ABSTRACT
In this work, we construct the generalized Green’s function for the initial value problem involving a partial differential operator specifically the heat operator in n , using Fourier transform method. The homogeneous and non-homogeneous heat equations are solved with specified initial condition. In particular employing the Duhamel’s principle which is a procedure for expressing the solution of a non-homogeneous problem as an integral of the solutions to the homogeneous problem in the way that the source term is interpreted as the initial condition, we obtained the solution of the non-homogeneous problem. We also study the properties of the Green’s function and for the two dimensional case, the function is plotted in three dimensions using MATLAB. Our result shows that the Green’s function constructed is unbounded in any neighbourhood of the origin. We also applied the concept of Green’s function to electrostatics. In particular we solved a generalized problem involving a unit source charge positioned at a specific point in space. We then solve the Poisson’s equation using the Green’s function solution giving us an inverse law for the associated electrostatic potential.
TABLE OF CONTENTS
Title page i
Declaration ii
Certification iii
Dedication iv
Acknowledgements v
Table of contents vi
Abstract viii
CHAPTER 1: INTRODUCTION
1.1 Background to the Study
1.2 Statement of the Problem 3
1.3 Motivation of the Study 4
1.4 Aim and Objectives 4
1.2 Significance of the Study 5
1.3 Justification of the Study 6
1.4 Scope and Limitation of the Study 6
CHAPTER 2: REVIEW OF RELATED LITERATURE
CHAPTER 3: METHODOLOGY
3.2 Green’s Theorem 12
3.3 Green’s Function 13
3.4 Operator Equations 13
3.4.1 The spectrum n of the operator L 14
3.5 Matrix Analogy 16
3.5.2 Properties of the Green’s function 17
3.6 The Heat Equation 17
3.7 Methods of Fourier Transform 18
3.7.1 Fourier transform with respect to x 19
3.7.2 Fourier transform of the partial derivative 19
3.7.3 Convolution 19
3.7.4 Theorem (convolution theorem) 19
3.7.5 Theorem (Fourier inversion theorem) 19
3.7.6 Example 19
3.7.7 Fundamental solution of the heat equation 23
3.8 Properties of the Fundamental Solution x,t of the Heat Equation 22
3.9 Invariance Properties of the Heat Equation 22
3.10 The Dirac-Delta Function 23
3.10.2 Properties of the Dirac-delta function 24
CHAPTER 4: MAIN RESULS AND DISCUSSION
4.2 Fourier Transform in 27
4.2.1 Example 27
4.2.2 The Fourier inversion formula 27
4.2.3 Theorem (Fourier inversion theorem) 28
4.2.4 The Fourier transformation and convolution in 28
4.2.5 Theorem (Convolution) 28
4.3. Construction of Generalized Green’s Function for n -Dimensional Heat Operator t .
28
4.3.3 Plot of the Green’s function in three dimensions using MATLAB 32
4.4 The Non-Homogeneous Problem 33
4.4.1 Duhamel’s principle 34
4.4.3 Lemma 34
4.4.4 Theorem (the solution of the non-homogeneous Cauchy problem) 35
4.5 The Non-Homogeneous Heat Equation 35
4.6 Application of Green’s Function 38
4.6.1 Boundary value problems in electrostatics 38
CHAPTER 5: SUMMARY AND CONCLUSIONS
5.1 Summary 41
5.2 Conclusions 41
References 43
CHAPTER 1
INTRODUCTION
1.1 BACKGROUND OF THE STUDY
The concept of a Green’s function came to light in the 19th century while studying the classical problems of Mathematical Physics, (Green, 1852 and 1854), and its fundamental application to the theory of differential equations. Indeed, physics, mechanics and other natural sciences have been developed greatly in the past decade or so, and today they investigate such processes and phenomena that require rigorous mathematical tools in order to make sense of them. For instance, we have the thermostat problem (Amos, et al. 2011), heat condition and bio- reaction engineering problems (Kamynin, 1964) and problems arising in electrochemistry and microelectronics. The Green’s function method has been very popular in computing the solution of heat conduction problems for basic geometries, specifically in the considered regions involving classical boundary conditions. Application of the Green’s function to such problems can be found in the book by (Duffy, 2001). The advantage of the Green’s function method rests in the fact that the solution of the non-homogeneous problems can be neatly expressed in terms of the Green’s function (Beck, et al. 1992). Green’s functions provide a powerful tool to solve linear problems consisting of differential equations (partial or ordinary, with, possibly non- homogeneous term) and enough initial- and/or boundary conditions (also possibly non- homogeneous) so that this problem has a unique solution. The Green’s function is defined by a similar problem where all initial- and/or boundary conditions are homogeneous and the non-homogeneous term in the differential equation is a delta function. If one knows the Green’s function of a problem one can write down its solution in closed form as linear combinations of integrals involving the Green’s function and the functions appearing in the non-homogeneities. Green’s functions can often be found in an explicit way, and in this case it is very efficient to solve the problem in this way (Cabada and Saavedra, 2017). The Green’s function has been an interesting topic in modern physics and engineering, especially for the electromagnetic theory in various source distributions (charge, current, and magnetic current), various construct conductors, and dielectric. However, most of the problems can be solved without the use of Green’s functions the symbolic simplicity with which they could be used to express relationships makes the formulations of many problems simpler and more compact. Moreover, it is easier to conceptualize many problems; especially the dyadic Green’s function is generalized to layered media of planar, cylindrical, and spherical configurations (Huang, 2017).
In recent times the methods of Green’s function has been generalized to non-classical solutions of classical differential problems, (Anderson, 2003), of equal importance is the work of (Cabada, 2014). Green’s function is essentially an integral kernel that can be employed to solve a wider class of differential equations, including simpler examples such as ordinary differential equation (ODEs) with initial or boundary value conditions, as well as more difficult examples such as inhomogeneous partial differential equations (PDEs) with boundary conditions. Furthermore, Green’s functions allow for visual interpretations of the actions associated with an impulsive force or to a charge concentrated at a point, thus making them particularly useful in areas of applied mathematics.
More precisely, given a linear differential operator L = L ( x) acting on the collection of distributions over a subset W of some Euclidean space n , a Green’s function G = G ( x,t ) at the point xÎW corresponding to L is any solution of
Where, d denotes the delta function. Multiplying the above equation by a function g (t ) and integrating with respect to gives
The right hand side reduces merely to g ( x) due to properties of the delta function, and because L is a linear operator acting only on x and not ont, the left hand side can be rewritten as
This reduction is particularly useful when solving for u = u ( x) in differential equations of the form
where the above confirms that
It then follows that u has the specific integral form
where G ( x,t ) is the Green’s function.
Equation (1.6) illustrates both the intuitive physical interpretation of a Green’s function as well as a relatively simple associated differential equation. (Adrian, et al. 2010).
1.2 STATEMENT OF THE PROBLEM
Many methods have been applied in solving partial differential equations with initial and boundary conditions. In this work we shall exploit the method of Fourier transform in the construction of generalized Green’s function for the heat operator ¶_{t} - D n-dimensional Euclidean space n. Specifically, we consider the homogeneous and nonhomogeneous partial differential equations;
with appropriate initial condition, and where
1.3 MOTIVATION OF THE STUDY
Many researchers have used different methods in solving partial differential equations. This thesis is motivated by studying the work done by (Kukla and Siedlecka, 2013) and (Hassan, 2017) respectively who used the Green’s functions in solving the heat conduction problem in a composite cylinder and obtained a solution of one-dimensional eigenvalue problems and presented it in form of eigenfunction series and the later considered one –dimensional heat equation for the non-homogeneous problem with the appropriate boundary and initial conditions, using Laplace transform method and obtained a Green’s function. Unlike the methods used by other researchers here, we introduced the method of Fourier transform to construct the generalized Green’s function for the heat operator in n-dimensional Euclidean space.
1.4 AIM AND OBJECTIVES
The primary aim of this work is to construct a generalized Green’s function for partial differential operator (heat operator ) in n .
The objectives of this research work are;
I. To study the concept of Green’s functions
II. Use the constructed Green’s function to solve initial value problems.
III. To plot the Green’s function constructed in three dimensions using MATLAB
IV. Emphasize the importance and applications of Green’s function.
1.5 SIGNIFICANCE OF THE STUDY
The use of Green’s function is a fundamental procedure for computing the solution to differential equations, and also it is a useful tool that can be used to construct numerous important useful solutions of differential equations. This specific topic has applications in various areas of science and engineering, most especially in Mathematics, Physics such as in Electrostatic, quantum field and fluid mechanics.
In Mathematics, Green’s function technique is used to deal with non-homogeneous boundary value problems. In the derivation of the Green’s function, the non-homogeneous part is dirac- delta function which physically means an impulsive force. Thus, Green’s function deals with the physical situation in the presence of an impulsive load thus, making it useful in areas of applied Mathematics. (Beck, et al. 1992).
In Electrostatics Physics, it would be difficult to avoid the presence of differential equations. These Mathematical equations appear quite often and extremely varied in type. For instance, the equation of simple harmonic motion, (a homogeneous ordinary differential equation), Poisson’s equation (an inhomogeneous partial differential equation) etc., there exist many ways, both analytical and numerical of solving these equations and also the method of Green’s function are particularly useful for these.
In the quantum context, Green’s functions are correlation functions, from which it is possible to extract information from the system under study, such as the density of states, relaxation times and response functions.
In Fluid Mechanics, Green’s Function and source functions are used to solve 2-dimensions and 3-dimensions transient flow problems that may result from complex well geometries, such as partially penetrating vertical and inclined wells, hydraulically fractured wells. (Gringarten and Ramedy, 1973).
1.6 JUSTIFICATION OF STUDY
Green’s function is useful in obtaining closed form solutions to nonlinear partial differential equations, and as such can be used in the analysis of complex engineering problems involving such equations; hence we are justified in going into this research on Green’s function.
1.7 SCOPE AND LIMITATION OF THE STUDY
This work covers the solutions of partial differential equations using Green’s function. However, it is limited to the construction of Greens function for the heat operator ¶_{t} - D in n ..
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