Newton-Raphson based modified Laplace Adomian decomposition method for solving quadratic Riccati differential equations

S Kakkar,S Rani,Vinod Mishra,Dimple Rani,T Srivastava

doi:10.1051/matecconf/20165705001

Abstract

Numerical Laplace transform method is applied to approximate the solution of nonlinear (quadratic) Riccati differential equations mingled with Adomian decomposition method. A new technique is proposed in this work by reintroducing the unknown function in Adomian polynomial with that of well known Newton-Raphson formula. The solutions obtained by the iterative algorithm are exhibited in an infinite series. The simplicity and efficacy of method is manifested with some examples in which comparisons are made among the exact solutions, ADM (Adomian decomposition method), HPM (Homotopy perturbation method), Taylor series method and the proposed scheme.

Highlights

One of the most significant classes of nonlinear differential equation, Riccati differential equation (RDE) is considered as follows: dy q(t) y r(t) y 2 p(t), y(0) a (1)dt where q(t), r(t) and p(t) are the known scalar functions and a is an arbitrary constant
The RDE is named after the name of Italian nobleman Count Jacopo Francesco Riccati(1676-1754) [4]
We employ our method to different forms of Quadratic RDE

Summary

Introduction

One of the most significant classes of nonlinear differential equation, Riccati differential equation (RDE) is considered as follows: dy q(t) y r(t) y 2 p(t), y(0) a (1). ADM and LTDM in solving RDEs in [3] For solving these kinds of equations Yang et al [7] employed the hybrid functions and Tau method. A well known numerical algorithm Laplace transforms and Adomian decomposition method has conquered much importance in solving many linear and nonlinear problems which provides a series solution. For handling the solutions of nonlinear system of partial differential equation Laplace decomposition method and pade approximant is used in [21]. This method is utilized to solve many more problems like Singular initial value problems [13], Double singular boundary value.

Analysis of Method

Numerical Results

Conclusions