Key Features:
Format: PDF
Pages: 33
Content: Lagos State University Project Topic (Mathematics)
Date: May 2024
Abstract:
This study investigates the process of solving Volterra integral equations of the second kind, highlighting the transformation from differential equations to integral forms and the
analysis of the method of successive approximations. By converting differential equations into integral equations, a more feasible approach to solving initial value problems emerges.
Utilizing a specific example of the Volterra integral equation, we employ successive approximations to iteratively refine the solution. This method illuminates the interaction between differential and integral calculus, offering a clear pathway from initial conditions to a comprehensive solution describing function behavior over an interval. Additionally, this approach underscores the significance of comprehending both forms of equations in mathematical and physical systems study. Practically, the ability to solve Volterra integral equations holds value across various disciplines, including physics, engineering, and finance, where such equations are prevalent. The method of successive approximations serves as a fundamental and versatile technique in the mathematical sciences, providing a straightforward approach applicable to diverse problem domains.
Table of Content:
Contents
Certification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
1 INTRODUCTION 1
1.1 Background of Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Statement of Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Aim and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Significance of Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.6 Definition of Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 LITERATURE REVIEW 9
2.1 Review of Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Contribution and Implication of Researchers Mentioned . . . . . . . . . . . 12
2.3 Exploring Characteristics of Volterra Integral Equations . . . . . . . . . . . 13
2.4 Illustrating the Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Definitions and Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 Properties of Volterra Integral Equations . . . . . . . . . . . . . . . . . . . 18
2.7 Types and Kinds of Volterra integral equations . . . . . . . . . . . . . . . . 19
3 METHOD OF STUDY 20
3.1 Derivation of Method for solving Volterra integral equation . . . . . . . . . 21
3.2 Construction of Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.1 (Successive Approximation) . . . . . . . . . . . . . . . . . . . . . . 22
3.2.2 Homogeneous Linear Integral Equation . . . . . . . . . . . . . . . . 24
3.2.3 Symmetric Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Conversion of Initial Value Problem (IVP) to Volterra integral equation . . 25
3.4 Solution to Volterra Integral Equation Using the Method of Successive
Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
REFERENCES 32
ii
Introduction:
Chapter 1
INTRODUCTION
Volterra integral equations, introduced by Vito Volterra in the early 20th century, are
a fundamental tool in mathematical modeling, particularly for systems with historical
importance. They describe how a system’s current state depends not only on its present
input but also on past inputs, finding applications across diverse fields like physics, biology,
engineering, and economics. Their ability to capture memory dependence makes them
invaluable for modeling complex phenomena such as population dynamics and signal
processing. Additionally, their flexibility in handling non-linearity, delays, and complex
interactions enhances their utility in accurately representing real-world dynamics.
Recently, there has been a sudden interest in understanding and analyzing Volterra
integral equations, driven by their relevance in theoretical and applied settings. Advances
in computational techniques have enabled the development of efficient methods for solving
and simulating these equations with unprecedented accuracy and efficiency.
In this study, we aim to delve into the realm of Volterra integral equations, exploring
their characteristics, developing analysis methods for solving them, and investigating their
applications across scientific and engineering domains. Through rigorous analysis and
practical experimentation, we aspire to advance the understanding of these equations,
contribute to the development of innovative solutions to real-world problems, and unlock
their full potential for contemporary research and applications.
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