Analytical versus numerical solutions of the nonlinear fractional time–space telegraph equation

2021 ◽  
pp. 2150324
Author(s):  
Mostafa M. A. Khater ◽  
Dianchen Lu
Keyword(s):  
Analytical Solutions ◽  
Electromagnetic Waves ◽  
Numerical Scheme ◽  
Numerical Solutions ◽  
Telegraph Equation ◽  
Scatter Matrix ◽  
B Spline ◽  
Stable Property ◽  
Time Space ◽  
Matrix Distribution

In this paper, the stable analytical solutions’ accuracy of the nonlinear fractional nonlinear time–space telegraph (FNLTST) equation is investigated along with applying the trigonometric-quantic-B-spline (TQBS) method. This investigation depends on using the obtained analytical solutions to get the initial and boundary conditions that allow applying the numerical scheme in an easy and smooth way. Additionally, this paper aims to investigate the accuracy of the obtained analytical solutions after checking their stable property through using the properties of the Hamiltonian system. The considered model for this study is formulated by Oliver Heaviside in 1880 to define the advanced or voltage spectrum of electrified transmission, with day-to-day distances from the electrified communication or the application of electromagnetic waves. The matching between the analytical and numerical solutions is explained by some distinct sketches such as two-dimensional, scatter matrix, distribution, spline connected, bar normal, filling with two colors plots.

Solution of non-linear time fractional telegraph equation with source term using B-spline and Caputo derivative

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Abdul Majeed ◽  
Mohsin Kamran ◽  
Noreen Asghar
Keyword(s):  
Caputo Derivative ◽  
Numerical Solutions ◽  
Linear Time ◽  
Initial Boundary ◽  
Telegraph Equation ◽  
Test Problems ◽  
Von Neumann ◽  
B Spline ◽  
Nonlinear Part ◽  
Spline Basis

Abstract This article focusses on the implementation of cubic B-spline approach to investigate numerical solutions of inhomogeneous time fractional nonlinear telegraph equation using Caputo derivative. L1 formula is used to discretize the Caputo derivative, while B-spline basis functions are used to interpolate the spatial derivative. For nonlinear part, the existing linearization formula is applied after generalizing it for all positive integers. The algorithm for the simulation is also presented. The efficiency of the proposed scheme is examined on three test problems with different initial boundary conditions. The influence of parameter α on the solution profile for different values is demonstrated graphically and numerically. Moreover, the convergence of the proposed scheme is analyzed and the scheme is proved to be unconditionally stable by von Neumann Fourier formula. To quantify the accuracy of the proposed scheme, error norms are computed and was found to be effective and efficient for nonlinear fractional partial differential equations (FPDEs).

scholarly journals Collocation solutions for the time fractional telegraph equation using cubic B-spline finite elements

2019 ◽  
Vol 57 (2) ◽  
pp. 131-144
Author(s):  
Orkun Tasbozan ◽  
Alaattin Esen
Keyword(s):  
Finite Elements ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Telegraph Equation ◽  
Numerical Results ◽  
Fractional Partial Differential Equations ◽  
Spline Collocation ◽  
B Spline ◽  
Fractional Telegraph Equation

Abstract In this study, we investigate numerical solutions of the fractional telegraph equation with the aid of cubic B-spline collocation method. The fractional derivatives have been considered in the Caputo forms. The L1and L2 formulae are used to discretize the Caputo fractional derivative with respect to time. Some examples have been given for determining the accuracy of the regarded method. Obtained numerical results are compared with exact solutions arising in the literature and the error norms L 2 and L ∞ have been computed. In addition, graphical representations of numerical results are given. The obtained results show that the considered method is effective and applicable for obtaining the numerical results of nonlinear fractional partial differential equations (FPDEs).

scholarly journals A Galerkin Solution for Burgers' Equation Using Cubic B-Spline Finite Elements

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
A. A. Soliman
Keyword(s):  
Stability Analysis ◽  
Numerical Scheme ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Unconditionally Stable ◽  
B Spline ◽  
B Splines ◽  
Galerkin Solution ◽  
Set Up ◽  
Interpolation Functions

Numerical solutions for Burgers’ equation based on the Galerkins’ method using cubic B-splines as both weight and interpolation functions are set up. It is shown that this method is capable of solving Burgers’ equation accurately for values of viscosity ranging from very small to large. Three standard problems are used to validate the proposed algorithm. A linear stability analysis shows that a numerical scheme based on a Cranck-Nicolson approximation in time is unconditionally stable.

Modeling ELF waves in the non-uniform Earth-ionosphere cavity

2021 ◽  
Author(s):  
Tamas Bozoki ◽  
Erno Pracser ◽  
Gabriella Satori ◽  
Andrzej Kulak ◽  
Janusz Mlynarczyk ◽  
...  
Keyword(s):  
Electromagnetic Waves ◽  
Numerical Solutions ◽  
Model Development ◽  
Resonance Field ◽  
Low Frequency ◽  
Relative Difference ◽  
Telegraph Equation ◽  
Theoretical Description ◽  
Inversion Algorithm ◽  
The Earth

<p>Below 100 Hz, in the lowest part of the extremely low frequency (ELF, 3 Hz - 3 kHz) band lightning-radiated electromagnetic waves propagate with extremely low attenuation (roughly below 1 dB/Mm) within the Earth-ionosphere waveguide which makes possible the formation of global electromagnetic resonances, known as Schumann resonances (SRs). The most commonly used description of this resonance field assumes a uniform Earth-ionosphere cavity, i.e. that the propagation conditions for ELF waves are practically the same on the dayside and nightside hemispheres, which is the most vulnerable simplification of these models. </p><p>In this work we present two different forward models for SRs that take into consideration the day-night asymmetry of the Earth-ionosphere cavity and are based on the analytical and numerical solutions of the two-dimensional telegraph equation (TDTE). We present numerical tests showing that the two models produce practically the same output, i.e. the relative difference between them is less than 0.4%. The conspicuous conformity between the outputs establishes not only the correctness of the formalisms but the correctness of the implementations (the coding) as well. To the best of the authors’ knowledge this is the first work that verifies this conformity between the two independent solutions. </p><p>We also compare our stationary models with time-dependent solutions of the TDTE as the stationarity of the resonance field may represent the next most vulnerable simplification that needs to be dismissed to approach a more realistic theoretical description of SRs. All these steps in model development serve our aim to infer global lightning activity based on multi-station ELF measurements by applying a sophisticated inversion algorithm.</p>

scholarly journals Laplace Transform Collocation Method for Solving Hyperbolic Telegraph Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Adebayo O. Adewumi ◽  
Saheed O. Akindeinde ◽  
Adebayo A. Aderogba ◽  
Babatunde S. Ogundare
Keyword(s):  
Laplace Transform ◽  
Collocation Method ◽  
Analytical Solutions ◽  
Trial Function ◽  
Numerical Scheme ◽  
Telegraph Equation ◽  
One Dimensional ◽  
Laplace Transform Technique ◽  
Telegraph Equations ◽  
New Form

This article presents a new numerical scheme to approximate the solution of one-dimensional telegraph equations. With the use of Laplace transform technique, a new form of trial function from the original equation is obtained. The unknown coefficients in the trial functions are determined using collocation method. The efficiency of the new scheme is demonstrated with examples and the approximations are in excellent agreement with the analytical solutions. This method produced better approximations than the ones produced with the standard weighted residual methods.

scholarly journals Numerical solutions of electromagnetic wave model of fractional derivative using class of finite difference scheme

2021 ◽  
Author(s):  
Vijay Patel ◽  
Dhirendra Bahuguna
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Electromagnetic Waves ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Finite Difference Methods ◽  
Wave Model ◽  
Dielectric Media

In this article, a numerical scheme is introduced for solving the fractional partial differential equation (FPDE) arising from electromagnetic waves in dielectric media (EMWDM) by using an efficient class of finite difference methods. The numerical scheme is based on the Hermite formula. The Caputo’s fractional derivatives in time are discretized by a finite difference scheme of order O((k^(4-alpha)) & O(k^(4-beta)), 1

scholarly journals Five semi analytical and numerical simulations for the fractional nonlinear space-time telegraph equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mostafa M. A. Khater ◽  
Choonkil Park ◽  
Jung Rye Lee ◽  
Mohamed S. Mohamed ◽  
Raghda A. M. Attia
Keyword(s):  
Numerical Simulations ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Approximate Solutions ◽  
Telegraph Equation ◽  
Space Time ◽  
Numerical Techniques ◽  
B Spline ◽  
Wave Solutions ◽  
Adomian Decomposition

AbstractThe accuracy of analytical obtained solutions of the fractional nonlinear space–time telegraph equation that has been constructed in (Hamed and Khater in J. Math., 2020) is checked through five recent semi-analytical and numerical techniques. Adomian decomposition (AD), El Kalla (EK), cubic B-spline (CBS), extended cubic B-spline (ECBS), and exponential cubic B-spline (ExCBS) schemes are used to explain the matching between analytical and approximate solutions, which shows the accuracy of constructed traveling wave solutions. In 1880, Oliver Heaviside derived the considered model to describe the cutting-edge or voltage of an electrified transmission. The matching between solutions has been explained by plotting them in some different sketches.

A Numerical Scheme for a Class of Parametric Problem of Fractional Variational Calculus

2012 ◽  
Vol 7 (2) ◽  
Author(s):  
Om P. Agrawal ◽  
M. Mehedi Hasan ◽  
X. W. Tangpong
Keyword(s):  
Analytical Solutions ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Optimization Problems ◽  
Practical Interest ◽  
Variational Calculus ◽  
Functional Minimization ◽  
Orthogonal Functions ◽  
Order Problem ◽  
Parametric Problem

Fractional derivatives (FDs) or derivatives of arbitrary order have been used in many applications, and it is envisioned that in the future they will appear in many functional minimization problems of practical interest. Since fractional derivatives have such properties as being non-local, it can be extremely challenging to find analytical solutions for fractional parametric optimization problems, and in many cases, analytical solutions may not exist. Therefore, it is of great importance to develop numerical methods for such problems. This paper presents a numerical scheme for a linear functional minimization problem that involves FD terms. The FD is defined in terms of the Riemann-Liouville definition; however, the scheme will also apply to Caputo derivatives, as well as other definitions of fractional derivatives. In this scheme, the spatial domain is discretized into several subdomains and 2-node one-dimensional linear elements are adopted to approximate the solution and its fractional derivative at point within the domain. The fractional optimization problem is converted to an eigenvalue problem, the solution of which leads to fractional orthogonal functions. Convergence study of the number of elements and error analysis of the results ensure that the algorithm yields stable results. Various fractional orders of derivative are considered, and as the order approaches the integer value of 1, the solution recovers the analytical result for the corresponding integer order problem.

Transient Two-Dimensional Heat Conduction Problem With Partial Heating Near Corners

2016 ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
James V. Beck ◽  
Donald E. Amos
Keyword(s):  
Heat Conduction ◽  
Numerical Integration ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Thermal Protection ◽  
Heat Conduction Problem ◽  
Separation Of Variables ◽  
Two Dimensional ◽  
Rectangular Region ◽  
Long Distance

This paper provides a solution for two-dimensional heating over a rectangular region on a homogeneous plate. It has application to verification of numerical conduction codes as well as direct application for heating and cooling of electronic equipment. Additionally, it can be applied as a direct solution for the inverse heat conduction problem, most notably used in thermal protection systems for re-entry vehicles. The solutions used in this work are generated using Green’s functions. Two approaches are used which provide solutions for either semi-infinite plates or finite plates with isothermal conditions which are located a long distance from the heating. The methods are both efficient numerically and have extreme accuracy, which can be used to provide additional solution verification. The solutions have components that are shown to have physical significance. The extremely precise nature of analytical solutions allows them to be used as prime standards for their respective transient conduction cases. This extreme precision also allows an accurate calculation of heat flux by finite differences between two points of very close proximity which would not be possible with numerical solutions. This is particularly useful near heated surfaces and near corners. Similarly, sensitivity coefficients for parameter estimation problems can be calculated with extreme precision using this same technique. Another contribution of these solutions is the insight that they can bring. Important dimensionless groups are identified and their influence can be more readily seen than with numerical results. For linear problems, basic heating elements on plates, for example, can be solved to aid in understanding more complex cases. Furthermore these basic solutions can be superimposed both in time and space to obtain solutions for numerous other problems. This paper provides an analytical two-dimensional, transient solution for heating over a rectangular region on a homogeneous square plate. Several methods are available for the solution of such problems. One of the most common is the separation of variables (SOV) method. In the standard implementation of the SOV method, convergence can be slow and accuracy lacking. Another method of generating a solution to this problem makes use of time-partitioning which can produce accurate results. However, numerical integration may be required in these cases, which, in some ways, negates the advantages offered by the analytical solutions. The method given herein requires no numerical integration; it also exhibits exponential series convergence and can provide excellent accuracy. The procedure involves the derivation of previously-unknown simpler forms for the summations, in some cases by virtue of the use of algebraic components. Also, a mathematical identity given in this paper can be used for a variety of related problems.

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