^{1, 2}

^{2, 3}

^{1}

^{2}

^{3}

We combine the Adomian decomposition method (ADM) and Adomian’s asymptotic decomposition method (AADM) for solving Riccati equations. We investigate the approximate global solution by matching the near-field approximation derived from the Adomian decomposition method with the far-field approximation derived from Adomian’s asymptotic decomposition method for Riccati equations and in such cases when we do not find any region of overlap between the obtained approximate solutions by the two proposed methods, we connect the two approximations by the Padé approximant of the near-field approximation. We illustrate the efficiency of the technique for several specific examples of the Riccati equation for which the exact solution is known in advance.

It is well known that the Riccati equation as

Adomian and his coauthors have presented a systematic methodology for practical solution of linear or nonlinear and deterministic or stochastic operator equations, including algebraic equations, ordinary differential equations, partial differential equations, and integral and integrodifferential equations [

Several investigators have proposed a variety of approaches to solve the Riccati equation, approximately [

This paper is arranged as follows. In the next section, we present a brief review of the ADM for nonlinear IVPs. In Section

We review the salient features of the Adomian decomposition method in solving IVPs for first-order nonlinear ordinary differential equations as

We rewrite (

Several algorithms for the Adomian polynomials have been developed by Rach [

From (

We remark that the convergence of the Adomian decomposition series has been previously proven by several researchers [

In this section, we advocate Adomian’s asymptotic decomposition method for solving the Riccati equation. We remark that Adomian’s asymptotic decomposition method does not need use of the initial condition to obtain the asymptotic solution or the solution in the large, which is another, convenient advantage in computations using this technique. Rather than nested integrations as required by decomposition, we now have nested differentiations. In effect our aim is to solve for the solution by not inverting the linear differential operator

In this section, several numerical examples are given to illustrate the efficiency of our technique as presented in this paper. We remark that all calculations are performed by Mathematica package 8.

Consider the following Riccati equation:

The exact solution is known in advance to be

To apply Adomian decomposition method, equation (

Solving (

Upon substitution and using the form of the Adomian polynomials in (

Computation shows that this Adomian decomposition series has a finite radius of convergence. By plotting the curves of

We calculated the Padé approximant

The near-field approximation

The Padé approximant

Consider the following Riccati equation:

The exact solution is known in advance to be

The curves of the near-field approximation

We calculated the Padé approximant

The near-field approximation

The Padé approximant

In this work, we combined the ADM and the AADM to approximate the global solution of the Riccati equation. We evaluated the approximate solution by matching the Padé approximant of the near-field approximation derived from the ADM with the far-field approximation derived from the AADM. Furthermore we have shown that the AADM can be an important complement in analysis of the solution’s asymptote.

The authors declare that there is no conflict of interests regarding the publication of this paper.