Finite Difference Approximations To Solutions Of Partial Differential Equations

This book introduces finite difference methods for both ordinary differential equations (ODEs) and partial differential equations (PDEs) and discusses the similarities and differences between algorithm design and stability analysis for different types of equations. A unified view of stability theory for ODEs and PDEs is presented, and the interplay between ODE and PDE analysis is stressed. The text emphasizes standard classical methods, but several newer approaches also are introduced and are described in the context of simple motivating examples.

The book is intended for graduate students of Engineering, Mathematics and Physics. We have numerically solved Hyperbolic and Parabolic partial differential equations with various initial conditions using Finite Difference Method and Mathematica. Replacing derivatives by finite difference approximations in these differential equations in conjunction with boundary conditions and initial conditions lead to equations relating numerical solutions at various position and time. These relations are intricate in that numerical value of the solution at one particular position and time is related with that at several other position and time. We have surmounted the intricacies by writing programs in Mathematica 6.0 that neatly provide systematic tabulation of the numerical values for all necessary position and time. This enabled us to plot the solutions as functions of position and time. Comparison with analytic solutions revealed nearly perfect match in every case. We have demonstrated conditions under which the nearly perfect match can be obtained even for larger increments in position or time.

What makes this book stand out from the competition is that it is more computational. Once done with both volumes, readers will have the tools to attack a wider variety of problems than those worked out in the competitors' books. The author stresses the use of technology throughout the text, allowing students to utilize it as much as possible.

A comprehensive performance analysis of the Finite Difference Method for the solution of Partial Differential Equations. Providing an in-depth understanding of; Finite Difference Methods, their applications, theoretical basis, the full derivation of Taylor Series Expansions and the construction of a working Computational Domain Grid System. Furthermore, detailing and showing how to effectively employ the Finite Difference Method, through the implementation of Finite Difference Schemes, to obtain accurate, stable and consistent numerical solutions for Partial Differential Equations, which model a multitude of varying dynamic processes. Moreover, it contains a detailed, thorough performance analysis investigation of three different Finite Difference Method schemes, when they are employed to obtain accurate numerical solutions for a fluid flow heat transfer process that is modelled by a first order Partial Differential Equation. These three schemes are the Forward-Time-Backwards-Space, Lax and Lax Wendroff Finite Difference Method schemes. Additionally, it explains the criteria that is required for optimal scheme stability, consistency and convergence. A brief breakdown of what the book contains;* A Description of the processes required to conduct an effective performance analysis of Finite Difference Method Schemes. * It specifies and explains the Forward-Time-Backwards-Space, Lax and Lax-Wendroff Finite Difference Scheme equations.* Explanations of the concepts of Finite Difference Method Stability, Consistency and Convergence. * The full derivations of the Taylor Series Expansions of the Forward-Time-Backwards-Space, Lax and Lax-Wendroff Finite Difference Scheme equations.* The development of an effective Finite Difference Method Computational Grid System, that can be used to calculate accurate numerical solutions for Partial Differential Equations. * A comprehensive end-to-end performance analysis of the three schemes for a fluid flow heat transfer process.* A discussion of the usefulness of the Finite Difference Method for solving Partial Differential Equations.* An overview of how to select an optimal Finite Difference Method scheme for accurate numerical solutions.You will gain valuable knowledge of the Finite Difference Method and its applications, expanding your expertise and intellect in this area of mathematics. Additionally, it will enable you to develop a systematic understanding of how to use Finite Difference Schemes to solve Partial Differential Equations and obtain accurate numerical solutions for dynamic processes. The book is self-contained allowing you to understand and conduct a Finite Difference Method performance analysis, so that you can apply the concepts to any process that is modelled by hyperbolic Partial Differential Equations. Furthermore, it is particularly valuable to; academics, educators, scholars, engineering industry professionals, and students. Especially, postgraduate Master's and undergraduate students. Assisting those who work/operate/study in the fields of Aerodynamics, Mathematics, Aerospace, Fluid Dynamics and Fluid Mechanics. Overall, this book will save you countless hours of research and reading, since the information contained within is distilled, concentrated and assimilated in an effective manner to help you to develop a deep understanding regarding the performance of the Finite Difference Method.

This book develops a systematic and rigorous mathematical theory of finite difference methods for linear elliptic, parabolic and hyperbolic partial differential equations with nonsmooth solutions. Finite difference methods are a classical class of techniques for the numerical approximation of partial differential equations. Traditionally, their convergence analysis presupposes the smoothness of the coefficients, source terms, initial and boundary data, and of the associated solution to the differential equation. This then enables the application of elementary analytical tools to explore their stability and accuracy. The assumptions on the smoothness of the data and of the associated analytical solution are however frequently unrealistic. There is a wealth of boundary – and initial – value problems, arising from various applications in physics and engineering, where the data and the corresponding solution exhibit lack of regularity. In such instances classical techniques for the error analysis of finite difference schemes break down. The objective of this book is to develop the mathematical theory of finite difference schemes for linear partial differential equations with nonsmooth solutions. Analysis of Finite Difference Schemes is aimed at researchers and graduate students interested in the mathematical theory of numerical methods for the approximate solution of partial differential equations.

Finite element methods for approximating partial differential equations have reached a high degree of maturity, and are an indispensible tool in science and technology. This textbook aims at providing a thorough introduction to the construction, analysis, and implementation of finite element methods for model problems arising in continuum mechanics. The first part of the book discusses elementary properties of linear partial differential equations along with their basic numerical approximation, the functional-analytical framework for rigorously establishing existence of solutions, and the construction and analysis of basic finite element methods. The second part is devoted to the optimal adaptive approximation of singularities and the fast iterative solution of linear systems of equations arising from finite element discretizations. In the third part, the mathematical framework for analyzing and discretizing saddle-point problems is formulated, corresponding finte element methods are analyzed, and particular applications including incompressible elasticity, thin elastic objects, electromagnetism, and fluid mechanics are addressed. The book includes theoretical problems and practical projects for all chapters, and an introduction to the implementation of finite element methods.