Optimal Perturbation Iteration Method for Bratu-Type Problems

In this paper, we introduce the new optimal perturbation iteration method based on the perturbation iteration algorithms for the approximate solutions of nonlinear differential equations of many types. The proposed method is illustrated by studying Bratu-type equations. Our results show that only a few terms are required to obtain an approximate solution which is more accurate and efficient than many other methods in the literature.Comment: 11 pages, 3 Figure

Approximate Solution of Tuberculosis Disease Population Dynamics Model

We examine possible approximate solutions of both integer and noninteger systems of nonlinear differential equations describing tuberculosis disease population dynamics. The approximate solutions are obtained via the relatively new analytical technique, the homotopy decomposition method (HDM). The technique is described and illustrated with numerical example. The numerical simulations show that the approximate solutions are continuous functions of the noninteger-order derivative. The technique used for solving these problems is friendly, very easy, and less time consuming

Existence and Numerical Solution of the Volterra Fractional Integral Equations of the Second Kind

This work presents the possible generalization of the Volterra integral equation second kind to the concept of fractional integral. Using the Picard method, we present the existence and the uniqueness of the solution of the generalized integral equation. The numerical solution is obtained via the Simpson 3/8 rule method. The convergence of this scheme is presented together with numerical results

A Practical Method for Analytical Evaluation of Approximate Solutions of Fisher's Equations

In this article, a framework is developed to get more approximate solutions to nonlinear partial differential equations by applying perturbation iteration technique. This technique is modified and improved to solve nonlinear diffusion equations of the Fisher type. Some problems are investigated to illustrate the efficiency of the method. Comparisons between the new results and the solutions obtained by other techniques prove that this technique is highly effective and accurate in solving nonlinear problems. Convergence analysis and error estimate are also provided by using some related theorems. The basic ideas indicated in this work are anticipated to be further developed to handle nonlinear models

The Use of Fractional Order Derivative to Predict the Groundwater Flow

The aim of this work was to convert the Thiem and the Theis groundwater flow equation to the time-fractional groundwater flow model. We first derived the analytical solution of the Theim time-fractional groundwater flow equation in terms of the generalized Wright function. We presented some properties of the Laplace-Carson transform. We derived the analytical solution of the Theis-time-fractional groundwater flow equation (TFGFE) via the Laplace-Carson transform method. We introduced the generalized exponential integral, as solution of the TFGFE. This solution is in perfect agreement with the data observed from the pumping test performed by the Institute for Groundwater Study on one of its borehole settled on the test site of the University of the Free State. The test consisted of the pumping of the borehole at the constant discharge rate Q and monitoring the piezometric head for 350 minutes

Optimal perturbation iteration method for Bratu-type problems

In this paper, we introduce the new optimal perturbation iteration method based on the perturbation iteration algorithms for the approximate solutions of nonlinear differential equations of many types. The proposed method is illustrated by studying Bratu-type equations. Our results show that only a few terms are required to obtain an approximate solution which is more accurate and efficient than many other methods in the literature