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.
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
Chapter 5 NUMERICAL SOLUTION OF SIXTH ORDER BOUNDARY VALUE PROBLEMS USING SEVENTH AND EIGHTH DEGREE SPLINE FUNCTIONS
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
Chapter 6 OVERALL CONCLUSION AND SCOPE FOR THE FUTURE WORK
Conclusion
Future Work
Research Objectives and Themes
The primary objective of this work is to develop and analyze numerical methods for solving linear and nonlinear boundary value problems (BVPs) of varying orders using polynomial and non-polynomial spline functions. The research addresses the limitations of analytical methods in solving differential equations involving real-world phenomena, providing high-accuracy numerical approximations suitable for fields such as engineering, physics, and astrophysics.
- Numerical solution of second-order linear and nonlinear BVPs using non-polynomial spline techniques.
- Development of sixth and seventh-degree spline functions for solving fifth-order boundary value problems.
- Implementation of seventh and eighth-degree spline functions for the numerical solution of sixth-order BVPs.
- Comparative analysis of the proposed spline methods against existing numerical techniques, including finite difference and B-spline methods.
- Validation of numerical methods through illustrative examples with varying parameters and step sizes.
Excerpt from the Book
1.1 Background
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. The problems related to the rate of change of certain entity in applied mathematics, physical science and engineering fields yield differential equations with appropriate 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. For example, perhaps till now, the solution of differential equation is not known. 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 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.
Using the mathematical modeling 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 ordinary or partial differential equations. We can employ the numerical methods to obtain an approximate solution which is near to accurate solution. These methods can also be employed to obtain approximate solutions of problems encountered in various fields, such as weather prediction, aero dynamics, crash safety measures in automobiles, operational research, stock market and actuarial science.
Summary of Chapters
Chapter 1 INTRODUCTION: This chapter provides the theoretical background, literature survey, and foundational concepts like stability and spline functions relevant to the study of boundary value problems.
Chapter 2 SOLUTION OF LINEAR BOUNDARY VALUE PROBLEMS BY APPROACHING NON POLYNOMIAL SPLINE TECHNIQUES: This chapter details the application of non-polynomial spline methods to solve second-order linear boundary value problems, demonstrating their efficiency through numerical examples.
Chapter 3 SOLUTION OF NON LINEAR BOUNDARY VALUE PROBLEMS BY APPROACHING NON POLYNOMIAL SPLINE TECHNIQUES: This chapter extends the non-polynomial spline approach to nonlinear two-point boundary value problems and presents numerical validation.
Chapter 4 NUMERICAL SOLUTION OF FIFTH ORDER BOUNDARY VALUE PROBLEMS USING SIXTH AND SEVENTH DEGREE SPLINE FUNCTIONS: This chapter covers the development of sixth and seventh-degree splines specifically for fifth-order boundary value problems, often found in viscoelastic flow modeling.
Chapter 5 NUMERICAL SOLUTION OF SIXTH ORDER BOUNDARY VALUE PROBLEMS USING SEVENTH AND EIGHTH DEGREE SPLINE FUNCTIONS: This chapter focuses on solving sixth-order boundary value problems, commonly arising in astrophysics, using higher-degree spline functions.
Chapter 6 OVERALL CONCLUSION AND SCOPE FOR THE FUTURE WORK: This concluding chapter summarizes the key findings of the book and suggests directions for future research in spline-based numerical algorithms.
Keywords
Boundary value problems, Non-polynomial splines, Polynomial splines, Ordinary differential equations, Numerical analysis, Sixth-order BVPs, Fifth-order BVPs, Viscoelastic flows, Convergence, Stability analysis, Mathematical modeling, Spline approximation, Newton’s method, Higher-order BVPs, Astrophysics.
Frequently Asked Questions
What is the core focus of this work?
The work primarily focuses on the development and implementation of spline-based numerical methods to approximate the solutions of higher-order linear and nonlinear boundary value problems in ordinary differential equations.
What are the central themes of the research?
Central themes include the use of non-polynomial and polynomial splines, error analysis of numerical methods, the handling of different orders of differential equations (second, fifth, and sixth), and the verification of results against exact solutions.
What is the primary goal of the methods presented?
The goal is to provide high-accuracy numerical solutions for boundary value problems in cases where analytical solutions are unavailable, particularly for complex phenomena in physics and engineering.
Which scientific methods are primarily utilized?
The work utilizes spline interpolation techniques, including cubic, sextic, septic, and higher-degree polynomial splines, as well as non-polynomial splines, and employs Newton’s method for solving nonlinear systems.
What is covered in the main body of the book?
The main body consists of specific chapters detailing the approach for linear and nonlinear second-order BVPs, as well as distinct chapters for higher-order BVPs, including fifth and sixth-order problems, supported by comprehensive numerical illustrations.
How are the keywords for this study defined?
The keywords, such as Boundary Value Problems, Spline Approximation, and Numerical Analysis, characterize the study's focus on the intersection of numerical mathematics and its practical application to differential equations.
How does the choice of spline parameters impact accuracy?
The book demonstrates that the specific choice of parameters, such as alpha and beta in the spline formulas, significantly affects the rate of convergence and the overall precision of the numerical results.
What makes the non-polynomial spline approach advantageous?
Non-polynomial splines incorporate trigonometric basis functions, which can offer enhanced accuracy and better flexibility in representing specific mathematical behaviors compared to traditional pure polynomial splines.
- Quote paper
- Dr. Parcha Kalyani (Author), 2014, Spline Solutions of Higher Order Boundary Value Problems, Munich, GRIN Verlag, https://www.grin.com/document/541611
