Some Computational Homotopy,Variational And Iterative Methods For Engineering And Applied Science Problems

The equations governing the engineering and applied sciences problems lead to the formation of ordinary, partial or fractional differential equations and different types of linear and nonlinear equations in general. In addition, the differential equations play an important role in modeling complicated physical, chemical and biological phenomenon such as vibrations, reaction process, ecological systems etc. The concept of differential equations has motivated a huge size of research work in the last several centuries. Moreover, it is worth mentioning that the backlog of differential equations is arising in applications of applied science and engineering, are of either first or second order, e.g homogeneous or non-homogeneous advection problem, foam drainage equation, rigid rod on a circular surface, duffing oscillator and Van der Pol’s oscillator problems. Third order equations occur in fluid mechanics problems, e.g., boundary layer Blasius and Falkner-Skan equations. Fourth order equations explain the rectangular plate and clamped beam problems. Equations of order5and>5are not very common. Hence, there is growing need to find the solution of these differential equations. However, from the last two decades with the swift improvement of nonlinear science and engineering phenomena, several computational, analytical and numerical solution techniques are developed and implemented by various scientists and engineers to tackle with ordinary, partial or fractional differential equations. These engineering and applied sciences model problems are highly nonlinear and nonlinear problems are non-integrable and not solvable in general, apart from a few exceptions, can not be solved analytically by using traditional methods. Unfortunately, due to the inborn difficulties of the nonlinear problems, most method of solution applies only in exceptional situation. Some of these highly nonlinear problems are solved numerically by using finite element, finite difference, polynomial and non polynomial spline, Sink Galerkin, collocation, inverse scattering methods, successive approximations and Taylor collocation. But most of these numerical techniques have their restrictions coupled with some inbuilt insufficiencies like linearization, discretization, impractical assumptions, 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 equations. These numerical methods divided into two main parts one is analytic-numeric (or series methods) and second is purely numerical methods. The plan of this thesis is to develop and implement the computational homotopy, variational and iterative methods to obtain the exact, approximate and numerical solutions of nonlinear ordinary, partial and fractional differential and nonlinear equations arising in several areas of sciences and engineering. Motivated from the above applications, we split our dissertation in four parts, in first part we propose and employ the homotopy methods to solve the nonlinear ordinary and partial differential equation. Some of problems arise in physics, biology and fluid mechanics, in second part we introduce variational approaches to deal with nonlinear problems, third section is devoted to the study of iterative methods. In the last part of the present study is the extension of applications of novel analytic-numeric methods for fractional differential equations related to physics, applied and engineering sciences. Particular attention is therefore focused on the numerical solution of these methods instead of the theoretical aspects.First part of our dissertation is based on three chapters, in chapter one, we give concise background of homotopy methods. In the section1.2of this chapter, we propose a new homotopy approach coupled with the Laplace transformation for solving the nonlinear ordinary or partial differential equations, namely homotopy perturbation transform method (HPTM). The equations are Laplace transformed and the nonlinear terms are represented by He’s polynomials. The solutions are obtained in the form of fast convergent series with elegantly assessable terms. This method, in compare to standard perturbation techniques, is appropriate even for systems without any small/large parameters and therefore it can be applied more extensively than traditional perturbation techniques. A good agreement of the novel method solution with the existing solutions is presented graphically and in tabulated forms to study the efficiency and accuracy of HPTM. Chapter two extend the application of HPTM, existing homotopy perturbation method (HPM) and finite difference technique for novel mathematical modeling of laminar, steady, incompressible boundary layer flow problems. The selective transformation reduces the boundary layer partial differential equations into ordinary differential equations. The resulting nonlinear differential equations are solved for velocity and temperautre profiles using the HPTM-Pade’, homotopy perturbation and the finite difference methods. Graphs are portrayed for the effects of some values of parameters. Moreover, comparison of the present solution is made with the existing solution and excellent agreement is noted. In third chapter, we propose another novel homotopy approach using auxiliary parameter, Adomian polynomials and Laplace transformation for nonlinear differential equations. This method is called the Auxiliary Laplace Parameter Method (ALPM). The nonlinear terms can be easily handled by the use of Adomian polynomials. Comparison of the present solution is made with the existing solutions and excellent agreement is achieved. The beauty of this proposed method is its capability of combining two powerful methods via auxiliary parameter for obtaining fast convergence for any kind of ordinary or partial differential equation. The fact that the proposed technique solves nonlinear problems without any discretization or restrictive assumptions can be considered as a clear advantage of this algorithm over the other numerical methods.Second part of our study is based on Chapter four. The purpose of Chapter4applies the variational approaches to solid mechanics, geophysical, physics and micro-electromechanical systems problems. The first contribution of this chapter is to find the maximum deflection of a rectangular clamped plate using Ritz, Galerkin and Kantorovich methods. Stresses are found out and numerical results are plotted for a square plate in the form of curves for different Poisson’s ratio. The results of the present problem are in good agreement with those reported earlier but, with a simple and practical approach. That is why this work is good as compared to other results in previous literature. In the second contribution of chapter4, variational approaches to new soliton solutions and in third and fourth problems, natural frequency, angular frequency and the initial amplitude of nonlinear oscillator equations are illustrated including the Ritz method, Hamiltonian approach and amplitude-frequency formulation. Numerical results for various instances are presented and compared with those obtained by variational methods, exact and existing literature. The comparisons show effectiveness, efficiency and robustness of these methods.The objective of third part of our thesis is to propose the iterative methods for nonlinear equations and nonlinear differential equations. This part is structured into three chapters. Following this chapter, in Chapter5, we present a family of iterative methods for solving nonlinear equations. It is proved that these methods have the convergence order of eight. These methods require three evaluations of the function, and only use one evaluation of first derivative per iteration. The efficiency of the method is tested on a number of numerical examples. On comparison with the eighth order methods, the new family of iterative methods behaves either similarly or better for the test examples. The motivation of Chapter6is to propose a new method, namely difference kernel iterative method to solve the ordinary and partial differential equations. In this method, we have reduced the multiple integrals into a single integral and expressed it in terms of difference kernel. To make the calculation easy and convenient we have used Laplace transform to solve the difference kernel. The objective of Chapter7is three fold:first, to formulate the MHD flow over a nonlinear stretching sheet with slip condition; second, to suggest a novel modified Laplace iterative method (MLIM) for governing flow problem by suitable choice of an initial solution and third, the convergence of the obtained series solution is properly checked by using the ratio test. The method is based on the application of Laplace transform to boundary layers in fluid mechanics. The obtained series solution is combined with the diagonal Pade’approximants to handle the boundary condition at infinity. The convergence analysis elucidates that the modified Laplace iterative method (MLIM) gives accurate results. An excellent agreement between the MLIM and existing literature is achieved.The last part is consisting on chapter8. In this chapter, we have solved some fractional mathematical models appears in applied sciences by means of newly developed analytic-numeric methods via a modified Riemann-Liouville derivative. We conclude this thesis and give possible directions for future research in chapter9.