# Bi-Finite Difference Method to Solve Second-Order Nonlinear Hyperbolic Telegraph Equation in Two Dimensions

Authors: K.R. Raslan; Khalid. K Ali; Hind K Al-Jeaid; M.A. Shaalan
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This study introduces a computational scheme by the bi-finite difference method (Bi-FDM) to solve the hyperbolic telegraph equation in two dimensions. The proposed numerical method converts nonlinear two-dimensional hyperbolic telegraph equation of second order to difference equations that can be solved by the Mathematica program. Consistency and stability of the proposed scheme are discussed and found to be accurate ofOhx2+hy2+Δτ2 and conditionally stable, respectively. The efficiency and accuracy of Bi-FDM have been shown by comparing the numerical results of the presented problems with the exact solutions and other numerical techniques.


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## Body

## 1. Introduction

Forδ>0 and γ>0, we consider the following second-order nonlinear hyperbolic telegraph equation in two dimensions on region R=x,y:x∈0,1,y∈0,1 fVττ=Vxx+Vyy−2δVτ−γ2V+Gx,y,τ,Vor temporal coordinate with seconds τ>0 [20]:(1)Vττ=Vxx+Vyy−2δVτ−γ2V+Gx,y,τ,V,with the initial conditions,(2)Vx,y,0=f1x,y,Vτx,y,0=f2x,y,x∈0,1,y∈0,1,and the boundary conditions,(3)V0,y,τ=f3y,τ,V1,y,τ=f4y,τ,y∈0,1,τ>0,Vx,0,τ=f5x,τ,Vx,1,τ=f6x,τ,x∈0,1,τ>0.Equation (1) is in spatial-temporal spaces and represents damped wave equation at δ>0andγ=0 and telegraph equation at δ>0andγ>0. The two-dimensional nonlinear hyperbolic telegraph (1) of second-order under conditions (2) and (3) arises in electric voltage and current in a double conductor, signal analysis, wave propagation, random walk theory, and atomic physics [1–4]. Also, the hyperbolic partial differential equations are useful in explaining and understanding structures vibrations such as buildings, beams, and machines [5]. So, most previous branches of science can be studied by telegraph equation which is more appropriate in modeling reaction-diffusion than ordinary diffusion equation. In most of the literature, the numerical schemes for hyperbolic telegraph equations of second order in one dimension and multidimensions recently developed it as follows. Mohanty et al. studied the stability of numerical schemes of telegraph equations [6–8]. Hyperbolic telegraph equations of one dimension are solved by cubic and quartic B-spline collocation methods [9–11]. In [12, 13], Mohanty et al. used high-order approximation to solve two-dimensional quasilinear hyperbolic equations. Many authors solved two-dimensional linear hyperbolic equations [14–18]. Two-dimensional nonlinear hyperbolic equations of second order are studied using modified B-spline method [19], bi-cubic B-spline method [20], meshless local and stronge forms [21], and high-order difference schemes [22]. Smith introduced finite difference methods [23]. Raslan and Ali used the finite difference method to solve MHD-duct flow model [24]. Baleanu et al. applied a nonstandard finite difference scheme for fractional chaotic system [25]. Erturk et al. found some electrical engineering applications as capacitor microphone system [26] and description motion of beam on nanowire [27]. Hasan et al. [28] presented formulations of fractional optimal control of two spherical structures and one hollow cylindrical structure which represented fractional partial differential equations in two dimensions and can be solved via our presented scheme and will discuss it in the future work.The present study is organized as follows. In Section2, we present the bi-finite difference method of the proposed problem. We discuss the consistency and stability of the proposed scheme as given in Sections 3 and Section 4, respectively. Section 5 investigates some numerical problems and explains them in some tables and figures. Finally, Section 6 provides the conclusion for this paper.


## 2. Bi-Finite Difference Method

We divide the regionR into knots xi,yj, for i=0,1,…,n and j=0,1,…,m, in spatial directions with horizontal space in x− direction hx=xi−xi−1=1/n, i=1,2,…,n, and vertical space in y− direction hy=yj−yj−1=1/m, j=1,2,…,m, where n and m are + ve integers. Suppose that temporal direction with spacing Δτ>0 is τk=k∗Δτ, for k=0,1,…,l, where l is +ve integer. The finite difference method in two dimensions to find the approximate solution vx,y,τ corresponding to the exact solution Vx,y,τ can be written as(4)vxi,yj,τk=ci,jk,and its partial derivative up to second order in spatial and temporal directions at various grid points xi,yj,τk is as follows:(5)vxxi,yj,τk≃ci+1,jk−ci−1,jk2hx,vxxxi,yj,τk≃ci+1,jk+ci−1,jk−2ci,jkhx2,vyxi,yj,τk≃ci,j+1k−ci,j−1k2hy,vyyxi,yj,τk≃ci,j−1k+ci,j+1k−2ci,jkhy2,vτxi,yj,τk≃ci,jk+1−ci,jk−12Δτ,vττxi,yj,τk≃ci,jk+1+ci,jk−1−2ci,jkΔτ2.Equation (1), at points xi,yj,τk, for i=0,1,…,n, j=0,1,…,m, and k=0,1,…,l, where Vxi,yj,τk, is written as Vi,jk:(6)Vττi,jk=Vxxi,jk+Vyyi,jk−2δVτi,jk−γ2Vi,jk+Gi,jk.Conditions (2) and (3) become(7)Vi,j0=f1xi,yj,Vτi,j0=f2xi,yj,V0,jk=f3yj,τk,Vn,jk=f4yj,τk,Vi,0k=f5xi,τk,Vi,mk=f6xi,τk.Substituting (5) into (6), we obtain as follows:(8)ci,jk+1+ci,jk−1−2ci,jkΔτ2=ci+1,jk+ci−1,jk−2ci,jkhx2+ci,j−1k+ci,j+1k−2ci,jkhy2−2δci,jk+1−ci,jk−12Δτ−γ2ci,jk+Gi,jk,where (8) with conditions (7) can be solved.


## 3. Local Truncation Error

In this section, we introduce the local truncation error of our scheme.Theorem 1.
The local truncation error of finite difference scheme (8) is Ohx2+hy2+Δτ2.Proof.
By using Taylor’s expansion in (8), we investigate the local truncation error in two-dimensional space and time as follows:(9)Tci,jk=∂2ci,jk∂t2−∂2ci,jk∂x2−∂2ci,jk∂y2+2δ∂ci,jk∂t+γ2ci,jk−Gi,jk+hx212∂4ci,jk∂x4+hy212∂4ci,jk∂y3+δΔτ23∂3ci,jk∂t3+Δτ212∂4ci,jk∂t4+…,(10)Tci,jk=hx212∂4ci,jk∂x4+hy212∂4ci,jk∂y3+δΔτ23∂3ci,jk∂t3+Δτ212∂4ci,jk∂t4+⋯.
Hence,Tci,jk⟶0 as hx2,hy2,Δτ2⟶0.
Then, the local truncation error of finite difference scheme (8) is (11)Ohx2+hy2+Δτ2.


## 4. Stability of Bi-Finite Difference Scheme

In this section, we test the stability of the difference scheme.Theorem 2.
The difference scheme (8) is conditionally stable.Proof.
Firstly, supposeci,jk=ζkeIϑihxeIφjhy, and substituting with it in (8), we use the following linearization of the nonlinear term Gi,jk=MVi,jk, where M is locally constant; we obtain(12)ζk+1eIϑihxeIφjhy+ζk−1eIϑihxeIφjhy−2ζkeIϑihxeIφjhy=Δτhx2ζkeIϑi+1hxeIφjhy+ζkeIϑi−1hxeIφjhy−2ζkeIϑihxeIφjhy+Δτhy2ζkeIϑihxeIφj+1hy+ζkeIϑihxeIφj−1hy−2ζkeIϑihxeIφjhy−δ∗Δτζk+1eIϑihxeIφjhy−ζk−1eIϑihxeIφjhy−γ2−MΔτ2ζkeIϑihxeIφjhy,where I=−1, and we divided equation (12) by ζk−1eIϑihxeIφjhy as(13)1+δ∗Δτζ2+Bζ2+1−δ∗Δτ=0,where(14)B=−2+γ2−MΔτ2+2Δτhx21−cosϑhx+2Δτhy21−cosφhy.
Then,(15)ζ=−B±B2+4δ2Δτ2−421+δΔτ≤1.
IfB⩽1+δ∗Δτ and δ∗Δτ⩽1, so, the bi-finite difference scheme is conditionally stable under the condition B⩽2.


## 5. Numerical Problems

In this section, we test some numerical experiments for several two-dimensional hyperbolic telegraph problems using the bi-finite difference scheme (Bi-FDM). We calculateL∞-errors, L2-errors, root mean square errors (RMSE), and relative errors (RE) to show the efficiency of the proposed numerical scheme by comparing the exact solutions and the numerical solutions in our presented method and other different methods. The formulas of the used errors in two-dimensional space at different levels τ are given by [16](16)L∞−error=maxi,j=0n,mVi,jk−vi,jk,L2−error=hx∗hy∑i=0n∑j=0mVi,jk−vi,jk2,Root mean square errorRMSE=∑i=0n∑j=0mVi,jk−vi,jk2n∗m,Relative errorRE=∑i=0n∑j=0mVi,jk−vi,jk2∑i=0n∑j=0mVi,jk2.Problem 1.
We consider two-dimensional second-order nonlinear hyperbolic telegraph equation [20]:(17)Vττ−Vxx−Vyy+2δVτ+γ2V−2cosτ−sinτsinxsiny=0.
Subject to the initial and boundary conditions, forx,y∈0,1 and τ⩾0,(18)Vx,y,0=sinxcosy,Vτx,y,0=0,V0,y,τ=0,V1,y,τ=sin1cosycosτ,Vx,0,τ=sinxcosτ,Vx,1,τ=sinxcos1cosτ.and the exact solution is Vx,y,τ=sinxcosycosτ, where δ=γ=1.
KeepingΔτ=0.01, Table 1 presents L2,L∞ errors and relative error for equal grid sizes hx=hy=0.1, while Table 2 shows the same errors for different grid sizes hx=0.1andhy=0.2 at different time levels. Our results are compared with the obtained results in [19, 20]. Obviously, our numerical results are preferable. Figure 1 and 2 show the surface plots of numerical and exact solution at τ=1,3 for hx=hy=0.05andΔτ=0.001, respectively. The maximum absolute error plots are observed that, with increase in time, the errors are decreasing, as shown in Figure 3.Table 1
L2,L∞, and relative errors at different time levels for Problem 1.
τL2 error (Bi-FDM)L∞ (E)error (Bi-FDM)R (E)E (Bi-FDM)L2 (E)[20]L∞ (E)[20]R (E)E [19]11.197–52.1390–57.1789–52.2277–45.6153–45.976E-324.129–68.9229–63.2132–51.0859–43.9413–48.501E-341.091–52.1178–55.4081–52.7579–46.7468–4- - - - -71.229–52.3968–55.2785–53.1130–47.7897–43.157E-3101.346–52.6265–55.1976–53.4167–48.6373–42.287E-3Table 2
L2,L∞, and relative errors for Problem 1 at different time levels with hx=0.1,hy=0.2, and.Δτ=0.01.
τL2 error (Bi-FDM)L∞ error (Bi-FDM)RE (Bi-FDM)L2 [20]L∞ [20]13.0401E-055.3037E-051.7200E-041.2561E-044.0150E-0433.8642E-057.5546E-051.1931E-042.4419E-047.1428E-0457.6946E-061.5049E-058.2919E-055.7786E-051.9111E-04Figure 1
Numerical and exact surface plots atτ=1 for Problem 1.Figure 2
Numerical and exact surface plots atτ=3 for Problem 1.Figure 3
Maximum error plots atτ=5,7,10 and x=0.5 with hx=hy=0.1andΔτ=0.01 for Problem 1.Problem 2.
We consider two-dimensional second-order nonlinear hyperbolic telegraph equation [20]:(19)Vττ−Vxx−Vyy+2δVτ+γ2V−−2δ+γ2−1e−τsinhxsinhy=0,with the initial and boundary conditions, for x,y∈0,1 and τ⩾0,(20)Vx,y,0=sinhxsinhy,Vτx,y,0=−sinhxsinhy,V0,y,τ=0,V1,y,τ=e−τsinh1coshy,Vx,0,τ=0,Vx,1,τ=e−τsinhxsinh1.
The exact solution isVx,y,τ=e−τsinhxsinhy, where δ=10 and γ=5.
Root mean square error and relative error for equal grid sizeshx=hy=0.1 and Δτ=0.01 are calculated at different time levels, as shown in Table 3. Comparison of obtained results with results of other methods in [19, 20] display that our numerical results are effective. Figure 4 appears the surface plots of numerical and exact solution at τ=2 for hx=hy=0.05andΔτ=0.001. Figure 5 presents error plots at various time levels. We find the maximum absolute errors are increasing with time increasing.Table 3
Root mean square and relative errors at different time levels for Problem2.
τRMSE (Bi-FDM)RE (Bi-FDM)RMSE [20]RE [19]0 (E).56.2971E-062.1684E-052.6594E-041.3787–515.5545E-063.1535E-051.9039E-043.5957–539.3544E-073.9242E-052.8681E-051.0266–651.2770E-073.9585E-053.8988E-067.0735–5108.6081E-103.9601E-052.6275E-081.6514–5Figure 4
Numerical and exact surface plots atτ=2 for Problem 2.Figure 5
Maximum error plots atτ=5,7,10 and x=0.5 with hx=hy=0.1andΔτ=0.01 for Problem 2.Problem 3.
We consider two-dimensional second-order nonlinear hyperbolic telegraph equation [20]:(21)Vττ−Vxx−Vyy+2δVτ+γ2V+3cosτ+2δsinτ−γ2cosτsinhxsinhy=0,with the initial and boundary conditions, for x,y∈0,1 and τ⩾0,(22)Vx,y,0=sinhxsinhy,Vτx,y,0=0,V0,y,τ=0,V1,y,τ=sinh1sinhycosτ,Vx,0,τ=0,Vx,1,τ=sinhxsinh1cosτ.
The exact solution isVx,y,τ=sinhxsinhycosτ, where δ=10 and γ=5.
We tabulate root mean square errors and relative errors forhx=hy=0.05andΔτ=0.001 at different time levels in Table 4. The acquired results are better than acquired results in [19, 20]. Surface plots of and exact and numerical solution at τ=1 for hx=hy=0.05andΔτ=0.001 are as observed in Figure 6.Table 4
Root mean square and relative errors at different time levels for Problem3.
τRMSE (Bi-FDM)RE (Bi-FDM)RMSE [20]RE [19]0 (E).51.6828E-064.3384E-068.9921E-051.3787–511.6932E-067.0904E-066.8547E-053.5957–522.3164E-071.2593E-062.9487E-054.4722–552.8018E-072.2346E-061.8225E-057.0735–5Figure 6
Numerical and exact surface plots atτ=1 for Problem 3.Problem 4.
We consider two-dimensional second-order nonlinear hyperbolic telegraph equation [20]:(23)Vττ−Vxx−Vyy+2δVτ+γ2V+2ex+y−τ=0.
Subject to the initial and boundary conditions, forx,y∈0,1 and τ⩾0,(24)Vx,y,0=ex+y,Vτx,y,0=−ex+y,V0,y,τ=ey−τ,V1,y,τ=e1+y−τ,Vx,0,τ=ex−τ,Vx,1,τ=ex+1−τ.
The exact solution isVx,y,τ=ex+y−τ, where δ=γ=1.
We computeL2,L∞ norms and relative errors for equal grid sizes hx=hy=0.05 and Δτ=0.001 at different time levels, as shown in Table 5. Table 5 clarifies our obtained results are in good agreement with results in [19, 20]. Figure 7 illustrates surface plots of numerical solution with exact solution at τ=1, for hx=hy=0.1andΔτ=0.01. Figure 8 shows error plots are decreasing versus time increasing at various time levels for hx=hy=0.1,Δτ=0.01 and hx=hy=0.05,Δτ=0.001, respectively.Table 5
L2,L∞, and relative errors at different time levels for Problem 4.
τL2 error (Bi-FDM)L∞ error (Bi-FDM)RE (Bi-FDM)L2 (E)[19]L∞ (E)[20]R (E)E [19]1 (E)2.309E-53.786E-51.8424–53.2351–31.0000–31.2906–439.048E-72.458E-65.3347–63.1028–41.3654–49.1555–556.557E-71.347E-62.8567–52.4495–52.8590–55.3354–5103.940E-108.052E-102.5475–63.6505–69.2105–81.1812–5Figure 7
Numerical and exact surface plots atτ=1 for Problem 4.Figure 8
Maximum error plots atτ=2,3,4 and x=0.6 for Problem 4.Problem 5.
We consider two-dimensional second-order nonlinear hyperbolic telegraph equation [20]:(25)Vττ−Vxx−Vyy+2δVτ+γ2V−1+γ2x2y2sinhτ−2δx2y2coshτ+2x2+y2sinhτ+sinx2y2sinhτ−sinV=0,with the initial and boundary conditions, for x,y∈0,1 and τ⩾0,(26)Vx,y,0=0,Vτx,y,0=x2y2,V0,y,τ=0,V1,y,τ=y2sinhτ,Vx,0,τ=0,Vx,1,τ=x2sinhτ,and the exact solution is Vx,y,τ=x2y2sinhτ, where δ=10 and γ=50.
We evaluateL2,L∞ errors for hx=hy=0.1andΔτ=0.01 at different time levels. We get precise results by comparing with results in [20] as in Table 6, and Figure 9 clears surface plots of numerical and exact solution at τ=1 for hx=hy=0.1andΔτ=0.01.Table 6
L2 and L∞ errors at different time levels for Problem 5.
τL2 error (Bi-FDM)L∞ (E)error (Bi-FDM)L2 (E)[20]L∞ [20]13.0796–81.2502–76.2420E-052.3975E-0427.5325–83.0573–71.8389E-047.3788E-0432.0166–78.1908–75.1608E-042.7000E-0351.4865–66.0327–63.6000E-032.0900E-02Figure 9
Numerical and exact surface plots for Problem5 at τ=1.


## 6. Conclusion

This study employed the bi-finite difference method (Bi-FDM) as a collocation method to solve a nonlinear hyperbolic telegraph equation of second order in two dimensions. The local truncation error of the proposed scheme is derived. The numerical scheme has conditionally stable. Five different forms of the equation under study were studied to verify the accuracy of the proposed method, and theirL2,L∞ norms were calculated. According to the obtained results, the proposed numerical scheme is efficient and in good agreement with other works in [19, 20]. In addition, the Bi-FDM allows us to produce an extremely better estimation of the hidden aspects for a system of the different applications under study.



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