Preface

Differential equations form the backbone of various physical systems occurring in a wide range of science and engineering disciplines viz. physics, chemistry, biology, economics, structural mechanics, control theory, circuit analysis, biomechanics, etc. Generally, these physical systems are modeled either using ordinary or partial differential equations (ODEs or PDEs). In order to know the behavior of the system, we need to investigate the solutions of the governing differential equations. The exact solution of differential equations may be obtained using well-known classical methods. Generally, the physical systems occurring in nature comprise of complex phenomena for which computation of exact results may be quite challenging. In such cases, numerical or semi-analytical methods may be preferred. In this regard, there exist a variety of standard books related to solution of ODEs and PDEs. But, the existing books are sometimes either method or subject specific. Few existing books deal with basic numerical methods for solving the ODEs and/or PDEs whereas some other books may be found related with semi-analytical methods only. But, as per the authors' knowledge, books covering the basic concepts of the numerical as well as semi-analytical methods to solve various types of ODEs and PDEs in a systematic manner are scarce. Another challenge is that of handling uncertainty when introduced in the model. Moreover, some books include complex example problems which may not be convincing to the readers for ease of understanding. As such, the authors came to the realization of need for a book that contains traditional as well as recent numerical and semi-analytic methods with simple example problems along with idea of uncertainty handling in models with uncertain parameters. With respect to student-friendly, straightforward, and easy understanding of the methods, this book may definitely be a benchmark for the teaching/research courses for students, teachers, and industry. The present book consists of 21 chapters giving basic knowledge of various recent and challenging methods. The best part of the book is that it discusses various methods for solving linear as well as nonlinear ODEs, PDEs, and sometimes system of ODEs/PDEs along with solved example problems for better understanding. Before we address some details of the book, the authors assume that the readers have prerequisite knowledge of calculus, basic differential equations, and linear algebra.

As such, the book starts with Chapter 1 containing preliminaries of differential equations and recapitulation of basic numerical techniques viz. Euler, improved Euler, Runge-Kutta, and multistep methods for solving ODEs subject to initial conditions. Chapter 2 deals with the exact solution approach for ODEs and PDEs. In this chapter, we address two widely used integral transform methods viz. Laplace and Fourier transform methods for solving ODEs and PDEs. Another powerful approximation technique, weighted residual method (WRM), is addressed in Chapter 3 for finding solution of differential equations subject to boundary conditions referred to as boundary value problems (BVPs). In this regard, this chapter is organized such that various WRMs viz. collocation, subdomain, least-square, and Galerkin methods are applied for solving BVPs. A new challenging technique viz. using boundary characteristic orthogonal polynomials (BCOPs) in well-known methods like Rayleigh-Ritz, Galerkin, collocation, etc. has also been introduced in Chapter 4.

Due to complexity in various engineering fields viz. structural mechanics, biomechanics, and electromagnetic field problems, the WRMs over the entire domain discussed in Chapter 3 may yield better results when considered over discretized domain. In this regard, various types of finite difference schemes for ODEs and PDEs, and application of the finite difference method (FDM) to practical problems by using schemes like explicit and implicit have been presented in Chapter 5. Finite element method (FEM) serves as another powerful numerical discretization approach that converts differential equations into algebraic equations. The FDM discussed in Chapter 5 generally considers the node spacing such that the entire domain is partitioned in terms of squares or rectangles, but the FEM overcomes this drawback by spacing the nodes such that the entire domain is partitioned using any shape in general. As such, Chapter 6 is mainly devoted to the FEM and especially Galerkin FEM. Effectiveness of the FEM is further studied for static and dynamic analysis of one-dimensional structural systems. Chapter 7 gives an idea of widely used numerical technique named finite volume method (FVM). Accordingly, brief background, physical theory, and algorithm for solving particular practical problem are addressed in this chapter. A brief introduction to another numerical discretization method known as boundary element method (BEM) is addressed in Chapter 8 along with BEM algorithm and procedure to find fundamental solution.

Some problems are nonlinear in nature resulting in governing nonlinear differential equations. Recently, research studies have been done for solving nonlinear differential equations efficiently and modeling of such differential equations analytically is rather more difficult compared to solving linear differential equations discussed in Chapters 1-8. So, this book may also be considered as a platform consisting of various methods that may be used for solving different linear as well as nonlinear ODEs and PDEs. Though the computation of exact solutions for nonlinear differential equations may be cumbersome, a new class of obtaining analytical solutions, that is semi-analytic approach, has emerged. Generally, semi-analytic techniques comprise of power series or closed-form solutions which have been discussed in subsequent chapters. In this regard, Akbari-Ganji's method (AGM) has been considered as a powerful algebraic (semi-analytic) approach in Chapter 9 for solving ODEs. In the AGM, initially a solution function consisting of unknown constant coefficients is assumed satisfying the differential equation subject to initial conditions. Then, the unknown coefficients are computed using algebraic equations obtained with respect to function derivatives and initial conditions. Further, the procedure of exp-function method and its application to nonlinear PDEs have been illustrated in Chapter 10. Semi-analytical techniques based on perturbation parameters also exist and have wide applicability. As such, Chapter 11 addresses Adomian decomposition method (ADM) for solving linear as well as nonlinear ODEs, PDEs, and system of ODEs, PDEs. In this regard, another well-known semi-analytical technique that does not require a small parameter assumption (for solving linear as well as nonlinear ODEs/PDEs) is Homotopy Perturbation Method (HPM). The HPM is easy to use for handling various types of differential equations in general. As such, a detailed procedure of the HPM is explained and applied to linear and nonlinear problems in Chapter 12. Further, Chapter 13 deals with a semi-analytical method viz. variational iteration method (VIM) for finding the approximate series solution of linear and nonlinear ODEs/PDEs. Then, Chapter 14 confers homotopy analysis method (HAM), which is based on coupling of the traditional perturbation method and homotopy in topology. Generally, the HAM involves a control parameter that controls the convergent region and rate of convergence of solution. It may be worth mentioning that the methods viz. ADM, HPM, VIM, and HAM discussed in Chapters 11, 12, 13, and 14, respectively, not only yield approximate series solution (which converges to exact solution) but they may produce exact solution also depending upon the considered problem.

Emerging areas of research related to solution of differential equations based on differential quadrature and wavelet approach have been considered in Chapters 15 and 16, respectively. Chapter 15 contributes an effective numerical method called differential quadrature method (DQM) that approximates the solution of the PDEs by functional values at certain discretized points. In this analysis, shifted Legendre polynomials have been used for computation of weighted coefficients. Further, in order to have an overview of handling ODEs using Haar wavelets, a preliminary procedure based on Haar wavelet-collocation method has been discussed in Chapter 16. Other advanced methods viz. hybrid methods that combine more than one method are discussed in Chapter 17. Two such methods viz. homotopy perturbation transform method (HPTM) and Laplace Adomian decomposition method (LADM) which are getting more attention of researchers are demonstrated to make the readers familiar with these methods. Differential equations over fractal domain are often referred to as fractal differential equations. Recently, fractal analysis has become a subject of great interest in various science and engineering applications. Often, the differential equations over fractal domains are referred to as fractal differential equations. Accordingly, in Chapter 18, only a basic idea of fractals and notion of fractal differential have been incorporated.

Another challenging concept of this book is also to introduce a new scenario in which uncertainty has been included to handle uncertain environment. In actual practice, the variables or coefficients in differential equations exhibit uncertainty due...