Effects of Nonlinearity over Linearity by Using Homotopy Perturbation Method

 

S. K. Rana* and P. K. Sharma

Department of Mathematics, National Institute of Technology, Hamirpur-177005

*Corresponding Author E-mail: sanjeevrananit@gmail.com, psharma@nitham.ac.in.

ABSTRACT:

 

INTRODUCTION:

Non-linear differential equations have been the focus of many studies due to their frequent appearance in various applications in solid mechanics, fluid mechanics, Biology, Physics and engineering. The nature of interaction between thermal and elastic field is well explained by non-linear thermoelasticity. Various numerical techniques appeared to solve non-linear differential. Much effort was paid on existence, uniqueness and stability of the obtained solution. Recently, the focus is on numerical methods which do not require discretization of space and time variables or linearization of non-linear equations. Variational iteration method (VIM), Adomian’s decomposition method (ADM) and homotopy perturbation method (HPM) etc. are some such method used to solve non-linear differential equations. The basic idea of VIM is to construct a correction functional using a general Lagrange multiplier which can be identified optimally via varitional theory. Adomian’s decomposition method is to split the given equation into linear and nonlinear parts, invert the highest-order derivative operator contained in the linear operator in both sides and calculate Adomian’s polynomials. Sweilam [1] applied the VIM and ADM to solve numerically the harmonic wave generation in a non-linear, one-dimensional elastic half-space. Homotopy perturbation method [HPM] is a combination of homotopy in topology and classic perturbation techniques, provides us with a convenient way to obtain analytic or approximate solutions to a wide variety of problems arising in different fields.

 

 

Developing the perturbation method for different usage is very difficult because this method has some limitations and based on the existence of a small parameter. Therefore, many new methods have been recently introduced, some ways to eliminate the small parameter, such as artificial parameter method.

 

One of the semi-exact methods is HPM. He [2-8] has introduced it and successfully applied to solve different types of linear and nonlinear functional equations. This method has a useful feature, that it provides the solution in a rapid convergent power series with elegantly computable convergence of the solution. Problems with or without small parameters with the homotopy perturbation technique and the proposed method does not require small parameters in the equations, so the limitations of the traditional perturbation methods is eliminated. The initial approximation can be freely selected with possible unknown constants. The approximations obtained by this method are valid not only for small parameters, but also for very large values of parameters. Chun et al. [9] has obtained the exact solutions of nonlinear wave and diffusion equations without any restrictive assumption by homotopy method. Biazar and Ghazvini [10] studied the convergence of HPM and presented sufficient conditions for convergence of considered method. Babolian et al. [11] proposed some guidelines for beginners who intend to use HPM.    

 

In this paper, The HPM is used to discuss the effect of nonlinearity over linearity, under same initial conditions for first and second order partial differential equations along with application of the method in linear and non linear differential equations. The analytic results obtained have been computed numerically and presented graphically.

 

2.   BASIC IDEA OF HOMOTOPY PERTURBATION METHOD:                  

To convey an idea of HPM, we consider a general differential equation of the type

subjected to the boundary condition

where  is a general differential operator, is boundary operator, is a known analytic function, is the boundary of the  domain and denotes differentiation along the normal vector drawn outwards from

This shows that continuously traces an implicitly defined curve from starting point to as the embedding parameter increases monotonically from zero to unity. In topology, this is known as deformation. The expressions  and  are called homotopic. According to the HPM, the parameter  is used as a small parameter, and the solution of (4) can be expressed as a series in, in the form                                         

When, Eq. (4) corresponds to the nonlinear differential Eq.(1), and (6) becomes the approximate solution of Eq. (1), that is

If Eq. (1) admits a unique solution, then this method produces the unique solution. If Eq. (1) does not possess unique solution, the HPM will give one of the solution and many other solutions are possible. The convergence of the series (7) has been discussed by He [2,3].

 

3.  IMPLEMENTATION OF HPM:

(a) Linear Partial Differential Equation

Ex.1 Consider linear partial differential equation

subjected to initial condition

We define the homotopy equation :

where ,  are initial and original  solutions respectively.

Assuming the solution of the form He [2]

Substituting solution (11) into Eq. (10) and comparing coefficients of terms with identical powers of, leads to:

 

Initial conditions in homotopic form may be expressed as

 

We take initial solution as  , in Eqs. (12),

If , then the approximate solutions of the form

Ex. 2 : Consider linear second order partial differential equation (one dimensional wave equation),

subjected to initial and boundary conditions

 

Construct the homotopy equation as

Substituting solution (11) into Eq. (17) and comparing coefficients of terms with identical powers of, leads to:

where is Kronecker delta.

Assuming initial solution   in Eqs. (18), on solving, we get,

If , then the approximate solutions of the form

 

 

(b)  Nonlinear Differential Equation

Ex. 3: Consider first order non-linear partial differential equation,

subjected to initial condition (9).

Construct the homotopy equation as

Substituting solution (11) into Eq. (21), and comparing coefficients of terms with identical powers of, leads to:

Initial conditions

Suppose initial solution  , in Eqs. (22), on solving, we get,

If, then the approximate solutions of the form

                                                                                                                                                                                    (23)

Ex. 4: Consider second order non-linear partial differential wave equation

subjected to initial condition and boundary conditions

Construct the homotopy equation as

Initial conditions

Suppose initial solution   in Eqs. (27), on solving, we get,

If , then the approximate solutions of the form

 

              

4. CONCLUSION:

From section 3, conclude that nonlinearity produces progessively more and more deformation in the wave profile as time increases. These results haven shown that the homotopy perturbation method is reliable and efficient in handling nonlinear problems.

 

5. References:

1.          N.H. Sweilam, Harmonic wave generation in non linear thermoelasticity by variational iteration method and Adomian’s method, J. Computational Appl.Maths. 207 (2007) 64-72. 

2.          J.H. He, Homotopy perturbation technique, Comp. Meth. Appl. Mech. Eng. 178 (1999) 257-262.

3.          J.H. He, A coupling method of homotopy technique and a perturbation technique for non-linear problems, Int. J. Non-Linear. Mech. 35 (2000) 37-43.

4.          J.H. He, Homotopy perturbation method: a new nonlinear analytical technique, Appl. Math. Comput., 135 (2003) 73–79.

5.          J.H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Appl.Math. Comput., 151(2004) 287–292.

6.          J.H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons Fractals, 26 (2005) 695–700.

7.          J.H. He, Homotopy perturbation method for bifurcation of nonlinear problems,  Internat. J. Nonlinear Sci. Numer. Simul., 6 (2005) 207–208.

8.          J.H. He, Homotopy perturbation method for bifurcation of nonlinear problems,  Internat. J. Nonlinear Sci. Numer. Simul., 6 (2005) 207–208.

9.          C. Chun, H. Jafari and Y. Kim, Numerical method for the wave and nonlinear diffusion equations with the homotopy method, Comput. Math. Appl., 57 (2009) 1226-1231.

10.       J. Biazar, H. Ghazvini, Convergence of homotopy perturbation method for partial differential equations, Nonlinear Analysis: Real World Appl., 10 (2009) 2633-2640.

11.       E.  Babolian, A. Azizi, J. Saeidian, Some notes on using the homotopy perturbation method for solving time-dependent differential equations, Math. Comput. Modelling, 50 (2009) 213-224.

 

Received on 20.12.2011        Accepted on 20.02.2012        

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