A New Tau Method for Solving Nonlinear Lane-Emden Type Equations via Bernoulli Operational Matrix of Differentiation

A new and efficient numerical approach is developed for solving nonlinear Lane-Emden type equations via Bernoulli operational matrix of differentiation. The fundamental structure of the presented method is based on the Tau method together with the Bernoulli polynomial approximations in which a new operational matrix is introduced. After implementation of our scheme, the main problem would be transformed into a system of algebraic equations such that its solutions are the unknown Bernoulli coefficients. Also, under several mild conditions the error analysis of the proposed method is provided. Several examples are included to illustrate the efficiency and accuracy of the proposed technique and also the results are compared with the different methods. All calculations are done in Maple 13

Numerical Solution of Nonlinear Fractional Volterra Integro-Differential Equations via Bernoulli Polynomials

This paper presents a computational approach for solving a class of nonlinear Volterra integro-differential equations of fractional order which is based on the Bernoulli polynomials approximation. Our method consists of reducing the main problems to the solution of algebraic equations systems by expanding the required approximate solutions as the linear combination of the Bernoulli polynomials. Several examples are given and the numerical results are shown to demonstrate the efficiency of the proposed method

A Quintic B-Spline Technique for a System of Lane-Emden Equations Arising in Theoretical Physical Applications

In the present study, we introduce a collocation approach utilizing quintic B-spline functions as bases for solving systems of Lane Emden equations which have various applications in theoretical physics and astrophysics. The method derives a solution for the provided system by converting it into a set of algebraic equations with unknown coefficients, which can be easily solved to determine these coefficients. Examining the convergence theory of the proposed method reveals that it yields a fourth-order convergent approximation. It is confirmed that the outcomes are consistent with the theoretical investigation. Tables and graphs illustrate the proficiency and consistency of the proposed method. Findings validate that the newly employed method is more accurate and effective than other approaches found in the literature. All calculations have been performed using Mathematica software

Numerical Solution of Nonlinear Fractional Volterra Integro-Differential Equations via Bernoulli Polynomials

This paper presents a computational approach for solving a class of nonlinear Volterra integro-differential equations of fractional order which is based on the Bernoulli polynomials approximation. Our method consists of reducing the main problems to the solution of algebraic equations systems by expanding the required approximate solutions as the linear combination of the Bernoulli polynomials. Several examples are given and the numerical results are shown to demonstrate the efficiency of the proposed method

Higher order multi-step Jarratt-like method for solving systems of nonlinear equations: application to PDEs and ODEs

This paper proposes a multi-step iterative method for solving systems of nonlinear equations with a local convergence order of 3m - 4, where in (>= 2) is the number of steps. The multi-step iterative method includes two parts: the base method and the multi-step part. The base method involves two function evaluations, two Jacobian evaluations, one LU decomposition of a Jacobian, and two matrix-vector multiplications. Every stage of the multi-step part involves the solution of two triangular linear systems and one matrix-vector multiplication. The computational efficiency of the new method is better than those of previously proposed methods. The method is applied to several nonlinear problems resulting from discretizing nonlinear ordinary differential equations and nonlinear partial differential equations. (C) 2015 Elsevier Ltd. All rights reserved.Postprint (updated version

Operational matrices for solving variable order differential equations

In this thesis, we extensively explore the role of matrices as substitutes for derivative and integral operators. By expressing an approximate solution of a partial differential in an implicit form involving polynomials, we demonstrate how to deduce novel composite operational matrices. We also show how to utilise the laws of matrix multiplication to come up with a single matrix that performs the role of differentiation and integration. In conjunction with the Garlekin technique, we apply these composite matrices to numerically solve partial differential equations. Through practical examples, we prove that these composite operational matrices are convenient in approximating the solution of partial differential equations using a computer algebra system like Mathematica.Mathematical Science

Wavelet Theory

The wavelet is a powerful mathematical tool that plays an important role in science and technology. This book looks at some of the most creative and popular applications of wavelets including biomedical signal processing, image processing, communication signal processing, Internet of Things (IoT), acoustical signal processing, financial market data analysis, energy and power management, and COVID-19 pandemic measurements and calculations. The editor’s personal interest is the application of wavelet transform to identify time domain changes on signals and corresponding frequency components and in improving power amplifier behavior