Some of the problems of real world phenomena can be described by differential equations involving the ordinary or partial derivatives with some initial or boundary conditions. To interpret the physical behavior of the problem it is necessary to know
the solution of the differential equation. Unfortunately, it is not possible to solve some of the differential equations whether they are ordinary or partial with initial or boundary conditions through the analytical methods. When, we fail to find the solution
of ordinary differential equation or partial differential equation with initial or boundary conditions through the analytical methods, one can obtain the numerical solution of such problems through the numerical methods up to the desired degree of
accuracy. Of course, these numerical methods can also be applied to find the numerical solution of a differential equation which can be solved analytically.
Several problems in natural sciences, social sciences, medicine, business management, engineering, particle dynamics, fluid mechanics, elasticity, heat transfer, chemistry, economics, anthropology and finance can be transformed into boundary value problems using mathematical modeling. A few problems in various fields of science and engineering yield linear and nonlinear boundary value problems of second order such as heat equation in thermal studies, wave equation in communication etc.
Fifth-order boundary value problems generally arise in mathematical modeling of viscoelastic flows. The dynamo action in some stars may be modeled by sixth-order boundary-value problems. The narrow convecting layers bounded by stable layers which are believed to surround A-type stars may be modeled by sixth-order boundary value problems which arise in astrophysics. The seventh order boundary value problems generally arise in modeling induction motors with two rotor circuits. Various phenomena such as convection, flow in wind tunnels, lee waves, eddies, etc. can also be modeled by higher order boundary value problems.
Inhaltsverzeichnis (Table of Contents)
-
Chapter 1
INTRODUCTION
- 1.1. Background
- 1.2. Literature survey
- 1.3. Existence and uniqueness of two-point BVPs
- 1.4. Stability analysis
- 1.5. Convergence of two-point BVPs
- 1.6. Cubic spline and non-polynomial spline functions
- 1.7. Newton’s method
- 1.8. Perturbation problems
- 1.9. Scope of the work
-
Chapter 2
SOLUTION OF LINEAR BOUNDARY VALUE PROBLEMS BY APPROACHING
NON POLYNOMIAL SPLINE TECHNIQUES
- 2.1. Introduction
- 2.2. Description of the method
- 2.3. Numerical illustrations
-
Chapter 3
SOLUTION OF NON LINEAR BOUNDARY VALUE PROBLEMS BY
APPROACHING NON POLYNOMIAL SPLINE TECHNIQUES
- 3.1. Introduction
- 3.2. Description of the method
- 3.3. Numerical illustrations
-
Chapter 4
NUMERICAL SOLUTION OF FIFTH ORDER BOUNDARY VALUE PROBLEMS
USING SIXTH AND SEVENTH DEGREE SPLINE FUNCTIONS
- 4.1. Introduction
- 4.1.1. Cubic spline-Bickley’s method
- 4.1.2. The two-point second order boundary value problem
- 4.2. Numerical solution of fifth order boundary value problems using sixth degree spline function
- 4.2.1. Construction of sixth degree spline
- 4.2.2. Method of obtaining the solution of fifth order boundary value problem by sixth degree spline
- 4.2.3. Numerical illustrations
- 4.3. Numerical solution of fifth order boundary value problems using seventh degree spline function
- 4.3.1. Construction of seventh degree spline
- 4.3.2. Method of obtaining the solution of fifth order boundary value problem by seventh degree spline
- 4.3.3. Numerical illustrations
- 5.1. Introduction
- 5.2. Numerical solution of the sixth order boundary value problems using seventh degree spline functions
- 5.2.1. Construction of seventh degree spline
- 5.2.2. Method of obtaining the solution of sixth order boundary value problem by seventh degree spline
- 5.2.3. Numerical illustrations
- 5.3. Numerical solution of the sixth order boundary value problems using eighth degree spline functions
- 5.3.1. Construction of eighth degree spline
- 5.3.2. Method of obtaining the solution of sixth order boundary value problem by eighth degree spline
- 5.3.3. Numerical illustrations
- Conclusion
- Future Work
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
The book explores the use of spline methods to approximate solutions to various boundary value problems in ordinary differential equations. The primary focus is on applying both polynomial and non-polynomial spline techniques to linear and nonlinear boundary value problems of different orders.
- Application of non-polynomial spline functions to solve linear boundary value problems with constant and variable coefficients.
- Extending the non-polynomial spline method to solve nonlinear boundary value problems.
- Developing numerical methods to obtain solutions for fifth-order boundary value problems using sixth and seventh degree spline functions.
- Constructing and utilizing seventh and eighth degree spline functions to solve sixth-order boundary value problems.
- Comparison of the accuracy and efficiency of the developed spline methods with existing techniques and exact solutions.
Zusammenfassung der Kapitel (Chapter Summaries)
Chapter 1 provides a comprehensive overview of boundary value problems, their significance in various scientific disciplines, and a detailed review of existing numerical methods. It also introduces the concepts of spline functions and their properties, particularly focusing on cubic and non-polynomial splines.
Chapter 2 delves into the application of non-polynomial spline techniques for solving linear second-order boundary value problems. It presents a numerical method based on non-polynomial splines, demonstrating its efficiency and accuracy through numerical examples.
Chapter 3 extends the non-polynomial spline method to approximate solutions for second-order nonlinear boundary value problems. The chapter provides examples showcasing the method's effectiveness in handling various types of nonlinear problems.
Chapter 4 focuses on solving fifth-order boundary value problems, which often arise in viscoelastic flow modeling. The chapter presents numerical methods using sixth and seventh degree spline functions to approximate solutions, demonstrating their accuracy and efficiency.
Chapter 5 addresses sixth-order boundary value problems, relevant in astrophysics and other fields. It presents numerical methods based on seventh and eighth degree spline functions, highlighting their comparative performance and accuracy against other known techniques.
Chapter 6 summarizes the key findings and contributions of the book, drawing conclusions about the efficacy of the developed spline methods. It also explores potential directions for future research, including extending these methods to higher-order problems and partial differential equations.
Schlüsselwörter (Keywords)
The main keywords and topics covered in the book include boundary value problems, spline methods, non-polynomial splines, ordinary differential equations, numerical solutions, linear and nonlinear problems, higher-order problems, accuracy, efficiency, and applications in diverse scientific disciplines.
- Citation du texte
- Dr. Parcha Kalyani (Auteur), 2014, Spline Solutions of Higher Order Boundary Value Problems, Munich, GRIN Verlag, https://www.grin.com/document/541611
