Differential Quadrature Solution of Hyperbolic Telegraph Equation

Differential quadrature method (DQM) is proposed for the numerical solution of one- and two-space dimensional hyperbolic telegraph equation subject to appropriate initial and boundary conditions. Both polynomial-based differential quadrature (PDQ) and Fourier-based differential quadrature (FDQ) are used in space directions while PDQ is made use of in time direction. Numerical solution is obtained by using Gauss-Chebyshev-Lobatto grid points in space intervals and equally spaced and/or GCL grid points for the time interval. DQM in time direction gives the solution directly at a required time level or steady state without the need of iteration. DQM also has the advantage of giving quite good accuracy with considerably small number of discretization points both in space and time direction

High-Order Numerical Solution of Second-Order One-Dimensional Hyperbolic Telegraph Equation Using a Shifted Gegenbauer Pseudospectral Method

We present a high-order shifted Gegenbauer pseudospectral method (SGPM) to solve numerically the second-order one-dimensional hyperbolic telegraph equation provided with some initial and Dirichlet boundary conditions. The framework of the numerical scheme involves the recast of the problem into its integral formulation followed by its discretization into a system of well-conditioned linear algebraic equations. The integral operators are numerically approximated using some novel shifted Gegenbauer operational matrices of integration. We derive the error formula of the associated numerical quadratures. We also present a method to optimize the constructed operational matrix of integration by minimizing the associated quadrature error in some optimality sense. We study the error bounds and convergence of the optimal shifted Gegenbauer operational matrix of integration. Moreover, we construct the relation between the operational matrices of integration of the shifted Gegenbauer polynomials and standard Gegenbauer polynomials. We derive the global collocation matrix of the SGPM, and construct an efficient computational algorithm for the solution of the collocation equations. We present a study on the computational cost of the developed computational algorithm, and a rigorous convergence and error analysis of the introduced method. Four numerical test examples have been carried out in order to verify the effectiveness, the accuracy, and the exponential convergence of the method. The SGPM is a robust technique, which can be extended to solve a wide range of problems arising in numerous applications.Comment: 36 pages, articl

Numerical solution of second order linear hyperbolic telegraph equation

This paper is of about a numerical solution of the second order linear hyperbolic telegraph equation. To solve numerically the second order linear hyperbolic telegraph equation, the cubic B-spline collocation method is used in space discretization and the fourth order one-step method is used in time discretization. By using the fourth order one-step method, it is aimed to obtain a numerical algorithm whose accuracy is higher than the current studies. The efficiency and accuracy of the proposed method is studied by two examples. The obtained results show that the proposed method has higher accuracy as intended.This work has been supported by the Scientific Research Council of Eskisehir Osmangazi University under project No. 2018-2090.Publisher's Versio

Integration of the hyperbolic telegraph equation in (1+1) dimensions via the generalized differential quadrature method

AbstractThe 2D generalized differential quadrature method (hereafter called ((1+1)-GDQ) is introduced within the context of dynamical system for solving the hyperbolic telegraph equation in (1+1) dimensions. Best efficiency is obtained with a low-degree polynomial (n⩽8) for both time variable and x-direction. From realistic examples, some models are presented to illustrate an excellent performance of the proposed method, compared with the exact results

A finite difference method for Piecewise Deterministic Processes with memory

In this paper the numerical approximation of solutions of Liouville-Master Equations for time-dependent distribution functions of Piecewise Deterministic Processes with memory is considered. These equations are linear hyperbolic PDEs with non-constant coefficients, and boundary conditions that depend on integrals over the interior of the integration domain. We construct a finite difference method of the first order, by a combination of the upwind method, for PDEs, and by a direct quadrature, for the boundary condition. We analyse convergence of the numerical solution for distribution functions evolving towards an equilibrium. Numerical results for two problems, whose analytical solutions are known in closed form, illustrate the theoretical finding.Comment: 20 pages, 7 figure

JDNN: Jacobi Deep Neural Network for Solving Telegraph Equation

In this article, a new deep learning architecture, named JDNN, has been proposed to approximate a numerical solution to Partial Differential Equations (PDEs). The JDNN is capable of solving high-dimensional equations. Here, Jacobi Deep Neural Network (JDNN) has demonstrated various types of telegraph equations. This model utilizes the orthogonal Jacobi polynomials as the activation function to increase the accuracy and stability of the method for solving partial differential equations. The finite difference time discretization technique is used to overcome the computational complexity of the given equation. The proposed scheme utilizes a Graphics Processing Unit (GPU) to accelerate the learning process by taking advantage of the neural network platforms. Comparing the existing methods, the numerical experiments show that the proposed approach can efficiently learn the dynamics of the physical problem

Numerical Integral Transform Methods for Random Hyperbolic Models with a Finite Degree of Randomness

[EN] This paper deals with the construction of numerical solutions of random hyperbolic models with a finite degree of randomness that make manageable the computation of its expectation and variance. The approach is based on the combination of the random Fourier transforms, the random Gaussian quadratures and the Monte Carlo method. The recovery of the solution of the original random partial differential problem throughout the inverse integral transform allows its numerical approximation using Gaussian quadratures involving the evaluation of the solution of the random ordinary differential problem at certain concrete values, which are approximated using Monte Carlo method. Numerical experiments illustrating the numerical convergence of the method are included.This work was partially supported by the Ministerio de Ciencia, Innovacion y Universidades Spanish grant MTM2017-89664-P.Casabán, M.; Company Rossi, R.; Jódar Sánchez, LA. (2019). Numerical Integral Transform Methods for Random Hyperbolic Models with a Finite Degree of Randomness. Mathematics. 7(9):1-21. https://doi.org/10.3390/math7090853S12179Casabán, M.-C., Company, R., Cortés, J.-C., & Jódar, L. (2014). Solving the random diffusion model in an infinite medium: A mean square approach. Applied Mathematical Modelling, 38(24), 5922-5933. doi:10.1016/j.apm.2014.04.063Casabán, M.-C., Cortés, J.-C., & Jódar, L. (2016). Solving random mixed heat problems: A random integral transform approach. Journal of Computational and Applied Mathematics, 291, 5-19. doi:10.1016/j.cam.2014.09.021Casaban, M.-C., Cortes, J.-C., & Jodar, L. (2018). Analytic-Numerical Solution of Random Parabolic Models: A Mean Square Fourier Transform Approach. Mathematical Modelling and Analysis, 23(1), 79-100. doi:10.3846/mma.2018.006Saadatmandi, A., & Dehghan, M. (2010). Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method. Numerical Methods for Partial Differential Equations, 26(1), 239-252. doi:10.1002/num.20442Weston, V. H., & He, S. (1993). Wave splitting of the telegraph equation in R 3 and its application to inverse scattering. Inverse Problems, 9(6), 789-812. doi:10.1088/0266-5611/9/6/013Jordan, P. M., & Puri, A. (1999). Digital signal propagation in dispersive media. Journal of Applied Physics, 85(3), 1273-1282. doi:10.1063/1.369258Banasiak, J., & Mika, J. R. (1998). Singularly perturbed telegraph equations with applications in the random walk theory. Journal of Applied Mathematics and Stochastic Analysis, 11(1), 9-28. doi:10.1155/s1048953398000021Kac, M. (1974). A stochastic model related to the telegrapher’s equation. Rocky Mountain Journal of Mathematics, 4(3), 497-510. doi:10.1216/rmj-1974-4-3-497Iacus, S. M. (2001). Statistical analysis of the inhomogeneous telegrapher’s process. Statistics & Probability Letters, 55(1), 83-88. doi:10.1016/s0167-7152(01)00133-xCasabán, M.-C., Cortés, J.-C., & Jódar, L. (2015). A random Laplace transform method for solving random mixed parabolic differential problems. Applied Mathematics and Computation, 259, 654-667. doi:10.1016/j.amc.2015.02.091Casabán, M.-C., Cortés, J.-C., & Jódar, L. (2018). Solving linear and quadratic random matrix differential equations using: A mean square approach. The non-autonomous case. Journal of Computational and Applied Mathematics, 330, 937-954. doi:10.1016/j.cam.2016.11.049Casabán, M.-C., Cortés, J.-C., & Jódar, L. (2016). Solving linear and quadratic random matrix differential equations: A mean square approach. Applied Mathematical Modelling, 40(21-22), 9362-9377. doi:10.1016/j.apm.2016.06.01

Solution of the Second Order of the Linear Hyperbolic Equation Using Cubic B-Spline Collocation Numerical Method

Wave equation is one of the second order of the linear hyperbolic equation. Telegraph equation as a special case of wave equation has interesting point to investigate in the numerical point of view. In this paper, we consider the numerical methods for one dimensional telegraph equation by using cubic B-spline collocation method. Collocation method is one method to solve the partial differential equation model problem. Cubic spline interpolation is an interpolation to a third order polynomial. This polynomial interpolate four point. B-Spline is one of spline function which related to smoothness of the partition. For every spline function with given order can be written as linear combination of those B-spline. As we known that the result of the numerical technique has difference with the exact result which we called as, so that we have an error. The numerical results are compared with the interpolating scaling function method which investigated by Lakestani and Saray in 2010. This numerical methods compared to exact solution by using RMSE (root mean square error), L2 norm error and L_∞ norm error . The error of the solution showed that with the certain function, the cubic collocation of numerical method can be used as an alternative methods to find the solution of the linear hyperbolic of the PDE. The advantages of this study, we can choose the best model of the numerical method for solving the hyperbolic type of PDE. This cubic B-spline collocation method is more efficiently if the error is relatively small and closes to zero. This accuration verified by test of example 1 and example 2 which applied to the model problem