Application Of Differential Equations Homotopy Analysis Method

Like other areas of study such as Finance and Economics, Mathematics is highly significant for the development and analysis of devices, structures, quantities, space and changes and it lays the foundation of Engineering. It has vast practical applications such as Black-Scholes equation for derivative and the public-key cryptosystem invented by Rivest, Shamir and Adelman, which is now the basis for security has redefined Financial Markets and Internet Security. This essay will discuss about the Engineering challenges while facing with non-linear or non-homogenous partial differential equations and enlighten on the possibility of Homotopy as an application of solving certain Engineering challenges.

Differential equations are one of the most important features of Mathematics in Engineering such as found in Newton`s Second Law in dynamics (mechanics and law of cooling in thermodynamics, Euler-Lagrange and Hamilton`s equation in classical mechanics, Radioactive decay in nuclear physics, Wave equation, Maxwell`s equations in electromagnetism and Laplace`s equation which defines harmonic functions engineering. Differential equations, integral equations or combinations of them, integro-differential equations, are obtained in modeling of real-life engineering phenomena that are inherently nonlinear with variable coef?cients [2]. Such type of equations does not have an analytical solution and should therefore be calculated by numerical or semi-analytical methods.

The desired results by engineers are precise and accurate which are generally obtained by the numerical analysis. At present, they require very high level programming codes and powerful processing units to deal with numerical approach to the solution. However, semi-analytical are seen as much convenient while comparing with numerical methods as they give very close approximation to the actual values. Another significant advantage of using such a method is the flexibility of expediently applying it to complex problems. Several analytical methods including the linear superposition technique, the exponential-function method, the Laplace decomposition method, the matrix exponential method, the homotopy perturbation method, variational iteration methods and the Adomian decomposition method have been developed for solving linear or nonlinear non-homogeneous partial differential equations. However, Homotopy Analysis Method (HAM) has evolved over recent years is being used to find approximate solution of differential, integral and integro-differential equations.

Like other analytical methods, HAM is an approach to find non-linear ordinary differential equations. Homotopy is basically devised from topology and is regarded as the continuation of the function from topological space to another and employs the concept of the possibility of flexibly generate a convergent series solution for nonlinear systems. The idea of homotopy was introduced by Shijun Liao in 1992 [5]. According to him,the main advantage of HAM over other analytical approaches is it is a series expansion method with being entirely independent of small physical parameters. In addition to that, the HAM is an unified method for the Lyapunov artificial small parameter method, the delta expansion method, the Adomian decomposition method, and the homotopy perturbation method. The greater generality of the method often allows for strong convergence of the solution over larger special and parameter domains and provides with an excellent flexibility in the expression of the solution and how the solution is explicitly obtained. Moreover, in concurrence with symbolic computation, the HAM can be grouped together along with many other standard mathematical methods-such as numerical methods, series expansion methods, integral transform methods, and so forth [6].

As engineering rely significantly on differential equations of various kinds, in most applications of engineering such as nonlinear equations arising in heat transfer and electro-hydro-dynamic flow, it is difficult to solve them analytically [5]. Interesting fact about HAM is that it is not related to the problem associated with engineering, but it also extends to nuclear and astrophysics where it is used to find the solutions of particle transport equation transforms a boundary value problem for a stationary transport equation in a boundary problem for a differential system that is solved using the techniques of a new homotopy analysis method.

The HAM has produced significant results in recent years and it is now being considered as a better approach towards finding solutions of equations involving nonlinear complexities. According to Martin, he managed to transform a boundary value problem for a stationary transport equation in a boundary problem for a differential system that was solved using the techniques of a new homotopy analysis method [4]In a separate research by Martin in 2011, he concluded that with the help of Simpson`s formula for the approximation of the definite integral that appears in the stationary transport equation and a new homotopy perturbation method (NHPM), exact solution of integro-differential equation for the steps h less than or equal to 0.25 were obtained [4]. Following graph is taken from the results of the research performed by Domairry in 2008 over HAM and NHPM in non-linear heat transfer equation which shows very close approximations by Ham compared to numerical solutions. The results for various researches performed over different practical engineering applications shows that HAM is a flexible and convenient technique and has the potential of solving basic and complex nonlinear differential, integral and integro-differential equation compared to conventional methodology, as it provides results that are significantly close to results obtained from numerical iteration. The practical application of HAM highlights the importance of Mathematics in Engineering and its application in finding solutions associated with this field of science. References

[1] Antonio Mastroberardino, "Homotopy analysis method applied to electrohydrodynamic flow", Communications in Nonlinear Science and Numerical Simulation, Volume 16, Issue 7, July 2011, Page 2730

[2] M. Matinfar, M. Saeidy, J. Vahidi, "Application of Homotopy Analysis Method for Solving Systems of Volterra Integral Equations", Advances in Applied Mathematics and Mechanics, Vol. 4, No. 1, February 2012, page 36-38

[3] Olga Martin, "A homotopy perturbation method for solving a neutron transport equation", Applied Mathematics and Computation, Volume 217, Issue 21, 1 July 2011, Page 8572-73

[4] Olga Martin, "On the homotopy analysis method for solving a particle transport equation", Applied Mathematical Modelling, Volume 37, Issue 6, 15 March 2013, Page 3966

[5] S. Abbasbandy, "The application of homotopy analysis method to nonlinear equations arising in heat transfer", Physics Letters A, Volume 360, Issue 1, 2006, Page 109

[6] Shijun Liao, "Comparison between the homotopy analysis method and homotopy perturbation method", Applied Mathematics and Computation, Volume 169, Issue 2, 15 October 2005, Page 1187

Additional Sources

https://www.dpmms.cam.ac.uk/~wtg10/importance.pdf http://en.wikipedia.org/wiki/Differential_equation http://en.wikipedia.org/wiki/Homotopy http://en.wikipedia.org/wiki/Homotopy_analysis_method