Comparison of Numerical Solution of 1D Hyperbolic Telegraph Equation using B-Spline and Trigonometric B-Spline by Differential Quadrature Method

Objectives: This paper aims to compute the approximate solution of one dimensional (1D) hyperbolic telegraph equation with appropriate primary and limiting conditions. Methods/Statistical Analysis: To find the approximate solution, two different modified spline basis function are used with the differential quadrature method and splines are used to compute the weighting coefficients and thus the equation is transformed to a set of first order conventional differential equations which is further solved by the SSP-RK43 method. Three test problems of this equation are simulated to establish the precision and usefulness of the proposed scheme. Findings: The obtained numerical results are found to be good in terms of accuracy, efficiency and simplicity. To validate the computed results using proposed scheme, various comparisons at different time levels has been done in the form of and errors. These errors are compared and enlisted in the form of tables with computed errors enlisted in literature. Application/Improvements: Being an important equation of nuclear material science, one dimensional (1D) hyperbolic telegraphs equation needs to take care in the sense of better numerical solution. In this context, a successful effort has been done in this research article by proposing a hybrid numerical scheme.


Introduction
The telegraph equation is utilized to outline the reaction diffusion in numerous branches of emerging sciences. This mathematical equation premised for crucial equations of nuclear material science. It is used to demonstrate the vibrations of structures, e.g. structures, shafts, and machines. It likewise emerges in the investigation of throb blood stream in supply routes, in the 1D irregular movement of bugs along a hedge 1,2 and also play a significant part in demonstrating of numerous appropriate problems like signal investigation 3 , wave propagation 4 , random walk theory 5 etc. This mathematical equation is regularly utilized as a part of signal investigation for transmission and proliferation of electrical signs 6 furthermore has applications in different fields 7 . This equation is given as: With initial or starting conditions (ICs) , ] , Eq. (1) with the coefficients stand for a partial differential equation which is actually a damped wave equation and correspond to the telegraph mathematical equation if . The telegraph equation is likewise adverted to show the combination of dissemination and wave proliferation by the properties of constrained velocity to standard warmth or mass transport condition 1 .
In past few years, many numerical schemes to solve telegraph equation have been developed by many researchers. The existence of its double periodic solution was investigated in 8 . The unconditionally stable schemes for telegraph Eq. (1) was used in 9-11 , etc. Further, different conditionally stable finite difference schemes were implemented to solve the telegraph equation [12][13][14] . In 15 finite difference for approximation of spatial derivatives and the time derivative handled by collocation scheme is used. Chebyshev cardinal functions were used for numerical simulation of the equation in 16 . Various other numerical schemes were also developed or implemented to solve the telegraph equation, Some of them are: method using interpolating scaling functions 17 , Chebyshev Tau method 18 , Rothe-wavelet method 1 , semi-discretion methods 19 , explicit difference method 14 , DRBIE method 20 , collocation scheme along thin plate splines radial basis function (RBFCM) 21 , Polynomial Differential Quadrature Method (PDQM) 22 , Cubic B-spline Quasi-interpolation (CBQ) 23 , cubic B-spline collocation method (CBCM) 24 , Quartic B-spline Collocation Method (QBCM) 25 , and Collocation Method based on Modified Cubic B-spline (CMMCB) 26 , etc. This article is anxious with an approximate solution of 1D hyperbolic PDE with suitable starting and limiting conditions acquired by utilizing altered cubic B-spline and trigonometric B-spline with differential quadrature strategy, consequently alluded as (MCB-DQM) and (MTCB-DQM) 27 . The proposed method depends on differential quadrature technique with cubic-B-spline and trigonometric B-spline basis function. Eq. (1) is initially changed over into an arrangement of coupled Partial Differential Equations (PDEs) utilizing the transformation: . The MCB-DQM and MTCB-DQM are applied to change the PDEs into a system of first order ODEs, in time, which is solved by utilizing the SSP-RK43 scheme 28 . Three test problems of this equation have been considered to exhibit the exactness and utility of the proposed technique. The maximum absolute error and error in the MCB-DQM and MTCB-DQM approximate solution have been analogized with the error calculated by other numerical schemes accessible in the literature. The numerical results are seen to be in awesome simultaneousness with the exact solution.
The article is partitioned into various sections, in Section 2, the description of the modified cubic B-spline and modified trigonometric B-spline with Differential Quadrature Method (DQM) is given. Sections 3, discusses the procedure to utilize the proposed scheme. Three numerical test problems are given in to fabricate the significance and precision of the proposed technique in Section 4. In last section 5, summery of the paper is given in form of conclusion.

Description of the Method
The DQM was initially proposed by 29 . This strategy has been effectively connected to understand different one and two dimensional differential equations by utilizing various basis functions [30][31][32][33] .
In DQM the derivatives of some function which are present in the given PDE are replaced with their approximate values at desired different points. Because of the dependence on weight coefficients to domain grid points, we consider N equidistance nodal points on the real axis, that is with The solution at knot Is denoted by for . The approximate values of spatial derivatives are given as:

Differential Quadrature Method uses B-Spline
As B-spline have the certain nice properties like smoothness and competence to handle indigenous singularities, as B-spline basis functions are easy to implement so many researchers used cubic B-spline basis function to compute the approximate solution of physical models 26,27 definedas follows:  6) is reduced to linear equations as for (7) and represented as: weighting coefficient vector with respect to the points , are represented as , that is and the coefficient vector with respect to are evaluated as:

Differential Quadrature Method using Trigonometric B-Spline
The cubic trigonometric B-spline basis function for is characterized as: where , , and and (8) and is cubic trigonometric B-spline basis function with some geometric properties like Continuity, non-negativity and partition of unity.
The values of , and are given by Table 1, where: Both above defined basis functions can be improved so that the resulting matrix becomes diagonally dominant. The modification of these functions can be done as follows: 27 where form a basis on

Implementation of the Method
Using the transformation: the Eq. (1) is transformed to a set of PDEs as follows: (10) On substituting the values for the second order approximation of the space derivatives, obtained from MCB-DQM and MTCB-DQM, Eq. (10) rewritten as: for .
Hence, Eq. (11) reduces into a coupled system of firstorder ODEs in time, that is, and (12) where L represents the right hand side of ODEs.
The equation is solved subject to the BCs as defined in (3), and the ICs. (13) The resulting set of ODEs are solved by using SSP-RK43 scheme 28 given below: and consequently the solutions at the required time level are obtained.

Numerical Experiments
For numerical discussion, three test problems are considered to obtain the approximate solutions by MCB-DQM and MTCB-DQM. The and error norms are calculated using the exact solution and formulas to calculate these errors are defined in Eq. (14). and (14) where represent the numerical solution at node .  21 , and are reported in Table 2. It is evident that the errors are decreasing with increment in time, and the numerical results are more accurate than numerical solutions obtained by CMMCB 26 and RBFCM 21 . The physical conduct of solutions obtained by MCB-DQM and MTCB-DQM at various time levels is depicted in Figure 1.   Table  3. It is evident that the errors are decreasing as time increases (also, by decreasing the values of h), and computed results are more accurate than the results obtained in 21 Table 4 and it can be seen that the obtained results are better than the results given in 21,25,26 . The physical behaviour of the MCB-DQM and MTCB-DQM solutions at various time levels are shown in Figure 4.  23,25,26 and are reported in Table 5. It is evident from Table 5

Conclusion
Solution of the hyperbolic telegraph equation is obtained numerically in this paper using two different schemes named as, MCB-DQM and MTCB-DQM. These schemes are based on the DQM combined with modified cubic B-spline and modified cubic trigonometric B-spline as basic functions. On implementing the schemes and substituting the derivatives, set of ODEs are attained, which is solved using SSP RK43. The efficiency and precision of the proposed method is revealed by three test problems. The numerical result, and errors are compared with numerical solutions from literature and are found to be in decent agreement with formerly obtained results. The advantage of the developed methods is there ease to implement and reduced data complexity as compared to the present schemes.