Computational Homotopy And Iterative Methods For Applied Sciences And Engineering Problems

The governing problems in engineering and applied sciences lead to the formation of ordinary, partial and fractional differential equations with different types of linearity and nonlinearity. In order to model different complicated biological, chemical and physical phenomena differential equations play important role. A huge size of research work in the last several centuries progressed due to the motivation of differential equation. Moreover, differential equation arising in application of engineering and applied sciences problems is of first or second order, for example, Vander Pol’s oscillating problem, duffing oscillator, rigid rod on a circular face, vibration equation, foam drainage equation, homogeneous and non-homogeneous advection problem. Third order equation arises in fluid mechanics, for example, Falker-Skan equation and boundary layer Blasius equation. Clamped beam problem and rectangular plate problem explain the fourth order equations. While Kaup-Kupershmidt, Sawada-Kotera and KdV equation are the examples of fifth order but they are not very common. Hence, there is growing need to find the solution of these differential equations. In last few ’decades’ swift improvement in the field of engineering and nonlinear sciences, several numerical techniques, analytical and computational techniques are developed and implemented by various scientist and engineers to tackle with ordinary, partial or fractional differential equations. These engineering and model science problems are highly linear and nonlinear problems are non-integrable and not solvable in general, apart from a few exceptions, cannot be solved analytically by using traditional methods. Unfortunately, due to the inborn difficulties of nonlinear problems, most methods of solution apply only in exceptional situation. Some of these highly nonlinear problems are solved numerically by using finite element method, finite difference, inverse scattering methods, successive approximations, Taylor collocation, sink Galerkin, Collocation, polynomial and non-polynomial spline. But most of these numerical techniques have their restrictions and some deficiencies like impractical assumptions, discretization, linearization, huge computational work and non-compatibility with the flexibility of physical problems. As a consequence, some more computational homotopy, variational and iterative methods are introduced for solving the highly nonlinear ordinary, partial or fractional differential and nonlinear equation. These numerical methods are divided into two parts namely numerical and analytic-numeric method.The main objective of this thesis is to develop and implement the variational, transformed, iterative and computational homotopy method to obtain the numerical, approximate and exact solutions of nonlinear ordinary, partial and fractional differential equations arising in different areas of engineering and applied sciences. Motivated from application of above mentioned method, we split the dissertation into four parts. In the first part we propose and employ the combination of Elzaki transform and projected differential transform method to solve nonlinear ordinary, partial, fractional and fuzzy differential equations arises in applied sciences. In the second part we propose and employ some modified forms of He’s homotopy perturbation method (HPM) to solve nonlinear problems. Whereas in third part Elzaki projected differential transform method is applied to solve vibration equation and generalized Drinfeld-Sokolov equations.First part of our Dissertation is based on three chapters. In the first chapter in section 1.1 history of the transform method and their vast applications were discussed concisely. In section 1.2 of this chapter weintroduced and employ the new combination of Elzaki transform and projected differential transform method to solve the nonlinear ordinary, partial, fuzzy and fractional differential equations, namely Elzaki projected differential transform method (EPDTM). In the first chapter we employ EPDTM to solve fractional Telegraph equation to show the accuracy and efficiency of proposed method. Using this new innovative technique it is possible to find the exact solution or closed approximation solutions without using He’s and Adomian polynomials, which makes calculations simpler and easier. Error analysis shows that the solution obtained by aforementioned method rapidly converges towards exact solution. Numerical result shows the new technique is easily applicable and computationally simple. Several examples are solved to highlight the stability and accuracy of the method. Chapter two extended the application of newly proposed method to solve fuzzy quadratic Riccati differential equation. Fuzzy variables and parameters are denoted by convex normalized sets. EPDTM reduces the complex fuzzy differential equations into easily solvable differential equations. The resulting equations are than solved with EPDTM and results are presented in the graphic and tabular form. Moreover, good agreement has been found on comparison of the obtained results with other analytical methods. Numerical and graphical results how the capability and simplicity of the proposed method. Where as in third chapter the aforesaid method is applied to quadratic fractional Riccati equation and numerical examples are solved. Numerical results obtained with the proposed method are than compared with the results already obtained by other analytical and numerical methods, which shows the accuracy, stability and consistency of the proposed method. Numerical results are computed with less computational effort and show higher accuracy and presented in tabular form.In second part of the thesis we propose some new modified forms of He’s homotopy perturbation method which is useful tool to solve nonlinear ordinary, partial and fractional differential equations. In Chapter four we combine the Elzaki transform with homotopy perturbation method to solve Sine Gordon and Klein Gordon equation and obtained results are compared with the results of other analytical and transformed methods, observed a good agreement between the results obtained by EHPM and other analytical methods. In Chapter five we introduced a new concept for homotopy perturbation method to solve the nonlinear equation with two nonlinear terms by using two embedding parameter instead of one expanding parameters and used the idea of book keeping parameter method as well, which make it easier to handle a nonlinear equation with more nonlinear terms. This idea is employed to solve nonlinear cubic quintic oscillator and results obtained by HPM with expanding parameters are compared with energy balance method (EBM), He’s frequency formulation method, book keeping parameter method, variational iteration method and global error minimization method. Which shows that new modification in HPM is efficient and reliable tool to solve nonlinear differential equations with more nonlinear terms. In chapter six we employ the idea of homotopy perturbation transform method (HPTM), which is a combination of Laplace transformation and homotopy perturbation method (HPM) for solving nonlinear ordinary and partial differential equations. The equations are Laplace transformed and nonlinear terms are represented by He’s polynomials. The solutions are obtained in the series form, which converges at fast rate with easily computable terms. Comparison of standard perturbation method shows that HPTM is appropriate even for system without any small/large parameters and therefore it can be applied more extensively than traditional perturbation techniques. HPTM is used to obtain soliton solution of Kaup-kupershimdt (KK) equation, which is modified form of fifth order KdV equation. Numerical results are presented in tabular form to show the accuracy, efficiency and reliability of this method.The objective of third part of thesis is to discuss application of EPDTM for nonlinear ordinary and partial differential equations. In chapter 7, we use EPDTM to solve vibration equation for large domain and its efficiency is tested using the different initial values for numerical examples, obtained results are presented in graphical form to study the accuracy and efficiency of EPDTM. In chapter 8, a system of generalized Drinfeld-Sokolov (gDS) equations is considered, which can be used to model nonlinear wave processes in two component media. We apply Elzaki projected differential transform method for solving gDS. Compare the numerical results obtain by proposed method with the exact solution. The comparison shows that EPDTM is reliable, accurate, and effective and gives the convergent solution. An error analysis of the obtained solution is provided. We conclude this thesis and give possible future directions for future research in Chapter 9.