A Research on the Numerical Solution of Partial Differential Equations Using Matrix Theory

1.1 INTRODUCTION

The majority of the problems of physics and engineering fall naturally into one of three physical categories: equilibrium problems, eigen value problems and propagation problems. Equilibrium problems are problems of steady state in which the equilibrium configurations in a certain domain to be determined. Eigenvalue problems may be thought of as extension of equilibrium problems wherein critical values of certain parameters are to be determined in addition to the corresponding steady-state configurations. Propagation problems are initial value problems that have an unsteady state or transient nature [2]. Normally, in most cases, the dependent variable to any of these problems is expressed in terms of several independent variables. Such problems inherently give rise to the need for partial derivatives in the description of their behaviour. The study of differential equations arising from these problems constitutes the field of Partial Differential Equations. Mathematically, a partial differential equation ( henceforth abbreviated as p.d.e.) for a dependent variable u(x,y,...) is a relation of the form

(1.1.1)

where F is a given function of the independent variables x,y,... and the "unknown" function u and of a finite number of its partial derivatives. We call u as a solution of (1.1.1) if after substitution of u(x,y,...) and its partial derivatives (1.1.1) is satisfied identically in x,y,... in some region ft in the space of these independent variables. The independent variables x,y,... are real (unless if stated otherwise) and u and the derivatives of u occurring in (1.1.1) are continuous functions of x,y,... in the real domain ft. As in the theory of Ordinary Differential Equations ( henceforth abbreviated as o.d.e.) a p.d.e. (1.1.1) is said to be of order n if the order of the highest partial derivatives involved is n. Equation (1.1.1) is said to be linear if F is linear in the unknown function and its derivatives and quasi-linear if F is linear in at least the highest order derivatives [1]. For example, the equation

as a first order quasi-linear p.d.e., and the equation

(1.1.3)

is a second order linear p.d.e. Meanwhile the equation

(1.14)

is the example of a second order non-linear. In equation (1.1.2)—(1.1.4) above, x and y are the independent variables and u=u(x,y) is the dependent variable whose form is to be found. (Heat Equation) (1.1.5) is homogeneous, whereas

(1.1.6)

where f(x,t) is a given function, is an inhomogeneous or non- homogeneous equation. The problem of finding the solution to the p.d.e. is very difficult as the general method of solution is not available except for certain special types of linear or quasi-linear equations. Furthermore, the general solution of the linear p.d.e. which contains arbitrary functions is of little use, since it has to be made to satisfy other conditions called boundary conditions[7] which arise from the physical problem itself . Similar to o.d.e., if are n different solutions of a linear homogeneous p.d.e, in some given domain then

(1.1.7)

is also a solution in the same domain with as arbitrary constants.

As the p.d.e. arise from different categories of physical phenomenon [1,11] (e.g. steady viscous flow, resonance in electric circuits, propagation of heat), this suggests that the governing equations are also quite different in nature. Normally, they are classified in terms of their mathematical form such as elliptic, hyperbolic and parabolic equation, or in terms of the type of problems to which they apply i.e. the heat equation and the wave equation. The most general second order linear p.d.e. in two independent variables, is given by,

(1.2.1) (1.2.2)

is negative, positive or zero. This classification depends in general on the region of the (x,y)-plane under consideration. Thus it is possible for a p.d.e. to change its classification within the different regions of the domain for which the problem is defined. The differential equation

(1.2.3)

for instance, is elliptic for, hyperbolic forand parabolic along, This type of equation is said to be mixed type. For a linear p.d.e. in more than two independent variables, for systems and for non-linear p.d.e., a similar but complicated classification can be carried out. In the case of three independent variables, the terms elliptic parabolic and hyperbolic are replaced by their three dimensional analogous such as ellipsoidal, etc. The elliptic class is related to equilibrium problems and are usually described in terms of a closed region having boundary conditions prescribed at every point on the region‘s boundary. These are called the boundary value problems[7]. The parabolic and hyperbolic problems are of propagation type and normally have a prescribed boundary condition on some part of the boundaries and an initial condition along the other part. It can also have open-ended regions into which the solution propagates. In mathematical parlance such problems are known as initial (boundary) value problems. Another important aspect of the classification of the p.d.e. into hyperbolic, parabolic and elliptic is due to the characteristic equation [2]. Here we can see that at every point of the x-y plane there are two directions in which the integration of the p.d.e. reduces to the integration of an equation involving total differential only. In other words, the integration of an equation in certain directions is not complicated by the presence of partial derivatives in other directions. Let the derivatives in equation (1.2.1) be denoted by

given). Therefore, the differentials of p and q in the directions tangential to C are given by

(1.2.4)

and

(1.2.5)

where the p.d.e. (1.2.1) is written as

(1.2.6)

Elimination of r and t in (1.2.6) using (1.2.4) and (1.2.5) gives: i.e.

(1.2.7)

Now choose, the tangent to C at a point P(x,y) to satisfy

(1.2.8)

Therefore (1.2.7) leads to

(1.2.9)

which gives the relationship between the total differential dp and dq with respect to x and y. This shows that at every point P(x,y) of the solution domain there are two directions, given by the roots of equation (1.2.8), along which there is a relationship given by equation (1.2.9). The directions given by the roots of equation (1.2.8) are called the characteristic directions and the p.d.e. is said to be hyperbolic, parabolic or elliptic according to similar determinant requirement as in (1.2.2).

The solution of the p.d.e. has to be made to satisfy the boundary conditions which arise from the problem formulation. There are four main types of The First Boundary-Value Problem (the Dirichlet Problem), where the solution u has to satisfy the given values

(1.3.1)

on the boundary s. If the problem is called Homogeneous Dirichlet problem[7]. 1. The Second Boundary-Value Problem (the Neumann Problem), where the solution u has to satisfy the normal derivatives

(1.3.2)

on the boundary of that region. 2. The Third Boundary-Value Problem (Mixed or Robin's Problem),where the solution u has to satisfy a combination of u and its derivatives namely

(1.3.3)

on the boundary s.

In this case we seek the solution such that it satisfies the periodicity conditions, for example,

(1.3.4)

where I is called the period. The physical meaning of the first three boundary-value problems can be illustrated by the problem of steady-state temperature distribution. In the Dirichlet problem, the temperature is given on the boundary of a solid. In the Neumann problem the loss or gain of heat through the boundary is given (it is proportional to ). In this problem in order to keep a steady-state distribution of temperature, the net flow of thermal energy passing through the boundary of a solid must be equal to zero, i.e., (1.3.5) temperature of which is where h is the coefficient of thermal conductivity divided by the specific heat.

In the numerical solution of p.d.e.'s by the finite difference method or finite element method, the differential system is replaced by a matrix system. In this section some useful properties of a matrix are outlined. Notations 1.4.1 A square matrix of order n real number which is the element in the row and column of the matrix A. inverse of A transpose of A determinant of A I unit matrix of order n 0 null matrix • spectral radius of A . column vector with element row vector with element complex conjugate of norm of A norm of vector x permutation matrix which has entries of zeros and one only, with one non-zero entry in each row and column. Definitions 1.4.1 The matrix A is said to be i. non-singular if ii. diagonal if its only non-zero elements lie on the diagonal iii. symmetric if i.e. » orthogonal if diagonal if where each is a square matrix of order with upper triangular if for lower triangular if for sparse if many of the elements are zero. For the matrix A whose elements which are not necessarily real numbers we denote as the conjugate transpose of A. A is called Hermitian if i.e. if for all i and j, i,j=l,2,...,n. The definition of a Hermitian matrix implies that the diagonal elements of the matrix are real. A real symmetric matrix is always Hermitian, but a Hermitian matrix is symmetric only if it is real.

definite if for all (Note that the inner product of two complex vectors is where is the complex conjugate of y^). A is non- negative or semi-positive definite if for all with equality for at least one A is a band matrix of bandwidth if. for or If p=q=l, then A is tridiagonal and a pentadiagonal matrix can be obtained when p=q=2.

Definition 1.5.1 If A is a square matrix of order n and if is a non- zero vector such that whereis some number, then is said to be an eigenvector of A with corresponding eigenvalue[13].

satisfies the nth degree polynomial equation ; this equation is known as the characteristic equation of A. Proof: We seek a scalar and non-zero vector such that or Since this is a system of n simultaneous homogeneous equations in the n unknowns, (not all are zero), therefore must be singular, or in other words Theorem 1.5.2 (Gerschgorin's first theorem) The largest of the moduli of the eigenvalues of the square matrix A cannot exceed the largest sum of the moduli of the elements along any row or any column.

Let be an eigenvalue of A and is the corresponding eigenvector with components Then the equation is in detail given by

Let be the largest in modulus of Select the sth equation and divide by giving Since therefore In particular this holds for , As the eigenvalues of the transpose of A are the same as those of A, the proof is also true for columns. Theorem 1.5.3 (Gerschgorin‘s circle theorem)

(1.5.2)

Proof: By the previous proof Therefore, As is any eigenvalue, therefore,

(1.5.3)

Corollary 1: If A is a square matrix of order n and

(1.5.4)

Then

In this paper, it is shown that with the increasingly high standard of technology used in society today and the accelerating use of digital computers, the numerical solution of parabolic partial differential equations remains an important subject for research for the future. The discovery of the Group Explicit method is an important landmark in methods of solving partial differential equations as in principle the method is simple, stable, accurate and flexible in the sense that it allows the calculation of more than one point at a time - a new and original idea.

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Assistant Professor in Mathematics, Mata Gujri Khalsa College, Kartarpur, Jalandhar gursimranpal2491@gmail.com