Linear Integral Equations

Introduction

An integral function in Mathematical Sciences is an equation that contains an unknown function under the integral sign. Integrals in maths are one of the several types of mathematical sciences and an integral aspect of calculus. Integral equations are classified into two types: linear and nonlinear integral equations.

Discussion

Concept of Linear Integral equation

Integral equations appear in a wide range of scientific and engineering situations. Volterra or Integral equations of Fredholm can be used to solve a wide range of starting and boundary value issues. The potential hypothesis contributed the most to the rise of Integral equations. Models of mathematical physics, such as diffraction issues, quantum mechanics scattering, conformal mapping, Water waves also played a role in the development of integral equations. Integral equations and differential equations are used to explain many additional applications in science and engineering. 

Linear Integral Equation Types in Mathematical Sciences

1. Fredholm Linear Integral Equation

The most typical use of Fredholm Linear Integral Equations in integral equations is in the form of

 (x)u(x) = f(x) + abk(x,t)u(t)dt                 -(3)

where a and b are constants and the unknown function u(x) emerges linearly under the integral sign. The kernel of the equation is denoted by K(x,t).

If ((x)) is 1, then (3) equals 

u(x) = f(x) +ab k(x,t)u(t)dt ,                     -(4)

and this equation is known as the Fredholm integral equation of the second kind. 

If ((x)) is 0, then (3) gives

f(x) +ab k(x,t)u(t)dt = 0                           -(5), 

which is known as the Fredholm integral equation of the first kind in Mathematical Sciences.

1. Volterra Linear Integral Equations

The Volterra Linear Integral Equation in Mathematical Sciences is most commonly employed in the following format:

(x)u(x) = f(x) + abk(x,t)u(t)dt                             -(6)

where the integration limits are functions of x and the unknown function u(x) emerges linearly under the integral sign.

If the function (x) is 1, then (6) becomes 

u(x) = f(x) +ab k(x,t)u(t)dt = 0                        – (7), 

which is known as the Volterra integral equation of the second kind.

If (x) is zero, then (6) becomes 

f(x) + axk(x, t)u(t)dt = 0                                     – (8),

 which is the Volterra integral equation of the first kind.

1. Singular and non – singular integral equations

A Singular integral equation in Mathematical Sciences is written as an Integrals in maths with unbounded limits when the integral’s Kernel k(x,t) would become infinite at some point in the interval.

Example:

u(x) = f(x) +  -∞+∞ u(t)dt

The integral equation is said to be nonsingular if the kernel k(x,t)is bounded and continuous.

Methods of solving linear integral equations by means of Wiener Integrals

Here we are going to discuss the expressions for the solution in Mathematical Sciences, the resolvent kernel, and the Fredholm determinant of the integral equation in terms of Wiener integrals.

S(t) = x(t) +01K(t,s)  x(s) ds

The Wiener integral in integral equations is based on a measure defined by Wiener on the space C of all real functions x(t)continuous on the interval 0≤t≤1and vanishing at t = 0. We will also discuss the properties and their transformation under translations and linear transformations. 

Now, let’s have a look at the basic solution,

Given the integral equation

z(t)=F[x|t],

Were,

F[x| t] = x(t) + 01 K(t, s)x(s)ds,

Now suppose that the following conditions are satisfied:

1. z(t)∈c and z'(t) exists and is of bounded variation on the interval (0, 1).
2. K(t,s) = K1(t,s),                             0≤ts,

K(t,s) = K2(t,s)                       0≤ts,

K(t,s) =(K1+K2)/2                 t=s

where K1{t, s) is continuous, { 0≤ts, 0≤ts,} ; K(0, s) = 0,0≤ts, and K2(t,s) is continuous, { 0≤st,0≤t≤1,}. 

1. K(t,s) is absolutely continuous in t for all 0≤st, after the jump at t = s is removed by the addition of a step function.
2. For almost all s, ∂K(t, s)/∂t is essential of bounded variation — that is, there exists a measurable function K*i(t, s) which is of bounded variation int for each s and which for almost all (t, s)0≤t≤1 is equal to ∂K(t, s)/∂t.
3. K*i(t, s) can be so chosen that

01 sup0≤t≤1 K*i(t, s)ds<∞ ,

01 var0≤t≤1 K*i(t, s)ds<∞

1. J(s) = K2(s, s)-(s, s)is of bounded variation 

Now we will discuss the explicit form of the solutions in the following theorem:

Theorem Let z(t) and K (t, s) satisfy conditions 1-6 above. Then the solution of the equation given below with F[x|t] is given by the formula:

  x(t) = |D| exp(-01[z'(s)]2ds ) ⋅cwu(t)exp(201z'(s)dF[u|s] –(u))dwu;

Where,  

(u)=01[d/dξ 01K(,s)u(s)ds]2dξ +201[01∂K(,s)u(s)ds/∂ξ]du() +01Jξd[u2()].

Conclusion

If only linear operations are done on the unknown function in an integral equation, it is said to be linear. The unknown function u(x) occurring under the integral sign in a linear integral equation is expressed in the functional form F(u(x)) such that the power of u(x) equals unity. The most common kind of linear integral equation is of the form,

h(x)g(x) = f(x) + λ ∫ k(x,t)g(t)dt  

where the upper limit might be variable or constant. The functions h(x), f(x), and k(x,t) are known, whereas g(x) must be calculated, which is a non-zero real or complex parameter.