Abstract

The modified decomposition method (MDM) and homotopy perturbation method (HPM) are applied to obtain the approximate solution of the nonlinear model of tumour invasion and metastasis. The study highlights the significant features of the employed methods and their ability to handle nonlinear partial differential equations. The methods do not need linearization and weak nonlinearity assumptions. Although the main difference between MDM and Adomian decomposition method (ADM) is a slight variation in the definition of the initial condition, modification eliminates massive computation work. The approximate analytical solution obtained by MDM logically contains the solution obtained by HPM. It shows that HPM does not involve the Adomian polynomials when dealing with nonlinear problems.

1. Introduction

Over the years, many mathematical models of tumour growth have appeared in literature [1–3]. These problems and phenomena are modeled by partial differential equations (PDE) such as deterministic reaction-diffusion equations which are used to model the spatial spread of tumours both at early growth and later invasive stages [4, 5]. In most cases, these problems do not admit analytical solution. So these equations should be solved using some particular techniques. Chaplain [6] used numerical solution (finite difference method) to solve the above problem. However, this method involved linearization, discretization, and assumption. Therefore, the real problem has to undergo simplification before it can be solved. In recent years, much attention has been devoted to the newly developed methods to construct an analytical solution of equation such as Adomian decomposition method (ADM) [7] and homotopy perturbation method (HPM) [8]. Both methods yield rapidly convergent series solutions for linear and nonlinear equations. The advantages of these methods are that they provide direct scheme for solving the problem, that is, without the need for linearization and discretization. The accuracy of the ADM method was studied extensively by Hashim et al. [9] and compared with other methods [10, 11]. Anderson et al. [12] proposed a modification of the ADM by a slight variation from the standard ADM. The modified method (MDM) was established based on the assumption that the initial function can be divided into two parts and the success of the MDM depends mainly on the proper choice of the parts. In this paper, we present approximate analytical solution of tumour invasion and metastasis model [13] solved by MDM and HPM. The results from both methods are then compared and reveal their capability, effectiveness and convenience. Both methods give successive approximations of high accuracy solution.

2. Problem Formulation

Let us consider a system describing the interactions of the tumour cells (denoted by ), extra cellular matrix (ECM, denoted by ), and matrix degrading enzymes (MDE, denoted by ) is given by [13] where is the tumour cell random motility coefficient, is the MDE diffusion coefficient, is the haptotactic coefficient, and , , are the positive constants.

Non-dimensionalise of (1) by setting where is the tumour cell density, is the ECM density, is the MDE concentration, is the length scale, and is the time (, where is a reference chemical diffusion coefficient). By dropping the tildes for notational convenience, we obtain the scaled system of equations: where , , , , , and . The initial conditions of each equation are where is a positive constant.

The approximate solutions of (3)–(5) are obtained by integrating each equation once with respect to and using the initial condition. Hence we obtained In (7)–(9), we assume , and are bounded for all in , , and ,  for all , . The terms , , , , , and are Lipschitz continuous with

3. Mathematical Methods

3.1. Adomian Decomposition Method (ADM)

The Adomian decomposition method is applied in (3)–(5): where is integrable differential operator with .

Operating on both sides of (11)–(13) with the integral operator leads to where are the nonlinear terms. The solutions , , and can be decomposed by an infinite series as follows [7]: where , , and are the components of , , and that will elegantly be determined. The nonlinear term is decomposed by the following infinite series: where is called Adomian’s polynomial and defined by From the above consideration, the decomposition method defines the components , , and for by the following recursive relationships.

Anderson et al. [12] proposed that the construction of the zeroth component of the decomposition series can be defined in a slightly different way. The modified method (MDM) was established based on the assumption that if the zeroth component and the function is possible to divide into two parts such as and , one can formulate the recursive algorithm for and general term in a form of the modified recursive scheme as follows:

for ,

for ,

for , This type of modification is giving more flexibility to the ADM in order to solve complicated nonlinear differential equations. MDM scheme avoids the unnecessary computation especially in calculation of the Adomian polynomials. The computation of these polynomials will be reduced very considerably by using the MDM.

3.2. Homotopy Perturbation Method (HPM)

To solve (3)–(5) with the HPM method, we construct the following homotopy: or In HPM, the solutions of (25)–(28) are expressed as power series in : where is an embedding parameter and , , and are the arbitrary initial approximation satisfying the given initial condition. As approaches to 1, we obtained Substituting (29)–(31) into (25), Substituting (30)-(31) into (26), Substituting (29)–(31) into (27), Equating the coefficients of the terms in (32)–(34) with the identical powers of , we obtained the following.

From (32), From (33), From (34),

4. Existence and Convergence of MDM and HPM

Theorem 1. Let ; then (3)–(5) have a unique solution.

Proof. (I) Let and be two different solutions of (7) then from which we get . Since , , implies and completes the proof.
(II) Let and be two different solutions of (8) then from which we get . Since , , implies and completes the proof.
(III) Let and be two different solutions of (9); then from which we get . Since , , implies and completes the proof.