Hyperbolic Telegraph Equation Research Articles

Numerical solution of one- and two-dimensional Hyperbolic Telegraph equation via Cubic–Quartic Hyperbolic B-Spline DQM: a statistical validity

Numerical solution of one- and two-dimensional Hyperbolic Telegraph equation via Cubic–Quartic Hyperbolic B-Spline DQM: a statistical validity

A numerical technique based on Legendre wavelet for linear and nonlinear hyperbolic telegraph equation

A numerical technique based on Legendre wavelet for linear and nonlinear hyperbolic telegraph equation

Extension of Haar wavelet technique for numerical solution of three-dimensional linear and nonlinear telegraph equations

In this manuscript, a hybrid numerical technique is presented for solving three-dimensional hyperbolic telegraph equations. The proposed technique is based on the Haar wavelet collocation method and the finite difference method. In this technique, the space derivatives are estimated by truncated Haar wavelet series, while the time derivatives are approximated using finite difference method. The method is applied to some linear and non-linear three-dimensional hyperbolic telegraph equations. In the linear case, the resulting algebraic system is solved by the Gauss elimination technique. Whereas in the nonlinear case, the quasi-Newton linearization technique is utilized to remove the nonlinearity of the problem. The maximum absolute errors, root mean square errors and computational rate of convergence are calculated for a different number of collocation points at different time steps of the final time t=1. The stability analysis for the proposed method is also discussed. Overall, the recently devised numerical method boasts straightforward implementation, rapid convergence, and enhanced accuracy in addressing both linear and nonlinear problems.

Solving The Hyperbolic Telegraph Equation Using a Modified Adomian Decomposition Method with an Invertible Partial Differential Operator

In this paper, a modified Adomian decomposition method (MADM) for solving the hyperbolic telegraph equation is proposed. The MADM introduces a new inverse partial differential operator that can speed up the convergence rate of the standard ADM. We also present a technique for converting the equation to a special case form, which makes the MADM easier to implement. The proposed method was tested on six different linear and nonlinear telegraph equations in one and two dimensions. The results show that the method is accurate and efficient for solving the telegraph equation.

A Novel Solution Approach for Time-Fractional Hyperbolic Telegraph Differential Equation with Caputo Time Differentiation

In the current analysis, a specific efficient and applicable novel solution approach, based on a fractional power series technique and Laplace transform operator, is considered to predict certain accurate approximate solutions (ASs) for a time-fractional hyperbolic telegraph equation by aid of time-fractional derivatives in a Caputo sense. The solutions are obtained in a fractional Maclurian series formula by solving the original problem in the Laplace space aided by a limit concept having fewer small iterations than the classical fractional power series technique. To confirm applicability and feasibility of the proposed approach, three appropriate initial value problems are considered. Consequently, some simulations of gained outcomes are numerically and graphically implemented to support the effect of the fractional-order parameter on the geometric behavior of the obtained solutions. In addition, graphical representations are also fulfilled to verify the convergence analysis of the fractional series solutions of the classical solution. The proposed technique is therefore proposed to be a straightforward, accurate and powerful approach for handling varied time-fractional models in various physical phenomena.

Semi-analytical approximation of time-fractional telegraph equation via natural transform in Caputo derivative

Abstract In the present research study, time-fractional hyperbolic telegraph equations are solved iteratively using natural transform in one, two, and three dimensions. The fractional derivative is considered in the Caputo sense. These equations serve as a model for the wave theory process of signal processing and transmission of electric impulses. To evaluate the validity and effectiveness of the suggested strategy, a graphical comparison of approximated and exact findings is performed. Convergence analysis of the approximations utilising L ∞ {L}_{\infty } has been done using tables. The suggested approach may successfully and without errors solve a wide variety of ordinary differential equations, partial differential equations (PDEs), fractional PDEs, and fractional hyperbolic telegraph equations.

An iterative approach using Sawi transform for fractional telegraph equation in diversified dimensions

Abstract In the present study, 1D, 2D, and 3D fractional hyperbolic telegraph equations in Caputo sense have been solved using an iterative method using Sawi transform. These equations serve as a model for signal analysis of electrical impulse transmission and propagation. Along with a table of Sawi transform of some popular functions, some helpful results on Sawi transform are provided. To demonstrate the effectiveness of the suggested method, five examples in 1D, one example in 2D, and one example in 3D are solved using the proposed scheme. Error analysis comparing approximate and exact solutions using graphs and tables has been provided. The proposed scheme is robust, effective, and easy to implement and can be implemented on variety of fractional partial differential equations to obtain precise series approximations.

A comparative study for the numerical approximation of 1D and 2D hyperbolic telegraph equations with UAT and UAH tension B-spline DQM

Abstract Two numerical regimes for the one- and two-dimensional hyperbolic telegraph equations are contrasted in this article. The first implemented regime is uniform algebraic trigonometric tension B-spline DQM, while the second implemented regime is uniform algebraic hyperbolic tension B-spline DQM. The resulting system of ODEs is solved by the SSP RK43 method after the aforementioned equations are spatially discretized. To assess the success of chosen tactics, a comparison of errors is shown. The graphs can be seen, and it is asserted that the precise and numerical results are in agreement with one another. Analyses of convergence and stability are also given. It should be highlighted that there is a dearth of study on 1D and 2D hyperbolic telegraph equations. This aim of this study is to efficiently create results with fewer mistakes. These techniques would surely be useful for other higher-order nonlinear complex natured partial differential equations, including fractional equations, integro equations, and partial-integro equations.

A novel B-spline collocation method for Hyperbolic Telegraph equation

<abstract><p>The present study is concerned with the construction of a new high-order technique to establish approximate solutions of the Telegraph equation (TE). In this technique, a novel optimal B-spline collocation method based on quintic B-spline (QBS) basis functions is constructed to discretize the spatial domain and fourth-order implicit method is derived for time integration. Test problems are considered to verify the theoretical results and to demonstrate the applicability of the suggested technique. The error norm $ L_{\infty } $ and the rate of spatial and temporal convergence are computed and compared with those of techniques available in the literature. The obtained results show the improvement and efficiency of the proposed scheme over the existing ones. Also, it is obviously observed that the experimental rate of convergence is almost compatible with the theoretical rate of convergence.</p></abstract>

High order accurate method for the numerical solution of the second order linear hyperbolic telegraph equation

AbstractIn this study, the Galerkin finite element method is applied to get the numerical solution of the linear telegraph equation by using cubic B‐spline function. Differently from the existing studies, the fourth order one‐step method is used to discretize in time the telegraph equation. The efficiency and accuracy of the proposed method is studied by three examples. The obtained results are shown that the proposed method has a higher accuracy.

Fully hyperbolic convolutional neural networks

Convolutional neural networks (CNN) have recently seen tremendous success in various computer vision tasks. However, their application to problems with high dimensional input and output, such as high-resolution image and video segmentation or 3D medical imaging, has been limited by various factors. Primarily, in the training stage, it is necessary to store network activations for back-propagation. In these settings, the memory requirements associated with storing activations can exceed what is feasible with current hardware, especially for problems in 3D. Motivated by the propagation of signals over physical networks, that are governed by the hyperbolic Telegraph equation, in this work we introduce a fully conservative hyperbolic network for problems with high-dimensional input and output. We introduce a coarsening operation that allows completely reversible CNNs by using a learnable discrete wavelet transform and its inverse to both coarsen and interpolate the network state and change the number of channels. We show that fully reversible networks are able to achieve results comparable to the state of the art in 4D time-lapse hyper-spectral image segmentation and full 3D video segmentation, with a much lower memory footprint that is a constant independent of the network depth. We also extend the use of such networks to variational auto-encoders, where optimization begins from an exact recovery and we discover the level of compression through optimization.

A New Strategy for the Approximate Solution of Hyperbolic Telegraph Equations in Nonlinear Vibration System

This study examines a new approach for the approximate solution of hyperbolic telegraph equations emerging in magnetic fields and electrical impulse transmissions. We introduce a Laplace-Carson transform coupled with the homotopy perturbation method which is called the Laplace-Carson homotopy perturbation method (Lc-HPM). The most significant feature of this approach is that we do not require any restriction of variables and hypotheses to find the results of nonlinear problems. Further, HPM using He’s is applied to reduce the number of computations in nonlinear terms. We demonstrate some graphical results to show thatLc-HPM is a simple and suitable approach for linear and nonlinear problems.

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

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 of O h x 2 + h y 2 + Δ τ 2 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.

An Analytical Approach for Fractional Hyperbolic Telegraph Equation Using Shehu Transform in One, Two and Three Dimensions

In the present research paper, an iterative approach named the iterative Shehu transform method is implemented to solve time-fractional hyperbolic telegraph equations in one, two, and three dimensions, respectively. These equations are the prominent ones in the field of physics and in some other significant problems. The efficacy and authenticity of the proposed method are tested using a comparison of approximated and exact results in graphical form. Both 2D and 3D plots are provided to affirm the compatibility of approximated-exact results. The iterative Shehu transform method is a reliable and efficient tool to provide approximated and exact results to a vast class of ODEs, PDEs, and fractional PDEs in a simplified way, without any discretization or linearization, and is free of errors. A convergence analysis is also provided in this research.

Numerical and Approximate Solutions for Two-Dimensional Hyperbolic Telegraph Equation via Wavelet Matrices

Numerical and Approximate Solutions for Two-Dimensional Hyperbolic Telegraph Equation via Wavelet Matrices

General Solution of Telegraph Equation Using Aboodh Transform

This paper describes using the Aboodh integral transform in solving one-dimensional second-order hyperbolic telegraph equations with satisfying initial conditions by solving some telegraph equations with specific initial conditions and then generalizing the solution to the telegraph equation using the Aboodh integral transform.

A New Technique to Speed Up Group Methods for Solving Hyperbolic Telegraph Equations

Numerous methods have been introduced in the literature for numerical solution of two-dimensional hyperbolic differential equations (telegraph equations). Improved techniques using explicit group methods derived from the standard and skewed (rotated) finite difference operators have been developed over the last few years in solving the linear systems that arise from the discretization of the several types of partial differential equations. In this paper, we present a preliminary study of the formulation of new preconditioned scheme based on explicit group relaxationmethods for the difference solution of the telegraph equations. The efficient and robustness of these new formulations over the existing explicit group schemes demonstrated through numerical experiments.

The Fragile Points Method (FPM) to solve two-dimensional hyperbolic telegraph equation using point stiffness matrices

In this paper, the Fragile Points Method (FPM) has been extended to solve the two-dimensional hyperbolic telegraph equation with specified initial and boundary conditions. Based on a naturally partitioned domain with scattered nodes and the Voronoi Diagram, the discretized FPM equations are derived by a Galerkin weak-form in the spatial domain and a finite difference scheme in the time domain. For the spatial discretization, discontinuous point-based polynomial trial and test functions are utilized, with numerical fluxes to ensure the consistency. For the time-domain discretization, theorems of unconditional stability and convergence for the telegraph equations are given. Numerical examples confirm the accuracy and robustness of the developed numerical method with uniform or random nodes and with different time increments. And as compared to several other meshless methods for the telegraph equation, it is shown that the developed FPM method has a better computational efficiency. This is because that a single-point quadrature rule is sufficient for evaluating the weak-form integral which gives a symmetric and sparse stiffness matrix, with the specifically-designed piecewise-linear trial and test functions.

A multilayer perceptron neural network approach for the solution of hyperbolic telegraph equations

ABSTRACT Neural networks have been extensively used for solving differential equations in the past, but they rely mostly on computationally expensive gradient-based numerical optimization procedure for solving differential equations. In this work, we are introducing a faster way to train neural networks for solving differential equations based on extreme learning machine algorithm. This algorithm is much faster as compared to traditional approaches, and it also provides highly accurate results. Reliability of the approach is tested by solving various cases of the hyperbolic telegraph equations. Solutions so obtained are compared to the results existing in the literature for analysing the accuracy of the proposed approach.

An approximation to the solution of one-dimensional hyperbolic telegraph equation based on the collocation of quadratic B-spline functions

In this work, the collocation method based on B-spline functions is used to obtain a numerical solution for a one-dimensional hyperbolic telegraph equation. The proposed method is consists of two main steps. As the first step, by using a finite difference scheme for the time variable, a partial differential equation is converted to an ordinary differential equation by the space variable. In the next step, for solving this equation collocation method is used. In the analysis section of the proposed method, the convergence of the method is studied. Also, some numerical results are given to demonstrate the validity and applicability of the presented technique. The L∞, L2, and Root-Mean Square(RMS) in the solutions show the efficiency of the method computationally.