Some Computational Homotopy,Variational and Iterative Methods for Engineering and Applied Science Problems

摘要

The equations governing the engineering and applied sciences problems lead to the formation ofordinary, partial or fractional differential equations and different types of linear and nonlinearequations in general.In addition, the differential equations play an important role in modelingcomplicated physical, chemical and biological phenomenon such as vibrations, reaction process,ecological systems etc.The concept of differential equations has motivated a huge size of researchwork in the last several centuries.Moreover, it is worth mentioning that the backlog of differentialequations is arising in applications of applied science and engineering, are of either first or secondorder, e.g homogeneous or non-homogeneous advection problem, foam drainage equation, rigid rod ona circular surface, duffing oscillator and Van der Pol's oscillator problems.Third order equations occurin fluid mechanics problems, e.g., boundary layer Blasius and Falkner-Skan equations.Fourth orderequations explain the rectangular plate and clamped beam problems.Equations of order 5 and ＞ 5 arenot 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 engineeringphenomena, several computational, analytical and numerical solution techniques are developed andimplemented by various scientists and engineers to tackle with ordinary, partial or fractionaldifferential equations.These engineering and applied sciences model problems are highly nonlinearand 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 inborndifficulties 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, finitedifference, polynomial and non polynomial spline, Sink Galerkin, collocation, inverse scatteringmethods, successive approximations and Taylor collocation.But most of these numerical techniqueshave their restrictions coupled with some inbuilt insufficiencies like linearization, discretization,impractical assumptions, huge computational work and non-compatibility with the flexibility ofphysical problems.As a consequence, some more computational homotopy, variational and iterativemethods are introduced for solving the highly nonlinear ordinary, partial or fractional differential andnonlinear equations.These numerical methods divided into two main parts one is analytic-numeric (orseries methods) and second is purely numerical methods.
　　The plan of this thesis is to develop and implement the computational homotopy, variational anditerative 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 andengineering.Motivated from the above applications, we split our dissertation in four parts, in first partwe propose and employ the homotopy methods to solve the nonlinear ordinary and partial differentialequation.Some of problems arise in physics, biology and fluid mechanics, in second part we introducevariational approaches to deal with nonlinear problems, third section is devoted to the study of iterativemethods.In the last part of the present study is the extension of applications of novel analytic-numericmethods 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 thetheoretical aspects.
　　First part of our dissertation is based on three chapters, in chapter one, we give concisebackground of homotopy methods.In the section 1.2 of this chapter, we propose a new homotopyapproach coupled with the Laplace transformation for solving the nonlinear ordinary or partialdifferential equations, namely homotopy perturbation transform method (HPTM).The equations areLaplace transformed and the nonlinear terms are represented by He's polynomials.The solutions areobtained in the form of fast convergent series with elegantly assessable terms.This method, incompare to standard perturbation techniques, is appropriate even for systems without any small/largeparameters and therefore it can be applied more extensively than traditional perturbation techniques.Agood agreement of the novel method solution with the existing solutions is presented graphically andin tabulated forms to study the efficiency and accuracy of HPTM.Chapter two extend the applicationof HPTM, existing homotopy perturbation method (HPM) and finite difference technique for novelmathematical modeling of laminar, steady, incompressible boundary layer flow problems.Theselective transformation reduces the boundary layer partial differential equations into ordinarydifferential equations.The resulting nonlinear differential equations are solved for velocity andtemperautre 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 thepresent 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 andLaplace transformation for nonlinear differential equations.This method is called the AuxiliaryLaplace Parameter Method (ALPM).The nonlinear terms can be easily handled by the use ofAdomian polynomials.Comparison of the present solution is made with the existing solutions andexcellent agreement is achieved.The beauty of this proposed method is its capability of combining twopowerful methods via auxiliary parameter for obtaining fast convergence for any kind of ordinary orpartial differential equation.The fact that the proposed technique solves nonlinear problems withoutany discretization or restrictive assumptions can be considered as a clear advantage of this algorithmover the other numerical methods.
　　Second part of our study is based on Chapter four.The purpose of Chapter 4 applies thevariational approaches to solid mechanics, geophysical, physics and micro-electromechanical systemsproblems.The first contribution of this chapter is to find the maximum deflection of a rectangularclamped plate using Ritz, Galerkin and Kantorovich methods.Stresses are found out and numericalresults are plotted for a square plate in the form of curves for different Poisson's ratio.The results ofthe present problem are in good agreement with those reported earlier but, with a simple and practicalapproach.That is why this work is good as compared to other results in previous literature.In thesecond contribution of chapter 4, variational approaches to new soliton solutions and in third andfourth problems, natural frequency, angular frequency and the initial amplitude of nonlinear oscillatorequations are illustrated including the Ritz method, Hamiltonian approach and amplitude-frequencyformulation.Numerical results for various instances are presented and compared with those obtainedby variational methods, exact and existing literature.The comparisons show effectiveness, efficiencyand robustness of these methods.
　　The objective of third part of our thesis is to propose the iterative methods for nonlinearequations and nonlinear differential equations.This part is structured into three chapters.Followingthis chapter, in Chapter 5, 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 efficiencyof the method is tested on a number of numerical examples.On comparison with the eighth ordermethods, the new family of iterative methods behaves either similarly or better for the test examples.The motivation of Chapter 6 is to propose a new method, namely difference kernel iterative method tosolve the ordinary and partial differential equations.In this method, we have reduced the multipleintegrals into a single integral and expressed it in terms of difference kernel.To make the calculationeasy and convenient we have used Laplace transform to solve the difference kernel.The objective of Chapter 7 is three fold: first, to formulate the MHD flow over a nonlinear stretching sheet with slipcondition; second, to suggest a novel modified Laplace iterative method (MLIM) for governing flowproblem by suitable choice of an initial solution and third, the convergence of the obtained seriessolution is properly checked by using the ratio test.The method is based on the application of Laplacetransform to boundary layers in fluid mechanics.The obtained series solution is combined with thediagonal Pade' approximants to handle the boundary condition at infinity.The convergence analysiselucidates that the modified Laplace iterative method (MLIM) gives accurate results.An excellentagreement between the MLIM and existing literature is achieved.
　　The last part is consisting on chapter 8.In this chapter, we have solved some fractionalmathematical models appears in applied sciences by means of newly developed analytic-numercmethods via a modified Riemann-Liouville derivative.We conclude this thesis and give possibledirections for future research in chapter 9.

目录

声明

Acknowledgements

Abstract

Table of Contents

List of tables

List of figures

Chapter 1 Introduction about novel homotopy approach and its application

1．1 Background

1．2 Homotopy perturbation transform method (HPTM)

1．3 Application of partial differential equations

1．3．1 Example

1．3．2 Example

1．3．3 Example

1．3．4 Example

1．4 Application of ordinary differential equations

1．4．1 Example

1．4．2 Example

1．4．3 Example

Chapter 2 Numerical solutions of boundary layer flow problems

2．1 Introduction

2．2 Boundary layer flow equations

2．3 Padé Approximants

2．4 Problem formulation for nonlinear stretching Sheet with porous condition

2．4．1 Homotopy perturbation transform method(HPTM) solution

2．5 Formulation for thin film viscous flow over a shrinking／stretching sheet

2．5．1 Homotopy perturbation method (HPM) solution

2．6 Governing equations for the long slider problem

2．6．1 Homotopy perturbation method (HPM) solution

2．7 First-order chemical reaction problem for stretching／shrinking sheet

2．7．1 Homotopy method solution

2．7．2 Finite difference method solution

Chapter 3 Auxiliary Laplace parameter method (ALPM)

3．1 Introduction

3．2 Description of the method

3．3 Application

3．3．1 Example

3．3．2 Example

3．3．3 Example

Chapter 4 Variational approaches for versatile physical problems

4．1 Introduction

4．2 Basic knowledge for rectangular plate problem

4．2．1 Formulation of problem

4．2．2 Solution of the problem

4．2．3 Analysis

4．3 Zakharov equations

4．3．1 Solitary wave solution via semi-inverse method

4．4 Hamiltonian approach for rigid rod problem

4．5 Formulation of the clamped beam problem

4．5．1 Solution of the problem

Chapter 5 Iterative method for finding root of nonlinear equations

5．1 Basie Definitions

5．2 Main results

5．3 Convergence analysis

5．4 Numerical implementations

Chapter 6 Difference kernel iterative method

6．1 Introduction

6．2 Method description

6．3 Application of method

6．3．1 Example

6．3．2 Example

6．3．3 Example

6．3．4 Example

6．3．5 Example

Chapter 7 Modified Laplace iterative method for boundary layer flow problem

7．1 Introduction

7．2 Method description

7．3 Formulation for boundary layer slip condition problem

7．4 Solution of governing flow problem

7．5 Convergence of method

Chapter 8 Novel fractional analytic-numeric methods via Jumarie’s derivative

8．1 Basic definitions

8．2 Fractional analytic-numeric method Ⅰ

8．3 Application

8．3．1 Example

8．3．2 Example

8．4 Fractional analytic-numeric method Ⅱ

8．5 Application

8．5．1 Example

8．5．2 Example

8．5．3 Example

8．6 Fractional analytic-numeric method Ⅲ

8．7 Application

Chapter 9 Conclusion and future work

9．1 Conclusion

9．2 Directions for future work

Bibliography

Publications