the following formula:(19)Ciz=∑j=0igi,jTj∗z,where(20)gi,j=22π−1i−j2δj,i,jeven,ji,i,jodd,0otherwise.
And(21)δj=12,ifj=0,1,ifj>0.Proof.
See [15]Theorem 2.
The polynomialsCiz in (9) are connected with Ti∗z by two formulae as follows:(22)C2iz=22π∑r=0i−1i+rδrT2r∗z,where δr is defined in (12).
And(23)C2i+1z=22π2i+32i+1∑r=0i−1i+r2r+1T2r+1∗z.The next corollary shows the above-intended purpose.Corollary 1.
Chebyshev polynomials of the fifth type have the following trigonometric representations:(24)X2icosϕ=2πcos1+2icosϕ.
And(25)X2i+1cosϕ=2π4i2+8i+33+2icos2+2iϕcosϕ−cos3+2iϕcos2ϕ.Proof.
See [15]The following connection theorem is needed in the sequel. The following two theorems are important.Theorem 3.
The analytical formCiz is specifically given as(26)Ciz=∑r=0iϱr,izr,where(27)ϱr,i=22r+32ϕ2r!2∑j=r+12i2−1i2+j−rjδj2j+r−1!2j−r!,ieven,1ii+2∑j=r2i−12−1i+12+j−r2j+122j+r!2j−r+1!,iodd.Theorem 4.
The reflection relation (17) of the analytical relation may be stated as(28)zm=∑ℓ=0mqm,ℓCℓz,where(29)qm,ℓ=π2−2m−122m!2m2+m+1+ℓ2m−ℓ!ℓ+m+2!,ℓeven,ℓ2+ℓm−2+ℓ!m+2+ℓ!+2ℓ+1m+ℓ!m−ℓ!,ℓodd.Proof.
See [15]The derivative formula is also needed.Corollary 2.
These two identities hold for all nonnegative integerq:(30)DqCiz|t=1=αi,q,DqCiz|t=0=−1i+qαi,q,where(31)αi,q=∑j=0iα¯i,j,q.
And(32)α¯i,j,q=22−1i−j2Γq+j−q+j+1qΓjΓq+12δj,ifi,jeven,jii2+i,i,jodd,0,otherwise,where jq is the pochhammer symbol.Proof.
See [15]Theorem 5.
(See [20]). Let j≥1, the following relation holds(33)DCjξ=∑k=0j−1Mk,jCkξ.
AndMk,j are given by(34)Mk,j=22−j+kj+1k+1k+2,jevenkodd,j2+2j+2k+2j,joddj−k+12even,j2+2j−2k−2j,joddj−k−12even,0,otherwise.(35)∂u∂t=∑j=1N∑i=0N∑k=0j−1aijMKijCkxCjt,∂u∂X=∑j=0N∑i=1N∑k=0I−1aijMKjiCixCKt.Theorem 6.
(See [20]). For every j≥2, the following relation holds(36)D2Cjξ=∑k=0j+kevenj−2M¯k,jCkξ,where(37)M¯k,j=π22−jj!k+1!2j−32+3j+kj−k2−1!12−j+k+7+j−32Γ12−j+k+2+j2×_4F3−j2−k2,−j2+k2+1,3−j2−12,−j+12−12−j,3−j2−32,12−j+12|1.


## 2. Implementation of the Method

The aim of this section is to obtain two numerical algorithms for the solution of the first-order hyperbolic differential equation and the second-order convection-diffusion equation.


### 2.1. First-Order Hyperbolic Equation

Consider the following first-order hyperbolic partial differential equation(38)Dtv=ξ1Dxv+ξ2v+S,x∈0,1,t∈0,1,subject to the initial condition(39)vx,0=k0x,x∈0,1.And the boundary condition(40)v0,t=k1t,t∈0,1,where ξ1,ξ2 are two constants. We expand the exact solution v, the derivatives Dtv and Dxv by the fifth-type Chebyshev expansion as(41)vx,t≈vNx,t=∑i,j=0NaijCixCjt,∂vNx,t∂t=∑i,j=0NaijCix∂Cjt∂t,∂vNx,t∂x=∑i,j=0Naij∂Cix∂xCjt,RNx,t=DtvN−ξ1DxvN−ξ2vN−S.We apply the typical tau method and make use of the boundary conditions to get a system ofN+1×N+1 algebraic equations in the required double-shifted fifth-kind Chebyshev coefficients aij,i,j=0,1,…,N that can be solved using any standard iteration technique, such as the iteration method of Newton. It is therefore possible to evaluate the semianalytic solution vNx,t.


### 2.2. Second-Order Convection-Diffusion Equation

Consider the following second-order convection-diffusion equation:(42)∂u∂t+β∂u∂x=α∂2u∂x2,x,t∈0,1×0,1,subject to the initial condition(43)ux,0=fx,x∈0,1.And the boundary conditions(44)u0,=g0t,u1,t=g1t,t∈0,1,∂u∂t+β∂u∂x=α∂2u∂x2,x,t∈0,1×0,1,subject to the initial condition(45)ux,0=fx,x∈0,1.And the boundary conditions(46)u0,t=g0t,u1,t=g1t,t∈0,1,where α,β are two constants. We expand the exact solution in terms of the shifted fifth-kind Chebyshev polynomials and derivatives of u based on the above-mentioned derivatives theorems as(47)ux,t≃uNx,t=∑i=0N∑j=0NaijCixCj′t,∂2u∂x2=∑j=0N∑i=2N∑k=0i−2ai,jM¯k,jCkxCjt,Rx,t=∑J=0N∑i=0N∑K=0j−1aijMk,jCixCKt+∑J=0N∑i=0N∑K=0i−1βijMk,jCkxCjt−∑J=0N∑i=2N∑K=0i−1αaijM¯k,jCkxCjt.∑j=0N∑i=0Nai,jCixCj0=fx,∑j=0N∑i=0Nai,jCi0Cjt=g0t,∑j=0N∑i=0Nai,jCi1Cjt=g1t.Applying the inner product and using the orthogonality relation, we get(48)∑j=0N∑i=0N∑k=0j−1aijMk,jδi,mδk,nhihk+∑j=0N∑i=1N∑k=0i−1βaijMk,iδi,mδj,nhihj−∑j=0N∑i=2N∑k=0i−1αai,jMk,iδk,mδj,nhkhj=0,∑j=0N∑i=0Nai,jδimhiCj0=fm,∑j=0N∑i=0Nai,jδjnhjCi0=gn0,∑j=0N∑i=0Nai,jCi1δjnhj=gn1.The obtained system of algebraic equations is solved by the use of Gaussian elimination to get the unknown expansion coefficients and, hence, the numerical solution.


## 2.1. First-Order Hyperbolic Equation

Consider the following first-order hyperbolic partial differential equation(38)Dtv=ξ1Dxv+ξ2v+S,x∈0,1,t∈0,1,subject to the initial condition(39)vx,0=k0x,x∈0,1.And the boundary condition(40)v0,t=k1t,t∈0,1,where ξ1,ξ2 are two constants. We expand the exact solution v, the derivatives Dtv and Dxv by the fifth-type Chebyshev expansion as(41)vx,t≈vNx,t=∑i,j=0NaijCixCjt,∂vNx,t∂t=∑i,j=0NaijCix∂Cjt∂t,∂vNx,t∂x=∑i,j=0Naij∂Cix∂xCjt,RNx,t=DtvN−ξ1DxvN−ξ2vN−S.We apply the typical tau method and make use of the boundary conditions to get a system ofN+1×N+1 algebraic equations in the required double-shifted fifth-kind Chebyshev coefficients aij,i,j=0,1,…,N that can be solved using any standard iteration technique, such as the iteration method of Newton. It is therefore possible to evaluate the semianalytic solution vNx,t.


## 2.2. Second-Order Convection-Diffusion Equation

Consider the following second-order convection-diffusion equation:(42)∂u∂t+β∂u∂x=α∂2u∂x2,x,t∈0,1×0,1,subject to the initial condition(43)ux,0=fx,x∈0,1.And the boundary conditions(44)u0,=g0t,u1,t=g1t,t∈0,1,∂u∂t+β∂u∂x=α∂2u∂x2,x,t∈0,1×0,1,subject to the initial condition(45)ux,0=fx,x∈0,1.And the boundary conditions(46)u0,t=g0t,u1,t=g1t,t∈0,1,where α,β are two constants. We expand the exact solution in terms of the shifted fifth-kind Chebyshev polynomials and derivatives of u based on the above-mentioned derivatives theorems as(47)ux,t≃uNx,t=∑i=0N∑j=0NaijCixCj′t,∂2u∂x2=∑j=0N∑i=2N∑k=0i−2ai,jM¯k,jCkxCjt,Rx,t=∑J=0N∑i=0N∑K=0j−1aijMk,jCixCKt+∑J=0N∑i=0N∑K=0i−1βijMk,jCkxCjt−∑J=0N∑i=2N∑K=0i−1αaijM¯k,jCkxCjt.∑j=0N∑i=0Nai,jCixCj0=fx,∑j=0N∑i=0Nai,jCi0Cjt=g0t,∑j=0N∑i=0Nai,jCi1Cjt=g1t.Applying the inner product and using the orthogonality relation, we get(48)∑j=0N∑i=0N∑k=0j−1aijMk,jδi,mδk,nhihk+∑j=0N∑i=1N∑k=0i−1βaijMk,iδi,mδj,nhihj−∑j=0N∑i=2N∑k=0i−1αai,jMk,iδk,mδj,nhkhj=0,∑j=0N∑i=0Nai,jδimhiCj0=fm,∑j=0N∑i=0Nai,jδjnhjCi0=gn0,∑j=0N∑i=0Nai,jCi1δjnhj=gn1.The obtained system of algebraic equations is solved by the use of Gaussian elimination to get the unknown expansion coefficients and, hence, the numerical solution.


## 3. Discussion of Convergence and Error Analysis

The following lemma is needed.Lemma 1.
The fifth-kind Chebyshev polynomialsCℓt satisfy the following inequality:(49)Cℓt<2+ℓ2π,∀t∈0,t.Theorem 7.
Ifft∈Lw∗2I∗ (the set of all square Lebesgue integrable functions), f3t⩽L, and if its expansion is(50)ft=∑ℓ=0∞aℓCℓt.
The series in (50) uniformly converges to ft. Also, we have(51)aℓ<2πL2ℓ3,forallℓ>3.Theorem 8.
Letut satisfies the conditions of Theorem 7, uN+1t,uNt are two approximate solutions of ut, and we define e¯Nt=uN+1t−uNt, and then, we get the following estimation of the error:(52)e¯Nt2,w∗=O1N3,where e¯Nt2,w∗ means the L2−norm of e¯Nt.Theorem 9.
Letft ascertain conditions of Lemma 1, let eNt=∑ℓ=N+1∞aℓCℓt be the truncation error, and then, eNt satisfy the following estimate(53)eNt<3LN.Theorem 10.
Assume thatvx,t is separable C3 function with bounded third-order derivatives, and then, the coefficients in (31) satisfy:(54)ai,j<Ωi3j3,∀i,j>3,where Ω is a generic positive constant.Proof.
The proof is a direct consequence of Theorem6.Theorem 11.
Ifvx,t satisfies the conditions of Theorem 10 and vNx,t is the approximation of vx,t, we then get the following estimation of the error:(55)v−vN2,w∗=ON−3,where .2,w∗ denotes the L2-norm.Proof.
The proof is a direct consequence of Theorem6.Using the orthogonality ofCixCjt and with the aid of Theorem 9 and applying Parseval’s identity, we get the desired result.


## 4. Numerical Results

We offer some numerical tests in this section to illustrate the precision, efficacy, and the wide applicability of the proposed system. We compare our method with Laguerre-Gauss-Radau scheme [21] which shows that our method is very efficient. Therefore, we affirm that the proposed technique is more suitable for solving problems of this type.The following tables depict the absolute errors.(56)Ex,t=yN,Mx,t−yx,t,where yN,Mx,t is the numerical, and ,yx,t is the exact solution at the node x,t. The point-wise errors are evaluated by(57)E=maxx,t∈0,1×0,1Ex,t.Example 1.
(see[21]). Let us start with the following hyperbolic PDE:(58)Dty=Dxy+y+S,x∈0,1,t∈0,1,with the initials(59)y0,t=e−2t,yx,0=e−x,x∈0,1,t∈0,1,where(60)Sx,t=−2e−2t−x.
The smooth solution is given by(61)y=e−2t+x.
In Tables1–3, we compare between our method for the case N=16, t=0.1,0.5,1. These are obtained by the generalized collocation method Laguerre–Gauss–Radau [21]. In Figure 1, we depict the exact solution of Example 1, and for N=16, in Figure 2, we depict the maximum absolute error when N=16 (Table 4).Table 1
Comparison between the errors of Example1 when N=16 and t=0.1.
xBhrawy et al.Current method0.12.84.10−72.11.10−170.28.79.10−61.56.10−170.31.20.10−51.59.10−170.41.12.10−57.32.10−180.57.95.10−63.37.10−180.63.29.10−63.34.10−190.71.78.10−61.47.10−180.86.53.10−63.86.10−180.91.04.10−51.79.10−1511.32.10−51.27.10−13Table 2
The different errors of Example1 at N=16 and t=0.5.
xBhrawy et al.Current method0.18.95.10−65.00.10−180.24.10.10−64.64.10−190.31.39.10−52.22.10−180.42.07.10−51.16.10−190.52.47.10−52.69.10−150.62.62.10−61.90.10−130.72.55.10−53.52.10−130.82.31.10−53.53.10−110.91.93.10−52.43.10−1011.45.10−51.29.10−9Table 3
The different errors for Example1 at N=16 and t=1.
xBhrawy et al.Current method0.14.87.10−53.14.10−130.24.89.10−55.81.10−120.34.17.10−55.83.10−110.43.03.10−54.02.10−100.51.75.10−52.13.10−90.64.74.10−69.27.10−90.76.68.10−63.46.10−80.81.61.10−51.14.10−70.92.32.10−53.39.10−712.79.10−59.26.10−7Figure 1
The solutionyx,t for N=16 in Example 1.Figure 2
The error in Example1.Table 4
The maximum absolute error for Example 1.
M6810121416MAE1.73.10−51.35.10−76.67.10−103.76.10−115.14.10−90.00.10−0Example 2.
(see [21]).And, then the following hyperbolic PDE:(62)∂yx,t∂t=∂yx,t∂x+yx,t−sinte−t−x−coste−t−x,x∈0,1,t∈0,1,with the initials,(63)y0,t=sinte−t,t∈0,1,yx,0=0,x∈0,1.
The exact smooth solution is(64)yx,t=e−t−xsint.
Table5 presents the results developed by our method when N=16and the method in [21]; in Figure 3, we illustrate the solution yx,t, while in Figure 4, we depict the error when N=16.Table 5
Absolute errors for Example2 when N=16.
x,tθ1=ϑ1=−12,θ1=ϑ1=0,θ1=ϑ1=12,Current methodθ2=ϑ2=−12,θ2=ϑ2=0,θ2=ϑ2=12,0.1,0.19.61×10−74.22×10−71.95×10−71.98×10−190.2,0.26.37×10−75.92×10−77.08×10−72.35×10−190.3,0.39.63×10−76.47×10−71.67×10−71.17×10−190.4,0.41.44×10−68.90×10−77.52×10−76.03×10−200.5,0.52.04×10−69.36×10−72.47×10−77.97×10−170.6,0.61.28×10−84.81×10−81.35×10−76.08×10−140.7,0.72.23×10−61.23×10−68.48×10−72.73×10−120.8,0.87.97×10−74.56×10−71.16×10−73.94×10−110.9,0.93.33×10−61.55×10−66.54×10−72.51×10−10Figure 3
The solutionyx,t for N=16 in Example 2.Figure 4
The error in Example2.Example 3.
(see [22]).In this example, we consider(65)fx=exp−x+0.520.00125,g00,t=0.0250.000625+0.02texp−0.5−t20.00125+0.04t,g11,t=0.0250.000625+0.02texp−1.5−t20.00125+0.04t,α=0.01,β=1.0,with the exact smooth solution(66)ux,t=0.0250.000625+0.02texp−x+0.5−t20.00125+0.04t.
In Table6, we list the maximum pointwise error for different values of N=16. In Figure 5, we depict the maximum absolute error when N=16.Table 6
The different errors of Example1 at N=16 and t=0.5.
N246810121416MAE4.10.10−12.54.10−24.26.10−35.28.10−55.67.10−65.63.10−82.67.10−102.69.10−13Figure 5
The error in Example3.


## 5. Conclusions

A precise numerical technique for solving the hyperbolic partial differential equations is being constructed and applied in the present work. The fifth-type approach of Chebyshev spectral tau was used to simplify the solution of hyperbolic partial differential equations to a set of algebraic equations, which can be more conveniently solved. The numerical findings showed our method to be extremely effective and reliable.



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