Description
ABSTRACT
Differential equation describes exchange of matter, energy, information or any other quantities; often as they vary in time and /or space. Integral transform which is particularly useful in solving differential equations. Integral transform is a very powerful mathematical tool applied in various areas of engineering and science. In this paper, we present and employ differential transformation and Exponentially Fitted Collocation Approximate Technique (EFCAT) to solve linear Volterra and Fredholm integro-differential equations. To demonstrate the applicability of the present methods many examples were considered and the two methods were used and compared. It is observed that that differential transformation save time and space and it is easy to implement than Exponentially Fitted Collocation Approximate Technique (EFCAT).
CHAPTER ONE
INTRODUCTION
1.0 BACKGROUND OF THE STUDY
Integro-differential equation is an equation that involves both integrals and derivatives of a function. Integro-differential equations find special applicability within scientific and mathematical disciplines. It plays an important role in many branches of mathematical sciences and their applications in the theory of engineering, physics, mechanics, chemistry, astronomy, biology, economics, potential theory, and electrostatics. The theory and application of integro-differential equations are important roles in engineering and applied sciences. The existence and uniqueness of the solutions of integro differential equations usually discussed in terms of their kernel have been established in (Linz, 2015). The integro-differential equations are usually difficult to solve analytically thus, it requires suitable numerical techniques to obtain analytic-numeric solutions of Integro-differential equations. Therefore, several authors have proposed and applied different methods to obtain the solution of both linear and nonlinear IDEs such as, Adomian decomposition (Adomian, 2014), Homotopy perturbation method (He, 1999), variation iteration method (He, 1998), Chebyshev polynomial collocation (Taiwo et al., 2011), The Taylor expansion approach (Kanwall et al., 2018), Bessel or Chebyshev polynomial approach are used to solve integro-differential equations in (Yuzbasi et al., 2011) and just mention a few. In this study, two methods were chose to solve Integro-differential equations – Differential Transform Method (DTM) and Exponentially Fitted Collocation Approximate Technique (EFCAT).
The Differential Transform Method (DTM) is a method for solving a wide range of problems whose mathematical models yield equations or systems of equations involving algebraic, differential,integral and integro-differential equations. The concept of the differential transform was first proposed by Zhou (1986) where in both linear and non-linear initial value problems in electric circuit analysis (Corinthios et al., 2015) were solved. This method constructs an analytical solution in the form of polynomials. It is different from the high-order Taylor series method, which requires symbolic computation of the necessary derivatives of the data functions. The Taylor series method is computationally expansive for large orders. The differential transform is an iterative procedure for obtaining analytic Taylor series solutions of differential equations. In recent years the application of differential transform theory has been appeared in many researches.
The theory and application of integro-differential equation is an important subject with in applied mathematics. A large class of scientific and engineering problems modelled by partial differential equations can be expressed in various forms of differential or integro-differential equations in abstract spaces. Integro-differential equations include many physical phenomena such as heat flow in materials with memory, viscoelasticity, heat conduction and wave propagation. Quasilinear integro-differential equation is also a factor which describes the study of nonlinear behavior of elastic strings and nonlinear conservative law with memory. One of the most important fields of modern research is the distributed control systems which is exercised through the boundary in a different way (Sheng et al., 2010).
These are motivations to solve these kind of equations. Using the DTM and employ Exponentially Fitted Collocation Approximate Technique (EFCAT) several examples of linear and nonlinear integro-differential equations are tested and the results reveal that the DTM is very effective and simple.
1.2 PROBLEM STATEMENT
Generally, integro-differential equations are difficult to solve, thus this present work is to apply Differential Transform Method (DTM) and Exponentially Fitted Collocation Approximate Technique (EFCAT) to solve integro-differential equations, which promise to be a reliable, easy, fast and accurate numerical technique to obtain numerical solution of integro-differential equations. We obtain derivative of power series of function y(x) and substitute into the linear integro differential equation. Slightly perturbation and collocation are carried out which eventually transform to square matrix form and MAPLE 18 software is used to obtain the unknown constants.
1.3 AIM AND OBJECTIVES OF THE STUDY
The main aim of this study is to carry out comparative analysis of solving integro-differential equations using differential transformation method and exponentially fitted collocation approximation method. The objectives of the study are
i. To detect a reliable, easy, fast and accurate method of solving integro-differential equations.
- To become conversant with differential transformation and exponentially fitted collocation approximation method of solving integro-differential equations
iii. To ensure accuracy when solving integro-differential equations.
- To overcome the challenges found in integral equations.
- To illustrate the efficiency and reliability of using differential transformation and exponentially fitted collocation approximation method of solving integro-differential equations.
1.4 SCOPE OF THE STUDY
The scope this paper covers the use of differential transformation and exponentially fitted collocation approximation method in solving integro-differential equations. The collocated perturbed Integro-differential equations were transformed in to square matrix form which eventually solved using MAPLE 18 software. In order to investigate the accuracy of the solution with a finite number of computation length many examples were considered. To show the efficiency of the present method, numerical experiments are performed on some applied problems which have been solved by some existing methods and the numerical solutions are compared with available results in the literature and that of analytical solution. The numerical results obtain show the simplicity and efficiency of the method.
1.6 SIGNIFICANCE OF THE STUDY
This study will serve as a means of detecting the easiest and accurate method of solving integro-differential equation.
1.7 RESEARCH QUESTIONS
How do you solve an integro-differential equation?
What is Volterra integro-differential equations?
What is integro-differential operator?
1.8 DEFINITION OF TERMS
Differential Operator: differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function
Integro-Differential: integro-differential equation is characterized by the existence of one or more of the derivatives u′ (x), u″ (x), outside the integral sign. The Volterra integro-differential equations may be observed when we convert an initial value problem to an integral equation by using Leibnitz rule.
Leibniz Rule: leibniz rule is a rule that states that if two functions f(x) and g(x) are differentiable n times individually, then their product f(x). g(x) is also differentiable n times. The Leibniz Rule generalizes the product rule of differentiation.
Differential Equation: A differential equation is an equation that contains at least one derivative of an unknown function, either an ordinary derivative or a partial derivative.
