Abstract and Applied Analysis

Volume 2015, Article ID 712584, 9 pages

http://dx.doi.org/10.1155/2015/712584

## Power Series Solution for Solving Nonlinear Burgers-Type Equations

^{1}Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil^{2}Departamento de Ciências Exatas e Tecnológicas, Universidade Estadual de Santa Cruz, Campus Soane Nazaré de Andrade, Rodovia Jorge Amado, Km 16, Bairro Salobrinho, 45662-900 Ilhéus, BA, Brazil^{3}Departamento de Física, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, Apartado Postal 14-740, 07000 México, DF, Mexico

Received 20 October 2014; Revised 9 March 2015; Accepted 10 March 2015

Academic Editor: Fazal M. Mahomed

Copyright © 2015 E. López-Sandoval et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Power series solution method has been traditionally used to solve ordinary and partial linear differential equations. However, despite their usefulness the application of this method has been limited to this particular kind of equations. In this work we use the method of power series to solve nonlinear partial differential equations. The method is applied to solve three versions of nonlinear time-dependent Burgers-type differential equations in order to demonstrate its scope and applicability.

#### 1. Introduction

Power series solution (PSS) method is an old method that has been limited to solve linear differential equations, both ordinary differential equations (ODE) [1, 2] and partial differential equations (PDE) [3, 4]. Linear PDE have traditionally been solved using the separation of variables method because it permits obtaining a coupled system of ODE easier to solve with the PSS method. Some examples of these are the Legendre polynomials and the spherical harmonics used in Laplace’s equations in spherical coordinates or in Bessel’s equations in cylindrical coordinates [3, 4]. It is known that in nonlinear PDE (NLPDE) this procedure is not possible.

In this work we compare the spectral method (SM) with the PSS method solving three versions of nonlinear time-dependent Burgers-type equations [5] because we know that the SM is the more accurate numerical method. The SM with collocation points (SMCP) is a numerical technique applied to solve linear and nonlinear differential equations with high accurate approximations to the solution [6]. This has been used to solve PDE using polynomial interpolation function with an orthogonal basis such as Fourier, Chebyshev, or Legendre functions [7]. The SM has also been very successful to solve any kind of DE problems, including integro-differential problems [8], with Newman boundary values [9], and nonlinear PDE [10].

We use the symbolic computation package Matlab to obtain the algebraic operations for the truncated series approximation. This program helps to do easier the tedious algebraic operations.

#### 2. Power Series Solution Method

We know that almost the totality of the NLPDE does not have a solution with an analytic expression, that is, a solution in a closed form of known functions. Our goal is to construct a solution using a power series, taking advantage of the capacity of power series to represent any function with algebraic series developing the idea to construct an approximate solution [11–17]. It also has the possibility to approximate a solution, inclusive if an analytic form does not exist, in a similar way like the Taylor’s series approximate the functions. The existence of the PSS does not guarantee per se that the represented function has an exact approximation in distant points relative to the central value. However, considering that the PSS needs to satisfy the NLPDE, with initial values condition (IVC) or with boundary values conditions (BVC), therefore we can construct a well posed problem to obtain an accurate solution, constrained with all these limiting conditions [18]. Furthermore, the polynomial of the PSS is a smoothed function and this can guarantee the existence of a solution [18].

The PSS method represents a general solution with a series of unknown coefficients. When the PSS polynomial is substituted in the PDE we obtain a recurrence relation for the expansion coefficients. These coefficients should be expressed in function of the coefficients result from IVC or BVC. In this way, we obtain a system of equations depending on these initial value based coefficients. In order to obtain and solve a consistent algebraic system of equations, we also need the same number of coefficients and equations [11]. All these conditions, in the beginning, provide a guarantee that the PDE is a well posed problem; that is, existence, uniqueness, and smoothness of the solution are well defined [18].

Finally, the PSS method is a proposal to find a semianalytic solution as an asymptotic approximation (in space and time) of a finite series with minimal error in the expansion of terms of the series. From numerical analysis when a power series converges on an interval to a function , the radius of convergence is . In our work, the radius of convergence is defined by each interval where our error was estimated as we see below.

#### 3. Numerical Results

First, we consider the nonlinear time-dependent one-dimensional generalized Burgers-Huxley equation [5]:with the initial condition where , , , and are real parameters. With , this equation admits a travelling wave solution. Then readswith , where and represent the wave number and frequency of the travelling wave, respectively, working as unknown variables. Introducing (3) in (1) we obtain The ansatz for (4) will be a PSS asThe respective derivatives and nonlinear terms in (4) result in Substituting the series of (6) in (4), we obtain the recurrence relationSolving with Matlab until degree of PSS from (4), we obtain the following values for the coefficients: We will use the initial conditions to obtain the unknown coefficients (8). From the initial condition (2), we express as a polynomial series applying Taylor’s theorem. ThenMatching the coefficients of this polynomial with the coefficients (8) of our ansatz, we obtain the next values: , , , , , , , , and so forth.

With and values matched to their respective coefficients in (8), we obtain an algebraic system of 2 equations with two variables. Solving this one, we obtain the value of the unknown variables and :Then, the complete solution as PSS for the NLDE (1) readsAs it usually does when an approximate solution with PSS is obtained, a test of accuracy of the approximation must be performed. In this way, we calculate the absolute difference between exact and approximated solution defined as , where is the exact solution obtained from [5], and is the calculated solution (11), at the point , until the power degree , respectively. We compute the error, with the parameters values , , and , within the intervals and . This result is shown in Figure 1. This parameter set was selected because it is the same one used in [5] to do a comparison. The convergence of the power series, , depends on and also the coefficient , and then it is possible to adjust these ones to solve the NLDE and to find a solution that approximates its behavior to any distance and time, at less in the interval where we calculate the solution.