SOLUTION OF SOME DIFFERENTIAL EQUATIONS BY ADOMIAN DECOMPOSITION METHOD

Abstract

The aim of the present thesis is to study some ordinary and partial differen-tial equations using Adomian decomposition method. We have attempted to study a class of nonlinear singular boundary value problems and have given modifications to improve the accuracy of the solution obtained by this method. An attempt is also made to make the Aclornian method appropriate for boundary and initial value prob-lems. A further insight into the partial solutions in the decomposition method and comparison of Adomian method with new iterative method is also presented. The chapterwise summary of the thesis is as follows: Chapter 1 is introductory in nature in which a detailed description of the decom-position method is given. Application of the method in various branches of science and engineering are sketched. A brief account of the related studies made by various authors in the field and a summary. of the thesis are also presented. In Chapter 2, Adomian decomposition method has been applied to the single and ii coupled system of nonlinear singular boundary values problems of the form 0<x<1, 0>0, k>0 (1) y'(0) = 0 and ay(l) + by'(1) = y and L;y — f (y) = 0 0<x<1 (2) with boundary conditions dy =0 x=0 and —=B1(V)—y) x=1 dx dz where y = [y1(;z),'y2(x), ...'Yk()I',.f = [fi('J),.f2(y).....fk(J)j" and'' = ['01, 2, B1 is the Biot number, 'cb is a constant vector and the differential operator L is defined by rd( vd) Lmmx dxxdx y = 0, 1 or 2 depending on if the particle is slab, cylinder or spherical respectively. The boundary value problem (1) describes the steady-state oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics. This problem arises in the study of heat conduction in .the human brain and oxygen diffusion into body tissues. The equations (2), arise from Fick's law diffusion into a porous solid and nonlinear reaction within the solid. Numerical results obtained by ADM agrees with the results reported in the literature. A part of this chapter has been published in 111 the proceedings of Third International Conference on Mathematical Sciences, ICM 2008 held at UAE. In Chapter 3, we presents the techniques that can be applied to obtain the higher accuracy of the present method. We consider a class of singular boundary value problems (P(x)J')' _ .f (x, y), 0 < x <1 (3) y(0) = A, y(1) = B or y(0) = A, ay(1) + by'(1) = B with p(x) = x2'g(.x), 0 < -y < 1. This chapter also extends the method, developed in Chapter 2 for p(x) = x, to a general class p(x) = x~g(x). Numerical examples are presented and results are compared with previous known results. A part of this chapter has been published in Numerical Algorithms, Springer (2008) . In Chapter 4, Fractional differential equations, which are used to model problems in Physics, Finance and Hydrology [64,67] are studied. Fractional space derivatives may be used to formulate anomalous dispersion models, where a particle plume spreads at a rate that is different than the classical Brownian motion model. This chap-ter presents the analytical solutions of the space fractional diffusion and fractional integro-differential equations by Modified as well as Standard ADM. As the decom- position method does not require discretization of the variables, i.e. time and space, iv it is not affected by computational round off errors and one is not faced with necessity of large computer memory and time. Consequently, the computational size will be reduced. It is also worth noting that the advantage of the decomposition methodology is that it displays a fast convergence of the solution. It may be achieved by observing the phenomenon of self-cancelling noise terms and the splitting of the initial term into appropriate parts in Modified ADM. Therefore, the Modified ADM will further accelerate the convergence of the series. A part of this chapter has been published in Inter. J. of App!. v1ath. i\'Iech. (IJAMM) (2008). In Chapter 5, we outline a strategy to use Adomian decomposition method prop-erly for solving linear ordinary and partial differential equations with homogeneous and inhomogeneous boundary. It is shown that the Adomian decomposition method could not always satisfy all boundary conditions in solving partial differential equa-tions. Our fundamental goal in this Chapter is to present a further insight into partial solutions in the decomposition method, and the resolution of above cases by suitable transformation. The modifications are necessary in order to make the Adomian de-composition method efficient for boundary and initial value problems. The suggested modification is tested on heat and reaction diffusion equations with homogeneous and nonhomogeneous boundary conditions. Equations with or without force term are considered and the method performs extremely well in terms of efficiency and simplicity. A part of this chapter is communicated for publication. V Chapter 6, discuss a new formulation of Adomian polynomial independent of A. The new formula avoids the parameter which causes the decomposition series to appear to be a perturbation procedure which is an incorrect conclusion. Recently, Daftardar-Gejji and Jafari [35] have proposed a new technique, New iterative method (NIM) for solving nonlinear functional equations. They have shown that NIM yields better results than the existing Adomian decomposition method. We have shown that the NIM technique is nothing but the Adomian decomposition method where nonlinearity is defined by using the new formula of Adomian polynomials. Despite the advantages of NIM proposed by Daftardar-Gejji and Jafari, it still may sometimes be desirable to use ADM. This is illustrated with some examples where it is easy to work with ADM compare to NIM. A part of this chapter is communicated for publication. Finally, in Chapter 7, based on the present study, conclusions are drawn and future research work in this direction is suggested. The utility of the proposed modifications in Adomian decomposition method has been shown in the thesis.