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Methods for solving problems of mathematical physics can be divided into the following four classes.
Analytical methods (the method of separation of variables, the method of characteristics, the method of Green's functions, etc.) methods have a relatively low degree of universality, i.e. focused on solving rather narrow classes of problems.
Approximate analytical methods (projection, variational methods, small parameter method, operational methods, various iterative methods) are more versatile than analytical ones.
Numerical methods (finite difference method, direct method, control volume method, finite element method, etc.) are very universal methods.
Probabilistic methods (Monte Carlo methods) are highly versatile. Can be used to calculate discontinuous solutions. However, they require large amounts of calculations and, as a rule, they lose with the computational complexity of the above methods when solving such problems to which these methods are applicable.
Comparing methods for solving problems of mathematical physics, it is impossible to give unconditional primacy to any of them. Any of them may be the best for solving problems of a certain class.
The proposed method of moving nodes for boundary value problems of differential equations combines a combination of numerical and analytical methods. In this case, we can obtain, on the one hand, an approximately analytical solution of the problem, which is not related to the methods listed above. On the other hand, this method allows one to obtain compact discrete approximations of the original problem. Note that obtaining an approximately analytical solution of differential equations is based on numerical methods. The nature of numerical methods also allows obtaining an approximate analytical expression for solving differential equations
Dr. Dalabaev is a professor at the University of World Economy and Diplomacy, Department of Mathematic modelling & Information Systems, Tashkent, Uzbekistan. His research interests are in computational fluid dynamics, numerical methods, modelling two phase flow, particle motion in fluids, moving node method, approximate-analytical solution of a differential equation.
This chapter presents some facts known in the literature. The necessary materials on which the subject of this study is based are presented. The concept of a grid, grid functions, and finite-difference methods (FDMs), as well as the control volume method, is briefly presented. There are sufficient numbers of publications devoted to numerical methods for differential equations [1, 22, 25, 27, 28].
The grid method or difference method is one of the effective numerical methods for approximate solution of equations. In the grid method, the domain of continuous variation of arguments is replaced by a discrete set of points, which is called a grid. Grid functions are functions defined at the grid nodes. Derivatives appearing in differential equations and boundary conditions are replaced by difference relations. After discretization, the differential equation is replaced by a system of linear or nonlinear algebraic equations. The resulting systems are called difference schemes. Briefly, let us outline the basic concepts of the theory of difference schemes.
One-dimensional case: A grid on a segment x?[a,b] is any finite ordered set of points of this segment. Most often, a uniform grid of the form will be used:
?─h=xi=a+ih;i=0,N,
where h=(b-a)/N is the grid step.
Two-dimensional case: For a rectangle G?=?{a?=?x?=?b; c?=?y?=?d} we define a uniform grid as ?─h=?─1×?─2, where
Here h1=(b-a)/N1 and h2=(b-a)/N2 are the steps in the variables x and y. If h1=h2, then the grid is called square; otherwise - rectangular. In the general case, the grid may be nonuniform.
Let us consider the concept of a grid function. Let ?─h be a grid introduced in the one-dimensional case and xi be grid nodes. The function y=y(xi) of a discrete argument xi is called a grid function defined on the grid ?─h. A grid function on any grid is defined similarly. Grid functions can be considered as functions of an integer argument, which is the number of a grid node.
Let a grid ?─h be given. The set of all grid functions defined on the grid forms ?─h, a vector space. In the space of grid functions, one can define difference or grid operators.
The operator ? that transforms a grid function y into a grid function f=?y is called a difference or grid operator. The set of grid nodes used to write a difference operator is called the template of this operator. The simplest difference operator is the operator of difference differentiation of a grid function.
Let us consider the problem of approximately calculating the derivatives of a function u(x) defined on a segment [a,b]. We will assume that the function u(x) has a certain degree of smoothness, and then the first-order difference derivatives on the grid ?─h for the function u(x) are given in Tab. 1.1 on a uniform grid.
Tab. 1.1:Difference operators approximating the derivative.
If the point xi is fixed and the step h tends to zero, then each of the difference relations tends to the value of the derivative of the function u(x) at the point xi. Note that the derivative can be replaced by any difference relation, but the order of the error with such a replacement will be different.
Higher-order difference derivatives can be introduced using recurrent formulas. If in the formulas for the first-order difference derivative, instead of the values of the function at the grid node, we use the value of the difference derivative, and we obtain formulas for calculating the second-order difference derivative, for example:
The second-order difference derivative is not unique. Difference derivatives of any order are constructed similarly.
Examples of the simplest approximations of differential expressions by difference ones on a uniform grid are given. In the general case, the error depends both on the distribution of grid nodes and on the smoothness of the function.
The following presentation is devoted to the construction and study of difference boundary value problems for model equations of mathematical physics.
The FDM is one of the oldest numerical methods for obtaining approximate solutions of partial differential equations.
The FDM is a direct approximation of the resolving partial differential equations, which is performed by replacing partial derivatives with differences on regular or irregular grids defined on the problem domain. Thus, in this case, a transition is made from a system of partial differential equations to a system of algebraic equations with respect to unknown quantities at grid points. The solution to the system of equations is obtained after imposing the necessary initial and boundary conditions. Therefore, the FDM is also called the grid method.
The main principle of the FDM is to replace partial derivatives of an unknown function with finite differences. As a result of this procedure, a system of algebraic equations is obtained for unknown functions defined at the nodes of the grid into which the problem domain is divided. The solution of a system of algebraic equations, including boundary conditions specified at the boundary nodes, yields the desired values of the unknown function at all nodes that will satisfy both the resolving system of partial differential equations and the specified boundary conditions.
Thus, the fundamental characteristic of the FDM is that it uses direct discretization of the resolving differential equations by replacing partial derivatives with finite differences defined at adjacent grid nodes. Creating a finite-difference grid is only a convenient way to represent the values of the sought functions at specified points located at a sufficiently small distance from each other, such that the resulting errors in the solution are sufficiently insignificant and can be neglected.
In addition, the FDM does not use approximating or interpolating functions to represent differential equations in the vicinity of specified points, as is the case, for example, in the finite element method and the boundary element method.
The FDM thus represents the most direct and visual method for solving partial differential equations.
Consider a linear two-point boundary value problem.
Consider a two-point boundary value problem for a linear differential equation. Find a solution to the equation:
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