PROFESSOR SUDESHNA BANERJEA DEPARTMENT OF MATHEMATICS JADAVPUR UNIVERSITY KOLKATA 700032 WEST BENGAL, INDIA
E-mail: [email protected]
1 CHAPTER 1
INTEGRAL EQUATIONS: AN INTRODUCTION
MODULE 1
CLASSIFICATIONS OF INTEGRAL EQUATIONS
1.1 Introduction
1.2 What is an integral equation
1.3 What are linear and nonlinear integral equations
1.4 Various forms of linear integral equations
1.5 Regular and singular integral equations
1.3 Problems
2 1.1 INTRODUCTION
The theory of Integral Equations constitute an important topic in Mathematics. Many initial and boundary value problems associated with ordinary and partial dif- ferential equations can be reduced to integral equations of various types. Development of the theory of integral equation is closely associated with the history of mathematics, specifically applied mathematics. The origin of integral equation may be attributed to N.H.Abel who in 1826 first reduced the problem of finding the path of descent of a particle along a smooth vertical curve under the action of gravity in an interval of time, to an integral equation. Later in 1896, V. Volterra developed the general theory of solution of a class of linear integral equation where the upper limit of integral is variable. Such type of integral equations are known after him. I. Fredholm in 1900 developed the theory of integral equation in which the limits of integral are constants and these integral equations are known as Fredholm integral equations.
1.2 What is an integral equation
An equation involving an unknown function y(x) of single variable x with a ≤ x ≤ b (a, b being real constants) is said to be an integral equation for y(x), if y(x) appears under the sign of integration. Following are examples of integral equations.
Z b 1. K(x, t)y(t)dt = f(x), a ≤ x ≤ b a
Z b 2. y(x) + K(x, t)y(t)dt = f(x), a ≤ x ≤ b a
Z b 3. y(x) + K(x, t)[y(t)]2dt = f(x), a ≤ x ≤ b a
3 Z x 4. y(x) + K(x, t)y(t)dt = f(x), a ≤ x ≤ b a
Z x 5. K(x, t)y(t)dt = f(x), a ≤ x ≤ b a
1.3 Linear and nonlinear integral equations
The integral equation in which the unknown function appears linearly is called a LINEAR INTEGRAL EQUATION, otherwise it is a NONLINEAR INTEGRAL EQUATION. The integral equations in example 1,2,4,5 are linear whereas integral equation in example 3 is a nonlinear integral equation.
1.4 Various forms of linear integral equations
The general one dimensional linear integral equation for an unknown function ϕ(x) is of the form
Z b cϕ(x) + λ K(x, t)ϕ(t)dt = f(x), a ≤ x ≤ b. (1.1) a Here c is either 0 or 1 and K(x, t) is a known function which is known as the KERNEL of the integral equation. The other known function f(x) is called the FORCING TERM. The known constant λ is the parameter of the integral equation. •The integral equation, in which the forcing term f(x) is equal to zero, is called HOMOGENEOUS integral equation, where as for NONHOMOGENEOUS integral equation f(x) 6= 0. • The integral equation (1.1) in which c = 0 is called integral equation of FIRST KIND, i.e.
4 Z b K(x, t)ϕ(t)dt = f(x), a ≤ x ≤ b. a • The integral equation (1.1) in which c = 1 is called integral equation of SECOND KIND, i.e.
Z b ϕ(x) + λ K(x, t)ϕ(t)dt = f(x) a ≤ x ≤ b. a • If the limits of integration appearing in the integral equation (1.1) are constants, then the integral equation is known as FREDHOLM integral equation. • If any one limit of integration is known function of x, the corresponding integral equation (1.1) is called VOLTERRA integral equation.
1.5 Regular and singular integral equations
• If the kernel K(x, t) of integral equation (1.1) is such that
Z b Z b |K(x, t)|2dxdt a a has a finite value, then the kernel is known as REGULAR kernel and the corre- sponding integral equation is known as REGULAR integral equation. • If the kernel K(x, t) of integral equation (1.1) is of the form f(x, t) K(x, t) = (x − t)α where f(x, t) is bounded in [a, b] × [a, b] with f(x, x) 6= 0 and 0 < α < 1, then the integral equation (1.1) is known as WEAKLY SINGULAR integral equation. f(x,t) • If in the integral equation (1.1), K(x, t) = x−t , where f(x, t) is a differentiable function of x and t with f(x, x) 6= 0, then the integral equation (1.1) is known as SINGULAR integral equation, where the integral is to be understood in the sense of CAUCHY PRINCIPAL VALUE as given by
Z b Z x− Z b K(x, t)ϕ(t)dt = lim K(x, t)ϕ(t)dt + K(x, t)ϕ(t)dt . a →0+ a x+
5 • If the kernel K(x, t) in the integral equation (1.1) is of the form
f(x, t) K(x, t) = (x − t)2
where f(x, t) is a differentiable function with f(x, x) 6= 0, then the integral equa- tion (1.1) is known as HYPERSINGULAR integral equation. A Hypersingular inte- R b ψ(t) gral a (x−t)2 dt, a ≤ x ≤ b is understood in the sense of two sided HADAMARD FINITE PART integral of order 2 defined by
Z b ψ(t) Z x− ψ(t) Z b ψ(t) 2 dt = lim 2 dt + 2 dt a (x − t) →0+ a (x − t) x+ (x − t) ψ(x + ) + ψ(x − ) − . In the present lecture we shall restrict ourselves to the theory linear integral equa- tion.
Problems
1. Determine whether the following integral equations are linear or non linear, homogeneous or non homogeneous
R b 3 (a) u(x) = a sin(x + t)u (t)dt a < x < b
2 R b (b) u(x) = x + a sin(x + t)u(t)dt a < x < b
R x 2 2 (c) u(x) = 1 + 0 (x − t) u (t)dt 0 < x < 1 Answer:
a) nonlinear, homogeneous
b) linear, non homogeneous
c) non linear, non homogeneous
6 2. Classify the following integral equation as Fredholm or Volterra integral equa- tions.
2 R x a) u(x) = 1 + x + 0 (x − t)u(t)dt 0 < x < 1. R 1 b) u(x) = 2x + 0 xt u(t)dt 0 < x < 1. R x c) 0 (x − t)u(t)dt = x 0 < x < 1. R b d) a (x + 1)tu(t)dt = 1 a < x < b. Answer:
a) Volterra Integral Equation of Second kind
b) Fredholm Integral Equation of Second kind
c) Volterra Integral Equation of First kind
c) Fredholm Integral Equation of First kind
3. Determine whether the following integral equations are regular or singular.
R 1 a) u(x) + 0 u(t) sin xt dt = 1 0 < x < 1. R b u(t) b) a x−t dt = 1 a < x < b. R b c) a u(t) ln |x − t|dt = x a < x < b. R b u(t) d) a (x−t)2 dt = x a < x < b. R b u(t) e) 1 dt = (x − a) a < x < b. a (x−t) 3 Answer:
a) Regular Integral Equation
b) Cauchy type Singular Integral Equation.
c) Weakly Singular Integral Equation.
d) Hyper Singular Integral Equation.
e) Weakly Singular Integral Equation.
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