International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395 -0056
Volume: 04 Issue: 05 | May -2017
p-ISSN: 2395-0072
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Application of Homotopy Analysis Method for Solving various types of Problems of Partial Differential Equations V.P.Gohil1, Dr. G. A. Ranabhatt2 1,2Assistant
Professor, Department of Mathematics, Government Engineering College, Bhavnagar, India ---------------------------------------------------------------------***---------------------------------------------------------------------
Abstract - In this paper, various types of linear, non-linear,
By means of generalizing the traditional homotopy method, Liao constructed the so-called zero-order deformation equations
Key Words: homotopy analysis method, partial differential equation, linear, homogeneous, linear, non linear, homogeneous, non homogeneous
(1)
homogeneous, non homogeneous problems of partial differential equations discussed. Also shown that homotopy analysis method applied successfully for solving non homogeneous and non linear equations
Where
is an embedding operators,
are nonzero
auxiliary functions, is an auxiliary linear operator, are initial guesses of and are
1.INTRODUCTION In recent years, this method (HAM) has been successfully employed to solve many types of non linear, homogeneous or non homogeneous, equations and systems of equations as well as problems in science and engineering . Very recently, Ahmad Bataineh et al.([2]) presented two modi_cations of HAM to solve linear and non linear ODEs. The HAM contains a certain auxiliary parameter h which provides us with a simple way to adjust and control the convergence region and rate of convergence of the series solution. Moreover, by means of the so-called h -curve, it is easy to determine the valid regions of h to gain a convergent series solution. Thus, through HAM, explicit analytic solutions of non linear problems are possible. Systems of partial differential equations (PDEs) arise in many scientific models such as the propagation of shallow water waves and the Brusselator model of the chemical reaction-diffusion model. Very recently, Batiha et al. [2] improved Wazwazs [9] results on the application of the variational iteration method (VIM) to solve some linear and non linear systems of PDEs. In [8], Saha Ray implemented the modified Adomian decomposition method (ADM) for solving the coupled sine-Gordon equation.
2. HOMOTOPY ANALYSIS METHOD
unknown functions. It is important to note that, one has great freedom to choose auxiliary objects such as and in HAM. When
we get by (1),
Thus increase from 0 to 1, the solutions from initial guesses Expanding
to
.
in Taylor series with respect to , (2)
Where ,
(3)
If the auxiliary linear operator, initial guesses, the auxiliary parameter and auxiliary functions are properly chosen than the series eqution (2) converges at
We consider the following differential equations,
varies
. (4)
Where
are nonlinear operators that the represents the
whole equations, x and t are independent variables and are unknown functions respectively.
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This must be one of solutions of the original nonlinear equations. According to (3), the governing equations can be deduced from the zero-order deformation equations (1). ISO 9001:2008 Certified Journal
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