# A New Unconditionally Stable Method for Telegraph Equation Based on Associated Hermite Orthogonal Functions

Authors: Di Zhang; Fusheng Peng; Xiaoping Miao
Journal: Advances in Mathematical Physics
 (2016)

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## Abstract

The present paper proposes a new unconditionally stable method to solve telegraph equation by using associated Hermite (AH) orthogonal functions. Unlike other numerical approaches, the time variables in the given equation can be handled analytically by AH basis functions. By using the Galerkin’s method, one can eliminate the time variables from calculations, which results in a series of implicit equations. And the coefficients of results for all orders can then be obtained by the expanded equations and the numerical results can be reconstructed during the computing process. The precision and stability of the proposed method are proved by some examples, which show the numerical solution acquired is acceptable when compared with some existing methods.


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## Body

## 1. Introduction

In this work, the following1d telegraph equation is considered:(1)∂2u∂t2+2α∂u∂t+β2u=∂2u∂x2+fx,t,a≤x≤b,t≥0with the initial conditions(2)ux,0=h0x,∂ux,0∂t=h1x,a≤x≤band the Dirichlet boundary conditions(3)ua,t=g0t,ub,t=g1t,t≥0,where α>0 and β>0 are real constants. We now assume that g0(t), g1(t) and h0(x), h1(x) are continuous functions of t and x, respectively. The telegraph equation has been arisen in the propagation of electrical signals in wave phenomena and transmission line.During past years, much literatures have paid attention to the analysis and development of telegraph equation, see, for example, [1–4]. Mohanty et al. [5–7] developed the finite difference method for solving telegraph equation and it has been proved to be of high accuracy. Dehghan and Shokri [8] proposed a new scheme for 1d telegraph equation using collocation points and radial basis function. In [2], a high-order accurate scheme is introduced for telegraph equation. Dehghan and Ghesmati [9] developed the boundary integral equation scheme for solving second-order hyperbolic equation. Saadatmandi and Dehghan [10] proposed the Chebyshev Tau method for telegraph equation. In [11], the Chebyshev cardinal function is used for solving the 1d telegraph equation. L.-B. Liu and H.-W. Liu [12] proposed a numerical solution for telegraph equation by applying the trapezoidal formula and quartic spline. Authors of [13–16] developed the radial basis function scheme for solving the telegraph equation. In recent years, some numerical schemes were proposed to solve (1) based on B-spline collocation method [17–19], polynomial scaling functions [20, 21], shifted Gegenbauer pseudospectral method [22], and differential quadrature algorithm [23]. These schemes are conditionally stable.To construct an unconditional stability method to solve telegraph equation, Mohanty et al. [24–26] developed three-level alternating direction implicit scheme to solve telegraph equation. Borhanifar and Abazari [27] proposed the parallel difference method, which is unconditionally stable, to solve telegraph equation. In [28], Mohanty presented a new unconditionally stable method to solve telegraph equation, where two parameters were introduced. Gao and Chi [29] presented two semidiscretion schemes for solving the hyperbolic equations. H.-W. Liu and L.-B. Liu [30] developed the spline difference method to solve 1d problem. In [31], the compact difference scheme is proposed for solving the telegraph equation. Xie et al. [32] developed a compact difference and ADI scheme for telegraph equations with a fourth order in space. Some high-order methods, using the Padé approximation method [33] and cubic Hermite interpolation [34], were proposed for the telegraph equations.In this study, we attempt to solve the telegraph equation using associated Hermite (AH) orthogonal functions. The Hermite functions, which were widely studied in the Hermite spectral method (HSM), are constructed by Hermite basis functions with a translated and scaled weighting function [35]. Kavian and Funaro [36] considered the use of HSM combined with variable transformation technique to solve the diffusion problems in unbounded domains. In [37, 38], Guo et al. developed the Hermite spectral method and Hermite pseudospectral methods (HPSM), and Alıcı [39] used the HPSM to solve the 2d Schrödinger equation. To stabilize the methods, a time-dependent parameter [40–42] is introduced for traditional Hermite functions to construct a conditional stability method. Unfortunately, the Hermite orthogonal functions were not used to structure an unconditional stability method for telegraph equation.The fundamental aim of the literature is eliminating of the time variables in telegraph equation and then construct a new unconditionally stable method, the accuracy of which is absolutely independent of temporal step. In our presented method, the time variables in (1) are handled by the AH basis functions at the first step. Then, the Galerkin’s method and the central difference method are introduced to this handled equation; a series of equations without time variables can be obtained. Finally, we can solve this telegraph equation by using the chasing method and the numerical solution could be reconstructed by using coefficients of AH functions.


## 2. Associated Hermite Functions

AH functions can be expressed as(4)ϕpt=2pp!π1/2-1/2e-t2/2Hpt,where the Hermite polynomials Hp(t) is(5)Hpt=-1pet2dpdtpe-t2.The Hermite polynomials satisfy the following recursive relationship [43]:(6)H0t=1,H1t=2t,Hp+1t=2tHpt-2pHp-1t.Then we have the derivative relations of Hermite polynomials:(7)Hp′t=2pHp-1t,p≥1Hp′t=2tHpt-Hp+1t,p≥0.By introducing a time-translating parameter, AH functions could be transformed to a causal form:(8)ϕ¯pt~=2pp!σπ1/2-1/2e-t~/2Hpt~,where t~=(t-Tf)/σ; Tf and σ are the translating parameter and time-scaling parameter, respectively. A function with causal relationships can be expanded by choosing a suitable Tf and σ: (9)ux,t=∑p=0∞upxϕpt~.Combining (6) and (8), we have(10)ϕ¯0t~=12σπe-t2/2,ϕ¯1t~=12σπ2te-t2/2,ϕ¯p+1t~=t2p+1ϕ¯pt~-pp+1ϕ¯p-1t~,p≥1.Property (7) and the above formula lead to the derivation of ϕ¯p(t~):(11)ddtϕ¯pt~=-1σ12ϕ¯1t~,p=0,1σp2ϕ¯p-1t~-1σp+12ϕ¯p+1t~,p≥1.Combining (9) and (11), we have(12)∂ux,t∂t=∑p=0∞upxddtϕ¯pt~=u0xddtϕ¯0t~+∑p=1∞upxddtϕ¯pt~=u0x-12σ2ϕ¯1t~+∑p=1∞upxp2σ2ϕ¯p-1t~-p+12σ2ϕ¯p+1t~.Let m=p-1; we can obtain(13)∂ux,t∂t=u0x-12σ2ϕ¯1t~+∑m=0∞um+1xm+12σ2ϕ¯mt~-m+22σ2ϕ¯m+2t~=∑m=0∞um+1xm+12σ2ϕ¯mt~+u0x-12σ2ϕ¯1t~+∑m=0∞um+1x-m+22σ2ϕ¯m+2t~.Let u-1(x)=0; we have(14)u0x-12σ2ϕ¯1t~+∑m=0∞um+1x-m+22σ2ϕ¯m+2t~=k=m+2u0x-12σ2ϕ¯1t~+∑k=2∞uk-1x-k2σ2ϕ¯kt~=∑k=0∞uk-1x-k2σ2ϕ¯kt~.Then we can deduce the partial derivative of u(x,t)(15)∂ux,t∂t=∑m=0∞um+1xm+12σ2ϕ¯mt~+∑k=0∞uk-1x-k2σ2ϕ¯kt~=m=p,k=p∑p=0∞up+1xp+12σ2-up-1xp2σ2ϕ¯pt~.For the second partial derivative ofu(x,t), we have(16)∂2ux,t∂t2=∑p=0∞up+1xp+12σ2-up-1xp2σ2·ddtϕ¯pt~=m=p-1u1x12σ2-12σ2ϕ¯1t~+∑m=0∞um+2xm+22σ2-umxm+12σ2·m+12σ2ϕ¯mt~-m+22σ2ϕ¯m+2t~,where(17)u1x12σ2-12σ2ϕ¯1t~+∑m=0∞um+2xm+22σ2-umxm+12σ2·-m+22σ2ϕ¯m+2t~=k=m+2u1x12σ2-12σ2ϕ¯1t~+∑k=2∞ukxk2σ2-uk-2xk-12σ2·-k2σ2ϕ¯kt~=∑k=0∞ukxk2σ2-uk-2xk-12σ2·-k2σ2ϕ¯kt~Letu-2x=0,u-1x=0.From the above equations, we can obtain the first and the second partial derivative of u(x,t):(18)∂ux,t∂t=∑p=0∞up+1xp+12σ2-up-1xp2σ2ϕ¯pt~,∂2ux,t∂t2=∑m=0∞um+2xm+22σ2-umxm+12σ2·m+12σ2ϕ¯mt~+∑k=0∞ukxk2σ2-uk-2xk-12σ2·-k2σ2ϕ¯kt~=m=p,k=p∑p=0∞up+2xp+22σ2p+12σ2-upx2p+12σ2+up-2xp-12σ2p2σ2·ϕ¯pt~.


## 3. Description of the Method

### 3.1. Construction of Computing Matrix

The partial differential with respect tox can be written as(19)∂2u∂x2=∑p=0∞∂2∂x2upxϕ¯pt~.Rewriting the telegraph equation (1) using (9), (18), and (19), we can obtain(20)∑p=0∞up+2xp+22σ2p+12σ2-upx2p+12σ2+up-2xp-12σ2p2σ2·ϕ¯pt~+2α∑p=0∞up+1xp+12σ2-up-1xp2σ2ϕ¯pt~+β2∑p=0∞upxϕpt~=∑p=0∞∂2∂x2upxϕ¯pt~+∑p=0∞fpxϕ¯pt~.In (20), we introduced a Galerkin procedure [44] in order to eliminate the time variable terms ϕ¯p(t~). By multiplying each side of (20) by time-dependent term ϕ¯q(t~), we can obtain the following equation by integrating over time from 0 to ∞:(21)q+12q+22uq+2x-2q+12uqx+q-12q2uq-2x+2ασq+12uq+1x-q2uq-1x+σ2β2uqx=σ2∂2∂x2uqx+σ2fqx,where(22)fqx=∫-Tf/2Tf/2fx,tϕ¯qt~dt.Discretize (21) using the central difference scheme in space; we have(23)q+12q+22uq+2xi-2q+12uqxi+q-12q2uq-2xi+2ασq+12uq+1xi-q2uq-1xi+σ2β2uqxi=σ2Δxi2uqi+1-2uqi+uqi-1+σ2fqxi.From (23), one could conclude that the variable of the q-order is connected with adjacent fields, from q-2 to q+2. In (23), uq(x)i is moved to the left side of the equation and simplified to(24)ηui=λσ2Δxi2ui+1+ui-1+σ2fi,where(25)η=2σ2Δxi2+σ2β2-122ασ121222-2ασ122σ2Δxi2+σ2β2-322ασ222232⋱⋱⋱Q-32Q-22-2ασQ-222σ2Δxi2+σ2β2-2Q-322ασQ-12Q-22Q-12-2ασQ-122σ2Δxi2+σ2β2-2Q-12Q×Q.In (24), [λ]=IQ×Q is an unit matrix, and [u]i=ui0,⋯,uiQ-2,uiQ-11×QT is a Q-tupple matrix of AH space for numerical results.Rewriting (24) with a nested matrix, we have(26)Au=Fq=0,1,2,…,where(27)A=η-σ2Δxi2λ-σ2Δxi2λη-σ2Δxi2λ-σ2Δxi2λη-σ2Δxi2λ⋱⋱⋱-σ2Δxi2λη.In (26), u is a combination of numerical result with all orders and all points. F represents source coefficients in the computational domain. [A] is a banded sparse matrix with Q submatrix elements.


### 3.2. Method’s Treatment of Boundary and Initial Conditions

The application of presented method to telegraph equation based on a concealment condition that the initial valve of the equation should be 0; that ish0(x)=h1(x)=0. For the nonhomogeneous initial condition, we have(28)wx,t=ux,t-h0x-th1x.For the Dirichlet boundary conditions, there are a couple of ways to deal with it. The point on the boundary would be regarded as a point source which changes over time in the first approach. Then we can expand w(a,t), w(b,t), and f(x,t) using AH basis functions, respectively. However, this kind of method has the drawback of complicated procedure and a little loss in accuracy. In this work, the second approach has been adopted in which the Dirichlet boundary conditions are homogenized. We apply(29)vx,t=wx,t-wa,t+wb,t-wa,tb-ax.Introducing (28) and (29) to (1)–(3), we can obtain a telegraph equation with the homogeneous boundary and initial conditions(30)vttx,t+2αvtx,t+β2vx,t=vxxx,t+Mx,t,a≤x≤b,t≥0,vx,0=0,vtx,0=0,a≤x≤b,va,t=0,vb,t=0,t≥0.Rewriting (22) and (26) we have(31)Mqx=∫-Tf/2Tf/2Mx,tϕ¯qt~dt,(32)Av=Mq=0,1,2,….To solve (32), the lower-upper (LU) decomposition scheme is applied to decompose [A] initially, and iteration method is used to obtain the numerical results. The LU decomposition of [A] is handled only once when starting calculation process. Finally, one can obtain the numerical result from the expansion coefficients as(33)ux,t=∑q=0Q-1uqxϕ¯qt~.Contrary to the other numerical approach, this presented method has an implicit relationship in each variables, which can be reflected in the sparse matrix [A]. One should notice that (32) is independent of the temporal testing procedure ϕ¯p(t~). And the matrix [A], which does not contain the order q, remains unchanged in the calculation process. Therefore, the temporal step does not affect the stability of this method any more, and then an unconditionally stable scheme is structured. In this presented method, the time step size is only applied to compute the AH coefficients of the source term due to (22) and (31), which has been done at the beginning of the computation. We can choose a relatively small value of time step to calculate (22) or (31) accurately and describe the process of propagation of electrical signals clearly, and this procedure does not increase the computation time.


## 3.1. Construction of Computing Matrix

The partial differential with respect tox can be written as(19)∂2u∂x2=∑p=0∞∂2∂x2upxϕ¯pt~.Rewriting the telegraph equation (1) using (9), (18), and (19), we can obtain(20)∑p=0∞up+2xp+22σ2p+12σ2-upx2p+12σ2+up-2xp-12σ2p2σ2·ϕ¯pt~+2α∑p=0∞up+1xp+12σ2-up-1xp2σ2ϕ¯pt~+β2∑p=0∞upxϕpt~=∑p=0∞∂2∂x2upxϕ¯pt~+∑p=0∞fpxϕ¯pt~.In (20), we introduced a Galerkin procedure [44] in order to eliminate the time variable terms ϕ¯p(t~). By multiplying each side of (20) by time-dependent term ϕ¯q(t~), we can obtain the following equation by integrating over time from 0 to ∞:(21)q+12q+22uq+2x-2q+12uqx+q-12q2uq-2x+2ασq+12uq+1x-q2uq-1x+σ2β2uqx=σ2∂2∂x2uqx+σ2fqx,where(22)fqx=∫-Tf/2Tf/2fx,tϕ¯qt~dt.Discretize (21) using the central difference scheme in space; we have(23)q+12q+22uq+2xi-2q+12uqxi+q-12q2uq-2xi+2ασq+12uq+1xi-q2uq-1xi+σ2β2uqxi=σ2Δxi2uqi+1-2uqi+uqi-1+σ2fqxi.From (23), one could conclude that the variable of the q-order is connected with adjacent fields, from q-2 to q+2. In (23), uq(x)i is moved to the left side of the equation and simplified to(24)ηui=λσ2Δxi2ui+1+ui-1+σ2fi,where(25)η=2σ2Δxi2+σ2β2-122ασ121222-2ασ122σ2Δxi2+σ2β2-322ασ222232⋱⋱⋱Q-32Q-22-2ασQ-222σ2Δxi2+σ2β2-2Q-322ασQ-12Q-22Q-12-2ασQ-122σ2Δxi2+σ2β2-2Q-12Q×Q.In (24), [λ]=IQ×Q is an unit matrix, and [u]i=ui0,⋯,uiQ-2,uiQ-11×QT is a Q-tupple matrix of AH space for numerical results.Rewriting (24) with a nested matrix, we have(26)Au=Fq=0,1,2,…,where(27)A=η-σ2Δxi2λ-σ2Δxi2λη-σ2Δxi2λ-σ2Δxi2λη-σ2Δxi2λ⋱⋱⋱-σ2Δxi2λη.In (26), u is a combination of numerical result with all orders and all points. F represents source coefficients in the computational domain. [A] is a banded sparse matrix with Q submatrix elements.


## 3.2. Method’s Treatment of Boundary and Initial Conditions

The application of presented method to telegraph equation based on a concealment condition that the initial valve of the equation should be 0; that ish0(x)=h1(x)=0. For the nonhomogeneous initial condition, we have(28)wx,t=ux,t-h0x-th1x.For the Dirichlet boundary conditions, there are a couple of ways to deal with it. The point on the boundary would be regarded as a point source which changes over time in the first approach. Then we can expand w(a,t), w(b,t), and f(x,t) using AH basis functions, respectively. However, this kind of method has the drawback of complicated procedure and a little loss in accuracy. In this work, the second approach has been adopted in which the Dirichlet boundary conditions are homogenized. We apply(29)vx,t=wx,t-wa,t+wb,t-wa,tb-ax.Introducing (28) and (29) to (1)–(3), we can obtain a telegraph equation with the homogeneous boundary and initial conditions(30)vttx,t+2αvtx,t+β2vx,t=vxxx,t+Mx,t,a≤x≤b,t≥0,vx,0=0,vtx,0=0,a≤x≤b,va,t=0,vb,t=0,t≥0.Rewriting (22) and (26) we have(31)Mqx=∫-Tf/2Tf/2Mx,tϕ¯qt~dt,(32)Av=Mq=0,1,2,….To solve (32), the lower-upper (LU) decomposition scheme is applied to decompose [A] initially, and iteration method is used to obtain the numerical results. The LU decomposition of [A] is handled only once when starting calculation process. Finally, one can obtain the numerical result from the expansion coefficients as(33)ux,t=∑q=0Q-1uqxϕ¯qt~.Contrary to the other numerical approach, this presented method has an implicit relationship in each variables, which can be reflected in the sparse matrix [A]. One should notice that (32) is independent of the temporal testing procedure ϕ¯p(t~). And the matrix [A], which does not contain the order q, remains unchanged in the calculation process. Therefore, the temporal step does not affect the stability of this method any more, and then an unconditionally stable scheme is structured. In this presented method, the time step size is only applied to compute the AH coefficients of the source term due to (22) and (31), which has been done at the beginning of the computation. We can choose a relatively small value of time step to calculate (22) or (31) accurately and describe the process of propagation of electrical signals clearly, and this procedure does not increase the computation time.


## 4. Numerical Examples

In this part, the following four examples of the telegraph equation with exact solution have been solved by the proposed scheme. To measure the accuracy and versatility of this proposed scheme, the followingL2, L∞, and root-mean-square (RMS) errors are reported:(34)L2=uexact-unum2=h∑j=0Nujexact-ujnum2,L∞=uexact-unum∞=maxjujexact-ujnum,RMS=∑j=0Nujexact-ujnum2N+1.Example 1.
We consider the telegraph equation:(35)uttx,t+2αutx,t+β2ux,t=uxxx,t+β2-2sinh⁡xcos⁡t-2αsinh⁡xsin⁡t,over a region Ω=[0≤x≤1]×[t>0], with initial conditions(36)ux,0=sinh⁡x,utx,0=0and boundary conditions(37)u0,t=0,u1,t=sinh⁡1cos⁡t.The exact solution of this equation isux,t=sinh⁡xcos⁡t. The results for different value of α and β obtained by using the proposed method are compared with those obtained by Mohanty [28], H.-W. Liu and L.-B. Liu [30] and L.-B. Liu and H.-W. Liu [12], which are presented in Tables 1 and 2. The graph of analytical and numerical result at x = 0.25, x = 0.5 and x = 0.75 is shown in Figure 1.Table 1
Comparison of RMS error in Example1 with λ=1.6 at t = 2 s.
Methods h α = 10, β=3 α = 100, β=3 α = 10, β=5 α = 100, β=5 Method in [28] 1/8 0.7316 ( - 03 ) 0.1094 ( - 02 ) 0.2252 ( - 02 ) 0.2636 ( - 02 ) 1/16 0.1826 ( - 04 ) 0.6757 ( - 04 ) 0.8022 ( - 04 ) 0.1708 ( - 03 ) 1/32 0.3995 ( - 05 ) 0.2023 ( - 05 ) 0.3351 ( - 05 ) 0.8244 ( - 05 )  Method in [30] 1/8 0.2528 ( - 03 ) 0.5542 ( - 02 ) 0.1196 ( - 02 ) 0.3135 ( - 02 ) 1/16 0.3620 ( - 04 ) 0.4652 ( - 04 ) 0.9672 ( - 04 ) 0.8555 ( - 03 ) 1/32 0.5196 ( - 04 ) 0.2305 ( - 04 ) 0.3528 ( - 05 ) 0.3215 ( - 04 )  Method in [12] 1/8 0.3044 ( - 03 ) 0.5616 ( - 03 ) 0.2079 ( - 03 ) 0.5047 ( - 03 ) 1/16 0.3792 ( - 04 ) 0.7000 ( - 04 ) 0.2535 ( - 04 ) 0.6291 ( - 04 ) 1/32 0.4709 ( - 05 ) 0.8741 ( - 05 ) 0.3109 ( - 05 ) 0.7854 ( - 05 )  Proposed method 1/8 8.3635 ( - 06 ) 1.2673 ( - 04 ) 6.9791 ( - 05 ) 6.3081 ( - 05 ) 1/16 3.5515 ( - 06 ) 2.0919 ( - 05 ) 8.1144 ( - 06 ) 8.3177 ( - 06 ) 1/32 8.3032 ( - 07 ) 1.1860 ( - 06 ) 8.4148 ( - 07 ) 4.4470 ( - 06 )Table 2
Comparison of RMS error in Example1 with λ=3.2 at t = 2 s.
Methods h α = 10, β=2 α = 100, β=2 α = 10, β=5 α = 100, β=5 Method in [28] 1 / 8 0.4116 ( - 02 ) 0.6115 ( - 02 ) 0.1149 ( - 01 ) 0.1737 ( - 01 ) 1 / 16 0.7773 ( - 04 ) 0.8872 ( - 03 ) 0.2174 ( - 02 ) 0.2570 ( - 02 ) 1 / 32 0.2591 ( - 04 ) 0.5430 ( - 04 ) 0.7906 ( - 04 ) 0.1684 ( - 03 )  Method in [30] 1 / 8 0.5803 ( - 02 ) 0.2435 ( - 02 ) 0.2568 ( - 02 ) 0.1258 ( - 01 ) 1 / 16 0.1306 ( - 03 ) 0.5218 ( - 03 ) 0.2294 ( - 02 ) 0.3035 ( - 02 ) 1 / 32 0.5256 ( - 04 ) 0.4042 ( - 04 ) 0.6335 ( - 04 ) 0.2087 ( - 03 )  Method in [12] 1 / 8 0.1557 ( - 02 ) 0.2330 ( - 02 ) 0.8488 ( - 03 ) 0.2022 ( - 02 ) 1 / 16 0.1952 ( - 03 ) 0.2900 ( - 03 ) 0.1040 ( - 03 ) 0.2519 ( - 03 ) 1 / 32 0.2431 ( - 04 ) 0.3621 ( - 04 ) 0.1268 ( - 04 ) 0.3144 ( - 04 )  Proposed method 1 / 8 7.8006 ( - 05 ) 6.6504 ( - 04 ) 2.8559 ( - 04 ) 4.6424 ( - 04 ) 1 / 16 1.5861 ( - 05 ) 1.4478 ( - 04 ) 1.5184 ( - 05 ) 7.7917 ( - 05 ) 1 / 32 3.1948 ( - 06 ) 5.0823 ( - 06 ) 1.7082 ( - 06 ) 2.9197 ( - 05 )Figure 1
Comparison of analytical and numerical result of Example1 at different space level with h=0.01, k=0.1.Example 2.
In this case, the telegraph equation (1) with α = 4, β = 2 over a region [0,π] with the following boundary and initial conditions was considered: (38)ux,0=sin⁡x,utx,0=-sin⁡x,0≤x≤π,u0,t=uπ,t=0,t≥0.We havef(x,t)=2-2α+β2e-tsin⁡x. And the exact solution of this example is ux,t=e-tsin⁡x. The L2, L∞, and RMS errors for t = 0.5 and 1.0 are presented in Table 3 with h = 0.02 and k = 0.0001. The errors are compared with the numerical results acquired in Dehghan and Shokri [8], Sharifi and Rashidinia [18] and Mittal and Bhatia [19]. The RMS errors for t = 3.0 with k = 0.01 are shown in Table 4. The numerical solutions by this proposed method are compared with the results obtained by [9] and another unconditionally stable method in [31]. The space-time graph of numerical result up to t = 2 is shown in Figure 2.Table 3
Errors in numerical result of Example2 with h = 0.02, k = 0.0001 at t = 0.5 and t = 1.
Methods t = 0.5 t = 1.0 L 2 L ∞ RMS error L 2 L ∞ RMS error Method in [8] 7.9491 ( - 05 ) 8.3721 ( - 06 ) 6.3239 ( - 06 ) 1.4554 ( - 04 ) 1.5680 ( - 05 ) 1.1579 ( - 05 ) Method in [18] 8.7500 ( - 06 ) 9.9900 ( - 06 ) 7.1600 ( - 06 ) 5.0700 ( - 06 ) 5.1900 ( - 06 ) 7.1700 ( - 06 ) Method in [19] 2.3328 ( - 06 ) 1.8612 ( - 06 ) — 4.3667 ( - 06 ) 3.4839 ( - 06 ) — Proposed method 6.7195 ( - 07 ) 7.5836 ( - 08 ) 5.3289 ( - 08 ) 1.0890 ( - 06 ) 1.2291 ( - 07 ) 8.6308 ( - 08 )Table 4
RMS errors of Example2 with k = 0.01 at t = 3.0.
h Linear RBF [9] Cubic RBF [9] Method in [31] Proposed method 0.05 3.0100 ( - 04 ) 7.1250 ( - 05 ) 6.0410 ( - 07 ) 3.6559 ( - 07 ) 0.02 7.1280 ( - 05 ) 1.7120 ( - 05 ) 6.0110 ( - 07 ) 1.4537 ( - 07 ) 0.01 4.3200 ( - 05 ) 8.2180 ( - 06 ) 6.0001 ( - 07 ) 7.3283 ( - 08 )Figure 2
Space-time graph of numerical result up tot=2, with h=0.02 and k=0.01 for Example 2.Example 3.
In this case, we consider telegraph equation (1) with α = 10, β = 5 over region [0,2], and the following boundary and initial conditions:(39)ux,0=tan⁡x2,utx,0=121+tan2⁡x2,0≤x≤2,u0,t=tan⁡t2,u2,t=tan⁡2+t2,t≥0.In this problem, we havef(x,t)=α[1+tan2((x+t)/2)]+β2tan((x+t)/2). And the analytical solution of the example is f(x,t)=tan⁡(x+t)/2. The L2, L∞, and RMS errors at t = 0.2 and t = 0.4 with h = 0.001, k = 0.001 are shown in Table 5. And also the numerical results are compared with Sharifi and Rashidinia [18] and Mittal and Bhatia [19]. The space-time graphs of numerical result with h = 0.02 and k = 0.01 up to t = 0.5 and 1.0 are presented in Figure 3.Table 5
Errors in numerical result of Example3 with h=0.001, k=0.001 at different time levels.
Methods Time L 2 L ∞ RMS error Method in [19] t = 0.2 2.1800 ( - 04 ) 3.6103 ( - 04 ) — Method in [19] t = 0.4 5.6618 ( - 04 ) 1.0368 ( - 04 ) — Method in [18] t = 0.2 9.9900 ( - 06 ) 6.8300 ( - 05 ) 6.0700 ( - 06 ) Method in [18] t = 0.4 7.0700 ( - 06 ) 4.2800 ( - 05 ) 5.2700 ( - 06 ) Proposed method t = 0.2 7.2429 ( - 06 ) 2.5596 ( - 07 ) 1.6187 ( - 07 ) Proposed method t = 0.4 6.2790 ( - 06 ) 4.9275 ( - 07 ) 1.4033 ( - 07 )Figure 3
Space-time graphs of numerical solution of Example3 with h=0.02 and k=0.01 up to t=0.5 (a) and t=1 (b).
(a) (b)Example 4.
We consider telegraph equation withα = 0.5 and β = 1 (40)uttx,t+utx,t+ux,t=uxxx,t+2-2t+t2x-x2e-t+2t2e-t,in the interval 0≤x≤1, with initial conditions(41)ux,0=0,utx,0=0and boundary conditions(42)u0,t=0,u1,t=0.The analytical solution of this equation isu(x,t)=x-x2t2e-t. The CPU time and L2, L∞, and RMS errors for t = 1 and 5 are shown in Table 6 with h = 0.01 and k = 0.001. The numerical solution is compared with results of [8, 19]. It can be concluded that numerical solutions obtained by this presented scheme are good in comparison with [8, 19]. Moreover, our method has a higher efficiency than the other two methods when the computation time gets longer. The space-time graph of numerical result is shown in Figure 4.Table 6
L 2, L∞, and RMS errors information in Example 4 with h=0.01, k=0.001.
Methods Time L 2 L ∞ RMS error CPU time (s) Method in [8] t = 1 1.4386 × 10−4 1.8479 × 10−5 1.4315 × 10−5 0 Method in [8] t = 5 7.5545 × 10−5 1.0455 × 10−5 7.5170 × 10−6 2 Method in [19] t = 1 4.5526 × 10−5 5.9153 × 10−5 — 0.43 Method in [19] t = 5 3.0161 × 10−6 5.2032 × 10−6 — 1.46 Proposed method t = 1 1.7961 × 10−6 3.4326 × 10−7 1.7784 × 10−7 0.34 Proposed method t = 5 1.4776 × 10−6 2.2537 × 10−7 1.4631 × 10−7 0.35Figure 4
Space-time graph of numerical result up tot = 2, with h = 0.04 and k = 0.01 for Example 4.


## 5. Conclusion

In the literature, we have constructed a new unconditionally stable method to solve the telegraph equation. Since the time variables have been eliminated from computations, the convergence and precision of the presented method are independent of the temporal step. To demonstrate the stability and accuracy of this proposed method, four numerical examples are conducted and compared with the numerical results available in previous literatures. The comparison numerical results reveal the unconditional stability and high precision of the presented scheme in the current literature.The paper has provided a new mentality for solving1d telegraph equation and this method is now extended to multidimensional cases and other kinds of partial differential equations in a future study.



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