Numerical computation of Klein–Gordon equations arising in quantum field theory by using homotopy analysis transform method

AbstractIn this paper, we present a reliable algorithm based on the homotopy analysis transform method (HATM) to solve the linear and nonlinear Klein–Gordon equations. The Klein–Gordon equation is the equation of motion of a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. It describes the quantum amplitude for finding a point particle in various places, the relativistic wave function, but the particle propagates both forwards and backwards in time. The HATM is a combined form of the Laplace transform method and homotopy analysis method. The method provides the solution in the form of a rapidly convergent series. Some numerical examples are used to illustrate the preciseness and effectiveness of the proposed method. The results show that the HATM is very efficient, simple and can be applied to other nonlinear problems

Analytical solution for determination the control parameter in the inverse parabolic equation using HAM

In this article, the homotopy analysis method (HAM) for obtaining the analytical solution of the inverse parabolic problem and computing the unknown time-dependent parameter is introduced. The series solution is developed and the recurrence relations are given explicitly. Special attention is given to satisfy the convergence of the proposed method. A comparison of HAM with the variational iteration method is made. In the HAM, we use the auxiliary parameter ~ to control with a simple way in the convergence region of the solution series. Applying this method with severa

Symbolic computation of exact solutions for fractional differential-difference equation models

The aim of the present study is to extend the G'/G-expansion method to fractional differential-difference equations of rational type. Particular time-fractional models are considered to show the strength of the method. Three types of exact solutions are observed: hyperbolic, trigonometric and rational. Exact solutions in terms of topological solitons and singular periodic functions are also obtained. As far as we are aware, our results have not been published elsewhere previously

The Ablowitz-Ladik lattice system by means of the extended (G′ / G)-expansion method

We analyzed the Ablowitz-Ladik lattice system by using the extended (G′ / G)-expansion method. Further discrete soliton and periodic wave solutions with more arbitrary parameters are obtained. We observed that some previously known results can be recovered by assigning special values to the arbitrary parameters. © 2010 Elsevier Inc. All rights reserved

Mathematical Methods, Modelling and Applications

This volume deals with novel high-quality research results of a wide class of mathematical models with applications in engineering, nature, and social sciences. Analytical and numeric, deterministic and uncertain dimensions are treated. Complex and multidisciplinary models are treated, including novel techniques of obtaining observation data and pattern recognition. Among the examples of treated problems, we encounter problems in engineering, social sciences, physics, biology, and health sciences. The novelty arises with respect to the mathematical treatment of the problem. Mathematical models are built, some of them under a deterministic approach, and other ones taking into account the uncertainty of the data, deriving random models. Several resulting mathematical representations of the models are shown as equations and systems of equations of different types: difference equations, ordinary differential equations, partial differential equations, integral equations, and algebraic equations. Across the chapters of the book, a wide class of approaches can be found to solve the displayed mathematical models, from analytical to numeric techniques, such as finite difference schemes, finite volume methods, iteration schemes, and numerical integration methods