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The International journal of analytical and experimental modal analysis ISSN NO: 0886-9367

SOLUTION OF LINEAR BY

Dany Joy Department of , College of Trivandrum [email protected]

ABSTRACT The theory of integral is very useful tool to deal with problems in applied mathematics and Engineering applications. A large class of initial and boundary value problems can be converted to Volterra or Fredholm . Here I discussing the basic concept of integral equation and the solution of Volterra integral equation by using Laplace transforms. Constructive examples are also provided to illustrate the ideas. The results reveals that the transform method is very effective and convenient.

Keywords: Integral Equations, Volterra Integral equation, Fredholm Integral equation, Laplace transform method, Linear and non-Linear Integral

1. INTRODUCTION

Many of the Physical problems of Engineering Sciences which were solved with the help of Ordinary and Partial Differential equations. This can be done by better methods of Integral equations.

The Laplace transform is an named after its inventor Pierre-Simon Laplace. It transforms a of a real t (often time) to a function of a complex variable s (complex ). The transform has many applications in science and engineering. The current widespread use of the transform (mainly in engineering) came about during and soon after World War II replacing the earlier Heaviside operational . The advantages of the Laplace transform had been emphasized by Gustav Doetsch to whom the name Laplace Transform is apparently due.

2. INTEGRAL EQUATION

An Integral equation is an equation in which an unknown function appears under one or more integral signs. For example, for 푎 ≤ 푥 ≤ 푏, 푎 ≤ 푡 ≤ 푏, the equations

푏 푦(푥) = 푓(푥) + 휆 ∫ 퐾(푥, 푡)푦(푡)푑푡 푎

Where the function 푦(푥) is the unknown function while the functions 푓(푥) and 퐾(푥, 푡) are known functions and 퐾(푥, 푡) is called the Kernel of the integral equation and 휆, 푎 and 푏 are constants.

2.1 Linear and Non-Linear Integral Equations An Integral equation is called linear if only linear operations are performed in it upon the unknown function. An integral equation which is not linear is known as a non-linear integral equation. 푏 퐿(푦) = 푦(푥) − 휆 ∫ 퐾(푥, 푡)푦(푡) 푑푡 푎

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We can easily verify that 퐿 is a linear integral . In fact, for any constants 푐1 and 푐2, we have 퐿{푐1푦1(푥) + 푐2푦2(푥)} = 푐1퐿{푦1(푥)} + 푐2퐿{푦2(푥)} Which is well known general criterion for a linear operator

2.2 Classification of Integral Equation

2.2.1 Fredholm Integral Equation

A linear integral equation of the form 푏 푔(푥)푦(푥) = 푓(푥) + 휆 ∫ 퐾(푥, 푡)푦(푡) 푑푡 (1) 푎 Where 푎, 푏 are both constants, 푓(푥)푔(푥) and 퐾(푥, 푡) are known functions while 푦(푥) is unknown function and 휆 is a non-zero real or complex parameter, is called Fredholm Integral equation of third kind. The function 퐾(푥, 푡) is known as the kernel of the integral equation.

2.2.1.1 Fredholm integral equation of the first kind

A linear integral equation of the form (putting 푔(푥) = 0 in (1)) 푏 푓(푥) + 휆 ∫ 퐾(푥, 푡)푦(푡) 푑푡 = 0 (2) 푎 is known as Fredholm integral equation of the first kind

2.2.1.2 Fredholm integral equation of the second kind

A linear integral of the form (by setting 푔(푥) = 1 in (1)) 푏 푦(푥) = 푓(푥) + 휆 ∫ 퐾(푥, 푡)푦(푡) 푑푡 (3) 푎 is known as the Fredholm integral equation of the second kind

2.2.1.3 Homogeneous Fredholm Integral equation of the second kind

A linear integral of the form (by setting 푓(푥) = 0 in (3)) 푏 푦(푥) = 휆 ∫ 퐾(푥, 푡)푦(푡) 푑푡 (4) 푎 is known as the Homogeneous Fredholm integral equation of the second kind

2.2.2 Volterra Integral Equation

A linear integral equation of the form 푥 푔(푥)푦(푥) = 푓(푥) + 휆 ∫ 퐾(푥, 푡)푦(푡) 푑푡 (5) 푎 Where 푎, 푏 are both constants, 푓(푥)푔(푥) and 퐾(푥, 푡) are known functions while 푦(푥) is unknown function and 휆 is a non-zero real or complex parameter, is called Volterra Integral equation of third kind. The function 퐾(푥, 푡) is known as the kernel of the integral equation.

Volume XI, Issue XII, December/2019 Page No:2674 The International journal of analytical and experimental modal analysis ISSN NO: 0886-9367

2.2.2.1 Volterra integral equation of the first kind

A linear integral equation of the form (putting 푔(푥) = 0 in (5)) 푥 푓(푥) + 휆 ∫ 퐾(푥, 푡)푦(푡) 푑푡 = 0 (6) 푎 is known as Volterra integral equation of the first kind

2.2.2.2 Volterra integral equation of the second kind

A linear integral of the form (by setting 푔(푥) = 1 in (5)) 푥 푦(푥) = 푓(푥) + 휆 ∫ 퐾(푥, 푡)푦(푡) 푑푡 (7) 푎 is known as the Volterra integral equation of the second kind

2.2.2.3 Homogeneous Volterra Integral equation of the second kind

A linear integral of the form (by setting 푓(푥) = 0 in (7)) 푥 푦(푥) = 휆 ∫ 퐾(푥, 푡)푦(푡) 푑푡 (8) 푎 is known as the Homogeneous Volterra integral equation of the second kind

2.3 Integral Equations of the Type

Consider an integral equation in which the kernel 퐾(푥, 푡) is dependent solely on the difference 푥 − 푡 𝑖푒 퐾(푥, 푡) = 퐾(푥 − 푡) Where 퐾 is a certain function of one variable. Then integral equations 푏 푦(푥) = 푓(푥) + 휆 ∫ 퐾(푥, 푡)푦(푡) 푑푡 (9) 푎 and 푥 푦(푥) = 푓(푥) + 휆 ∫ 퐾(푥 − 푡)푦(푡) 푑푡 (10) 푎 are called integral equations of the convolution type. 퐾(푥 − 푡) is called difference kernel.

Let 푦1(푥) and 푦2(푥) be two continuous functions defined for 푥 ≥ 0. Then the convolution of 푦1 and 푦2 is denoted and defined by 푥 푥 푦1 ∗ 푦2 = ∫ 푦1(푥 − 푡)푦2(푡) 푑푡 = ∫ 푦1(푡)푦2(푥 − 푡) 푑푡 (11) 0 0 The occurring in (11) are called the convolution integrals Note that the convolution defined by relation (11) is a particular case of the standard convolution.

∞ ∞ 푦1 ∗ 푦2 = ∫ 푦1(푥 − 푡) 푦2(푡)푑푡 = ∫ 푦1(푡)푦2(푥 − 푡) 푑푡 (12) −∞ −∞

By setting 푦1(푡) = 푦2(푡) = 0, for 푡 < 0 and 푡 > 푥, the integrals in (11) can be obtained from those in (12)

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3. Laplace Transform

3.1 Definition The Laplace transform of a function f(x), defined for all real numbers 푥≥ 0, is the function F(s), which is a unilateral transform defined by ∞ 퐹(푠) = ∫ 푓(푥)푒−푠푥푑푥 (13) 0 Where 푠 is a frequency parameter. 푠 = 휎 + 𝑖휔, with real numbers 휎 and 휔 An alternate notation for the Laplace transform is ℒ{푓}

3.2 Some important Laplace Transforms

3.2.1 푓(푥) = 1

∞ ℒ{푓(푥)} = ∫ 1. 푒−푠푥푑푥 0 푒−푠푥 ∞ = [ ] −푠 0 1 = 푠 3.2.2 푓(푥) = 푒푎푥

∞ ℒ{푓(푥)} = ∫ 푒푎푥. 푒−푠푥푑푥 0 ∞ 푒(푎−푠)푥 = [ ] (푎 − 푠) 0 1 = 푠 − 푎 Similarly we can find the laplace transform of all functions

3.2.3 Laplace Transform of several functions 1 (i) ℒ{1} = 푠 1 (ii) ℒ{푒푎푥} = 푠−푎 푛! (iii) ℒ{푥푛} = 푠푛+1 푎 (iv) ℒ{sin 푎푥} = 푠2+푎2 푠 (v) ℒ{cos 푎푥} = 푠2+푎2 푎 (vi) ℒ{sin ℎ푎푥} = 푠2−푎2 푠 (vii) ℒ{cos ℎ푎푥} = 푠2−푎2 (viii) ℒ{푦′(푥)} = 푠푌(푠) − 푦(0) (ix) ℒ{푦′′(푥)} = 푠2푌(푠) − 푠푦(0) − 푦′(0)

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3.2.4 Inverse Laplace Transforms In mathematics, the inverse Laplace transform of a function 퐹(푠) is the piecewise-continuous and exponentially-restricted real function 푓(푥) which has the property: 푓(푥) = ℒ−1{퐹(푠)}

3.2.5 Let 퐹(푠) and 퐺(푠) denote the Laplace transforms of 푓(푥) and 푔(푥) respectively. Then the product 퐻(푠) = 퐹(푠)퐺(푠) is the Laplace transform of the convolution of 푓(푥) and 푔(푥), is denoted by ℎ(푥) = (푓 ∗ 푔)(푥) and has the integral representation 푥 ℎ(푥) = (푓 ∗ 푔)(푥) = ∫ 푓(휏)푔(푥 − 휏)푑휏 0 or 푥 ℎ(푥) = (푔 ∗ 푓)(푥) = ∫ 푔(휏) 푓(푥 − 휏)푑휏 0

3.2.6 Laplace transform of Convolution Integral

Consider the Convolution integral 푥 ℎ(푥) = ∫ 푓(휏)푔(푥 − 휏)푑휏 0 = 푓(푥) ∗ 푔(푥) ℒ{ℎ(푥)} = ℒ{푓(푥) ∗ 푔(푥)} = 퐹(푠). 퐺(푠)

4. Solution of Integral Equation by Laplace Transform method

Consider the non homogeneous Volterra integral equation of convolution type 푥 푦(푥) = 푓(푥) + 휆 ∫ 퐾(푥 − 푡)푦(푡) 푑푡 (14) 푎 Now we can convert this integral into convolution 푥 ∫ 퐾(푥 − 푡)푦(푡) 푑푡 = 퐾(푥) ∗ 푦(푥) 푎

Now (14) becomes 푦(푥) = 푓(푥) + 휆퐾(푥) ∗ 푦(푥) Taking Laplace Transform on both sides, we get ℒ{푦(푥)} = ℒ{푓(푥)} + 휆ℒ{퐾(푥) ∗ 푦(푥)} 푌(푠) = 퐹(푠) + 휆퐾(푠). 푌(푠) 푌(푠) − 휆퐾(푠). 푌(푠) = 퐹(푠) 푌(푠)[1 − 휆퐾(푠)] = 퐹(푠) 퐹(푠) 푌(푠) = [1 − 휆퐾(푠)]

Now the solution 푦(푡) is obtained by taking inverse laplace on both sides

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Example 1: 푥 푦(푥) = 푥 + ∫ sin(푥 − 푡) 푦(푡) 푑푡 0 = 푥 + sin 푥 ∗ 푦(푥) Taking laplace on both sides, we get ℒ{푦(푥)} = ℒ{푥} + ℒ{sin 푥 ∗ 푦(푥)} 1 푌(푠) = + ℒ{sin 푥}. ℒ{푦(푥)} 푠2 1 1 = + . 푌(푠) 푠2 푠2 + 1 1 1 푌(푠) [1 − ] = 푠2 + 1 푠2 푠2 1 푌(푠) [ ] = 푠2 + 1 푠2 푠2 + 1 푌(푠) = 푠4 1 1 푌(푠) = + 푠2 푠4 Taking inverse on both sides, we get 푥3 푦(푥) = 푥 + 3!

Example 2: 푥 푦(푥) = 1 − 2푥 − 4푥2 + ∫ [3 + 6(푥 − 푡) − 4(푥 − 푡)2]푦(푡) 푑푡 0 푦(푥) = 1 − 2푥 − 4푥2 + [3 + 6푥 − 4푥2] ∗ 푦(푥) Taking laplace on both sides, we get ℒ{푦(푥)} = ℒ{1 − 2푥 − 4푥2} + ℒ{[3 + 6푥 − 4푥2] ∗ 푦(푥)} 푌(푠) = ℒ{1} − 2ℒ{푥} − 4ℒ{푥2} + [3ℒ{1} + 6ℒ{푥} − 4ℒ{푥2}]푌(푠) 1 2 4.2! 3 6 4.2! = [ − − ] + [ + − ] 푌(푠) 푠 푠2 푠3 푠 푠2 푠3 푠2 − 2푠 − 8 3푠2 + 6푠 − 8 푌(푠) = [ ] + [ ] 푠3 푠3 3푠2 + 6푠 − 8 푠2 − 2푠 − 8 푌(푠) [1 − ] = [ ] 푠3 푠3 푠3 − 3푠2 − 6푠 + 8 푠2 − 2푠 − 8 푌(푠) [ ] = [ ] 푠3 푠3 푠2 − 2푠 − 8 푌(푠) = [ ] 푠3 − 3푠2 − 6푠 + 8 푠2 − 2푠 − 8 푌(푠) = [ ] (푠 − 1)(푠2 − 2푠 − 8) 1 푌(푠) = 푠 − 1 Taking inverse on both sides 푦(푥) = 푒푥

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Example 3: 푑2푦 + 푦 = 0, 푥 > 0, 푦(0) = 1 and 푦′(0) = 0 푑푥2

Taking Laplace on both sides, we get 푑2푦 ℒ { + 푦} = ℒ{0} 푑푥2 푑2푦 ℒ { } + ℒ{푦} = 0 푑푥2 푠2푌(푠) − 푠푦(0) − 푦′(0) + 푌(푠) = 0 푠2푌(푠) − 푠 + 푌(푠) = 0 푌(푠)[푠2 + 1] = 푠 푠 푌(푠) = 푠2 + 1 Taking inverse on both sides, we get 푦(푥) = cos 푡

5. Conclusion Many initial and boundary value problems associated with ordinary and partial differential equations can be cast into the integral equations of Volterra and Fredholm types, respectively. Integral Equation offers a powerful technique for solving a variety of practical problems in Science and Engineering. In this paper the author implemented the Laplace Transform method for solving the Linear Volterra Integral equations. Laplace transform is an alternative method for solving the integral equation. And also it is commonly used to solve electrical circuit and system problems.

6. REFERENCES [1] Christy Tom Mathews and Bijesh P Biju, Linear Integral equations and its applications, The International journal of analytical and experimental modal analysis 0886-9367 [2] A. Aghili and H. Zeinali, Solution to Volterra Singular Integral Equations and Non Homogenous Time Fractional PDEs 2219-7184 [3] Ram P Kanwal, Linear Integral Equations Theory and Technique, Academic press New York [4] G.E. Roberts and H. Kaufman, Table of Laplace Transforms, W.B. Saunders Co., Philadelphia, (1966). [5] R.S. Dahiya and M. Vinayagamoorthy, Laplace transfom pairs of n – dimensions and heat conduction problem, Math. Comput. Modelling, 13(10), 35-50.

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