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# SUCCESSIVE APPROXIMATION METHOD FOR FIRST ORDER PDE USING PICARD

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Format: MS WORD ::   Chapters: 1-5 ::   Pages: 56 ::   Attributes: solution and exercise ::   2,353 people found this useful

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CHAPTER ONE

INTRODUCTION

1. BACKGROUND OF THE STUDY

Engineering problems can be mathematically described by differential equations. Many initial and boundary value problems associated with differential equations can be transformed into problems of solving some approximate integral equations.

Heat transfer is described by theory of integral equations. Integral equation arising in heat transfer with smooth condition is valid for continuous media. The common methods for solving the equations of heat transfer are purely mathematical are among them; the finite difference techniques (FDT) , the regression analysis (RA) , the Adomian decomposition method (ADM) , the combined Laplace-Adomian method (CLAM) , the homotopy analysis method (HAM) [9, 10], the differential transformation method (DTM) , the spline-wavelets techniques (SWT) , the boundary element method (BEM) , the heatbalance integral method (HBIM) [14, 15], the variational iteration method (VIM) , the local fractional variational iteration method (LFVIM) , and the Picard successive approximation method (PSAM) . On the other hand, the nanoscale heat problem can be characterized as fractal behaviors. As usual, the materials are called the Cantor materials. Heat transfer in fractal media with nonsmooth conditions is a hot topic.

For example, the heat transfer equations in a medium with fractal geometry  and fractal domains  were considered. The local fractional transient heat conduction equations based upon the Fourier law within local fractional derivative arising in heat transfer from discontinuous media were presented in [21–24].

First order initial value problems of the y’ (x) = f(x, y(x)), y(x0)), y(x0) = y0 can be rewritten as an integral

Y(x) = y0 + dt this integral formulation can be used to construct a sequence of approximate solutions to the IVP The basic idea is that given an initial guess of the approximate solution to the IVP, say Ø0(x), an infinite sequence of function { Øn(x)} is constructed according to the rule

Øn+1 = y0 + ) dt

That is the nth approximation is inserted into the right –hand side of the integral equation in place of the exact solution y’ (x) and used to compute the (n+1)st element of the sequence. This process is called picard method.

1. STATEMENT OF THE PROBLEM

Fractional calculus was successfully used to deal with the real world problems. There is its limit that the operators do not deal with the local fractional continuous functions (nondifferential functions). Hence, the local fractional Fourier flux is not handled by using some approaches from the classical and fractional operators. It is to this regard that study is based on successive approximation of first order PDE using picard.

1.3 AIM AND OBJECTIVES OF THE STUDY

The main aim of the research work is to carry out a successive approximation of first order PDE using picard. The objectives of the study are:

1. To examine whether successive approximation for first order PDE using Picard approximation is effective
2.  To determine a Local Fractional Volterra Integral Equation of the second Kind for PDE using picard
3. To investigate the factors affecting the use of picard for successive approximation of PDE

1.4 RESEARCH QUESTIONS

The study came up with research work so as to ascertain the above objectives. The following research questions guide the objectives of the study:

1. Is the successive approximation for first order PDE using Picard approximation effective?
2.  How can the Local Fractional Volterra Integral Equation of the second Kind for PDE using picard be achieved?
3. What are the factors affecting the use of picard for successive approximation of PDE?

1.5 SIGNIFICANCE OF THE STUDY

The study on successive approximation of first order PDE using picard will be of immense benefit to the entire mathematics department in Nigeria. The study will offer an explicit solution for the successive approximation for first order PDF using picard. The study will be able to derive the Local Fractional Volterra Integral Equation for heat equations. The study will serve as a repository of information to other students and researchers that desire to carry out similar research on the above topic. Finally the study will contribute to the body of the existing literature on successive approximation of first order PDE using picard

1.6 SCOPE OF THE STUDY

The study will simply focus on successive approximation of first order PDE using picard

1.7 DEFINITION OF TERMS

PDE: partial differential equation

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 Format: ms word Chapter: 1-5 Pages: 56 Attribute: solution and exercise Price: ₦3,000