INTEGRAL EQUATION AND ITS CONVERSIONS

CHAPTER ONE BACKGROUND OF THE STUDY 1.0 INTRODUCTION The theory and applications of integral equations, or, as it is often called, the inversion of definite integrals, have come suddenly into prominence and have held during the last half dozen years a central place in the attention of mathematicians. By an integral equation a term first suggested by du Bois-Reymond in (1888) is understood to be an equation in which the unknown function occurs under one or more signs of definite integration. The theory of integral equations has been an active field of research for many years and is inextricably related with other areas of Mathematics such as complex analysis, mathematical analysis, function theory, functional analysis and numerical analysis. Integral Equations arise naturally in applications, in many areas of Mathematics, Engineering, Science and Technology and have been studied extensively both at the theoretical and practical level.Jin Liang, Ti-Jun Xiao (2011). Mathematicians have so far devoted their attention mainly to two peculiarly simple types of integral equations, - namely: the linear equations of the first and second kinds. In this respect integral equations are in striking contrast to the closely related differential equations, where the passage from ordinary to partial differential equations is attended with very serious complications. Furthermore, Fredholm and Volterra integral equation arise from different origins and applications, such as boundary value problems as in Fredholm equations, and from initial value problems as in Volterra equations Wazwaz A. (2005).In many instances,the integral equation originated from the conversion of a boundary value problem or an initial value problem associated with a partial or an ordinary differential equation, but many problems lead directly to integral equations and cannot be formulated in terms of differential equations.Due to this close connection between differential and integral equations, one can often convert between the two.For example, one method of solving a boundary value problem is by converting the differential equation with its boundary conditions into an integral equation and solving the integral equation. In addition, because one can convert between the two, differential equations in physics such as Maxwell'sequations often have an analogintegral and differential form Admin (2022). STATEMENT OF THE PROBLEM The subject integral equation is one of the most useful mathematical tools in both pure and applied mathematics. It has numerous applications in many physical problems. Many initial and boundary value problems associated with ordinary differential equation (ODE) and partial differential equation (PDE) can be transformed into problems of solving some approximate integral equations.In this work, we wish to consider a typical integral equation in u(x)of the form u(x)=f(x)+λ∫_α(x)^β(x)▒K(x,t)u(t)dt (1) Where, K(x,t) is called the kernel of the integral equation,f(x) is a known function,α(x)and β(x) are the limits of integration and λ is a constant parameter which plays the same role as the eigenvalue in linear algebra and its conversion to initial value problems. MOTIVATION OF THE STUDY Most of the relevant differential equations are often difficult or impossible to solve exactly. My motivation to study integral equation is based on the fact that it is a powerful and elegant method for solving both ordinary and partial differential equations. Furthermore, it has wide range of applications in Physics and Engineering. AIM AND OBJECTIVES The aim of this work is to study integral equation and its conversion to initial or boundary value problems and vice versa. The objectives of this work are: To study the importance and concept of integral equations To study different types of integral equations, their properties and classifications. To know how to convert integral equation to initial or boundary value problems and vice versa. THE SIGNIFICANCE OF THE STUDY The integral equation is highly significant and relevant in a variety of sectors in real life, including engineering, where engineers use integral to determine the geometry of a building. In the field of mathematics such asincomplex and mathematical analysis, function theory, functional analysis and numerical analysis.In physics such aspotential theory, elasticity, acoustics, fluid mechanics, radiative transfer etc. SCOPE AND LIMITATIONS This work covers linear and nonlinear integral equations; however, it is limited to first and second order differential equations. RESEARCH METHODOLOGY The research was carried out by consulting relevant textbooks, lecture materials, journals, and the internet. DEFINITION OF TERMS We start by presenting some of the basic definitions used in the work. Definition1.6.1 (Differential Equation (DE)): A Differential equation is an equation which contains one or more variable (i.e. dependent variable) with respect to the other variable (i.e. independent variable). Example is; dy/dx=f(x,y) (1.6.1.1) Where x-is the independent variable, and y- the dependent variable, and dy/dx-the derivative. There are several types of differential equations: ordinary differential equations (ODEs), Partial differential equation (PDEs), linear differential equations (LDEs), Non-linear Differential equations (NLDEs), etc. Definition 1.6.2 (Ordinary differential Equation (ODE)): An ODE is an equation that contains only one independent variable and one or more of its derivative with its other variable. Example is: (d^2 y)/(dt^2 )=2dy/dt+5y=0 (1.6.2.1) Definition 1.6.3 (Order and Degree of a differential Equation): The order of a Differential Equation is the highest derivative that occurs in the equation, while the degree of a Differential Equation is the power of the highest order derivative, where the original equations is represented in the form of a polynomial equation such as:y^',y^'',y^''', and so on. For example: 〖(y^'')〗^3+3y^'+6y=2 (1.6.3.1) The above equation (1.6.3.1) is of order 2, and degree 3. Definition 1.6.4 (Initial and Boundary Value Problems): A differential Equation with conditions imposed at the same value of the independent variable is called an Initial Value Problem (IVP). Example y^''+ y^'+5x=0,x(0)=4,x^' (0)=1 (1.6.4.1) While a differential equation with conditions imposed at different values of the independent variable is called a Boundary Value Problem (BVP). Example y^''+ y^'+5x=0,x(0)=4,x^' (2)=1 (1.6.4.2) Definition 1.6.5 (Integral Equation):An integral equation is defined as an equation in which the unknown function u(x) to be determined appears under the integral sign. The most general form of a linear integral equation is h(x)u(x)=f(x)+∫_a(x)^b(x)▒K(x,τ)u(τ)dτfor all x∈[a,b] (1.6.5.1) in which u(x) is the function to be determined, f(x) is a known function and K(x,τ) is called the kernel of integral equation, a(x) and b(x) are the limits of integration. Definition 1.6.6 (Homogeneous and Non- Homogeneous integral equation): An integral equation is called homogeneous if the known function f(x) in (1.6.5.1) above is identically zero.i.e. h(x)u(x)=∫_a(x)^b(x)▒K(x,τ)u(τ)dτfor all x∈[a,b] (1.6.5.2) otherwise, it is called nonhomogeneous integral equation Definition 1.6.7 (Integral equation of the first kind):An integral equation is called an integral equation of the first kind if the unknown function appears only under the integral sign. Example u(x)=λ∫_a^b▒K(x,t)u(t)dt, x,t∈[a,b] (1.6.7.1) Definition1.6.8 (Integral equation of the second kind):An integral equation is called an integral equation of the second kind if the unknown function appears also outside the integral. Example u(x)=f(x)+λ∫_a^b▒K(x,t)u(t)dt, x,t∈[a,b] (1.6.8.1) Definition 1.6.9 (Linearity): An integral equation is linear if the unknown function u(x) and its integrals appear linear in the equation. Example u(x)=f(x)+λ∫_α(x)^β(x)▒K(x,t)u(t)dt (1.6.9.1) Definition 1.6.10 (Non-Linearity): An integral equation is nonlinear if the unknown function u(x) or its integrals appear nonlinear in the equation. Example u(x)=f(x)+λ∫_α(x)^β(x)▒〖K(x,t) u^2 (t)dt〗 (1.6.10.1) Definition 1.6.11 (Solution of an integral equation): A solution of an integral equation or an integro-differential equation on the interval of integration is a function u(x) such that it satisfies the given equation. Definition 1.6.12 (Differentiation): Differentiation is the process of finding the derivative of a function. Definition 1.6.13 (Integration): The process of integration reverses the process of differentiation. The process of finding a function from its derived function is called integration. Definition 1.6.13 (Function): A function is a set of ordered pairs, in which no two distinct elements have the same first entry. It can also be defined as a relation between the two sets Xand Y, where each member of the set X is related to one and only one member of the set Y. This relation is known as function from X to Ywritten f:X→Y. CHAPTER TWO REVIEW OF RELATED LITERATURE 2.0 INTRODUCTION Several researchers have done interesting works relating to this area of research. We will consider only a few of them:Sukavanam, N. (1984) Investigated Fredholm-type theory for third kind linear integral equations.Weimin H. (1994) investigated the uniqueness and smoothness results for second-kind Volterra Equations with weakly-Singular kernel.Shulaia, D. (2017), studied the integral equations of the third kind for the case ofpiecewise monotone coefficients.Jin, L. and Ti-Jun, X. (2011) in their work, established new and general existence theorem of Lipchitz continuous solutions to integral equations with infinite delay in Banach spaces, as well as of strong and classical solutions to the Cauchy problem for some evolution equations with infinite delay.Anupam, D., Bipan, H., Poom, K. (2019), proposed some new fixed point theorem involving measure of noncompactness and control function. Further, they proved the existence of a solution of functional integral equations in two variables by using the fixed point theorem in Banach Algebra, and also illustrate the results with the help of an example.Muhammad, N. and Muhammad A. (2016) in their work, introduced the notion of dualisticGeraghty’s type contractions, they proved some fixed point theorems for ordered mappings satisfying the above mentioned contractions. They discussed an application of their fixed point results to show the existence of solution of integral equations.Amar, D. et al (2021), studied the application of Petryshyn’s fixed point theorem to solvability for functional integral equations.Sahar, M. J. and Ahmed, H. H. (2021), in their work, addressed the integral equations in many issues and their applications.Soniya, S. et al. (2022), used the method of Petryshyn's fixed point theorem in Banach algebra in their work, investigated the existence of solutions for functional integral equations, which involves as specific cases many functional integral equations that appear in different branches of non-linear analysis and their applications. Finally, they recalled some particular cases and examples to validate the applicability of their study.Prit P. P., et al. (2023)used a computational method to find an approximate solution of the stochastic Volterra-Fredholm integral equation using the Walsh function approximation and its operational matrix.Kulikov, E.K., Makarov, A.A. (2023), in their work considered a numerical method for solving the Fredholm integral equation of the first kind. The essence of the method was to replace the original equation with the corresponding regularized equation of the second kind, which is then solved by the modified spline collocation method. CHAPTER 3 METHODOLOGY 3.0 INTRODUCTION In this chapter, we present all the basic tools necessary in the analysis of our proposed problem. Firstly, we shall adopt the work of [Jagbir S. (2021)] to give a definition of Leibnitz rule for differentiation under theintegral sign. 3.1 LEIBNITZ RULE The Leibniz rule for differentiation under the integral sign is given as d/dx [∫_α(x)^β(x)▒F(x,τ)dτ]=∫_α(x)^β(x)▒〖∂F/∂x dτ+F(x,β(x) ) dβ(x)/dx-F(x,α(x) ) dα(x)/dx〗 (3.1.1) In particular, we have d/dx [∫_a^x▒K(x,τ)u(τ)dτ]=∫_a^x▒〖∂K/∂x u(τ)dτ+K(x,x)u(x) 〗 (3.1.2) Suppose we want to differentiate the integral ∫_α(x)^β(x)▒G(x,t)dt with respect to x, applying the Leibniz rule gives d/dx ∫_α(x)^β(x)▒G(x,t)dt=G(x,β(x) ) dβ(x)/dx-G(x,α(x) ) dα(x)/dx+∫_α(x)^β(x)▒∂G/∂x dt (3.1.3) where, G(x,t) and ∂G/∂x are continuous functions in the domain xt-plane that contains the rectangular region R, a≤x≤b,t_0≤t≤t_1, and the limits of integration α(x) and β(x) are defined functions having continuous derivatives for a