Fourier Method in Initial Boundary Value Problems for Regions with Curvilinear Boundaries

The algorithm of the generalized Fourier method associated with the use of orthogonal splines is presented on the example of an initial boundary value problem for a region with a curvilinear boundary. It is shown that the sequence of finite Fourier series formed by the method algorithm converges at each moment to the exact solution of the problem – an infinite Fourier series. The structure of these finite Fourier series is similar to that of partial sums of an infinite Fourier series. As the number of grid nodes increases in the area under consideration with a curvilinear boundary, the approximate eigenvalues and eigenfunctions of the boundary value problem converge to the exact eigenvalues and eigenfunctions, and the finite Fourier series approach the exact solution of the initial boundary value problem. The method provides arbitrarily accurate approximate analytical solutions to the problem, similar in structure to the exact solution, and therefore belongs to the group of analytical methods for constructing solutions in the form of orthogonal series. The obtained theoretical results are confirmed by the results of solving a test problem for which both the exact solution and analytical solutions of discrete problems for any number of grid nodes are known. The solution of test problem confirm the findings of the theoretical study of the convergence of the proposed method and the proposed algorithm of the method of separation of variables associated with orthogonal splines, yields the approximate analytical solutions of initial boundary value problem in the form of a finite Fourier series with any desired accuracy. For any number of grid nodes, the method leads to a generalized finite Fourier series which corresponds with high accuracy to the partial sum of the Fourier series of the exact solution of the problem.

Abstract The algorithm of the generalized Fourier method associated with the use of orthogonal splines is presented on the example of an initial boundary value problem for a region with a curvilinear boundary. It is shown that the sequence of finite Fourier series formed by the method algorithm converges at each moment to the exact solution of the problem -an infinite Fourier series. The structure of these finite Fourier series is similar to that of partial sums of an infinite Fourier series. As the number of grid nodes increases in the area under consideration with a curvilinear boundary, the approximate eigenvalues and eigenfunctions of the boundary value problem converge to the exact eigenvalues and eigenfunctions, and the finite Fourier series approach the exact solution of the initial boundary value problem. The method provides arbitrarily accurate approximate analytical solutions to the problem, similar in structure to the exact solution, and therefore belongs to the group of analytical methods for constructing solutions in the form of orthogonal series. The obtained theoretical results are confirmed by the results of solving a test problem for which both the exact solution and analytical solutions of discrete problems for any number of grid nodes are known. The solution of test problem confirm the findings of the theoretical study of the convergence of the proposed method and the proposed algorithm of the method of separation of variables associated with orthogonal splines, yields the approximate analytical solutions of initial boundary value problem in the form of a finite Fourier series with any desired accuracy. For any number of grid nodes, the method leads to a generalized finite Fourier series which corresponds with high accuracy to the partial sum of the Fourier series of the exact solution of the problem.

Introduction Problem Statement
The method of separation of variables (Fourier method) allows finding partial solutions of many boundary value and initial boundary value problems for partial differential equations that allow separation of variables. The method is related to the Sturm-Liouville problem and, in many cases, to special functions at the stage of solving this problem. The classical Fourier method allows obtaining solutions to a wide class of problems, but its implementation for many types of problems, including problems whose statements contain irregular boundary conditions, even in cases where all parts of the boundary of the region are coordinate lines or surfaces, meets with significant difficulties. One of the ways to expand the scope of the classical Fourier method is to solve mathematical problems related to the nature of boundary conditions [1]. Special functions appear in the algorithm of the method when solving the Sturm-Liouville problem in cylindrical or spherical coordinate systems in cases of regions whose boundaries are coordinate lines or surfaces. In the general case of problems for areas with curvilinear boundaries, the use of special functions is inefficient. The classical Fourier method is applicable only for solving boundary value and initial boundary value problems for regions of classical shape, which is noted, for example, in solving contact problems [2] for elastic bodies with curvilinear boundaries. The solutions obtained by the classical Fourier method are given, in particular, in the articles [3,4], the application of the method is considered in many books, for example, in [5]. Other areas of development of mathematical tools for solving problems for areas with curvilinear boundaries are associated, first, with the creation and application of a number of methods other than the Fourier method, for example [2,6,7,8], and, secondly, with a modification of the Fourier method itself. This article aims to expand the scope of the classical Fourier method, defined by the application of a sequence of finite generalized Fourier series and orthogonal splines.
The next first initial boundary value problem is considered where S ∂ is a piecewise smooth curvilinear boundary of the region S , . According to the method of separating variables, the solution of problem (1) is sought as a product of two functions substituting which in the differential equation (1) leads to The boundary condition (1) taking into account (2) is found. Then, based on

The Algorithm of the Method
The first steps of the generalized Fourier method algorithm for regions with a curvilinear boundary coincide with similar steps of the classical Fourier method. The solution of problem (1) is also sought in the form of a product (2), whose substitution in the differential equation (1) leads to equation (3), and the boundary condition (1) gives (4). The same Sturm-Liouville boundary value problem (5) arises. From (3) also follows equation (6), the solution of which is related to the solution of the Sturm-Liouville problem (5) and taking into account two initial conditions (1).
The further steps of the algorithm of the generalized Fourier method, designed for solving initial boundary value problems in the case of regions with curvilinear boundaries, differ from the corresponding steps of the classical algorithm. They are related to the application of orthogonal splines in the construction of a sequence of approximate analytical solutions in the form of finite Fourier series.
Nontrivial solutions of the boundary value problem (5) are sought in the form where ij c are constant coefficients; -continuous orthogonal splines [9,10]: The orthogonal splines ( ) in the case of a rectangular uniform grid have the form [9] 26 Fourier Method in Initial Boundary Value Problems for Regions with Curvilinear Boundaries steps of grid, the nodes of a grid have coordinates The compact supports of tensor products of orthogonal splines on each grid are rectangular subdomains. Orthogonal splines also allow the use of triangular compact supports. The functions form two systems of orthonormal splines. The sum (7) is rewritten as where To determine the coefficients ij c of the approximate analytical solution (9) of the boundary value problem (5), the Reissner variational principle is used: components of normal to a boundary S ∂ . The variational principle is equivalent to problem (5) in mixed form , and after substituting (9) into 0 = R δ , it leads to a system of finite difference equations [10] for the internal nodes of the grid satisfies equations (5) in variational form, equation (6), as well as the boundary condition and two initial conditions  Consider equation (6) after substituting any found positive eigenvalue into it The general solution of this differential equation has the form are unknown constant coefficients. Therefore, the sum β α λ satisfies equation (5) in variational form, equation (6), and boundary condition (5). It remains to satisfy the initial conditions. The substitution of the sum (14) into the first initial condition (1) gives Multiplying both parts of the last equation by , integrating over the region S and using the orthogonality of the eigenfunctions allow writing Substituting the sum (14) in the second initial condition (1) gives

Multiplying both parts of this equation by
integrating over the region S allow writing Thus, the sum (11), whose coefficients are determined by formulas (12), (13), satisfies equation (5) in the variational form, equation (6), as well as the boundary condition (5) and two initial conditions (1), that is, the sum (11) is an approximate analytical solution of the problem (1) in the case of a region with a curvilinear boundary. The sum (11) for each fixed time value is a finite generalized Fourier series over the eigenfunctions of the boundary value problem in variational form.
The proof is completed. , satisfying the boundary condition (5), that is,

The Convergence of the Method
is equivalent to the boundary value problem (5). Here ∇ is the nabla operator. The variation of the functional is carried out on the set of functions satisfying the main boundary condition (5). The value of the second variation of the functional at its stationary point is positive, and therefore the functional has a minimum at the stationary point, and the solution of the variational problem, and hence the boundary value problem (5), are unique. In addition, the functional has a value equal to zero at a stationary point Thus, the problem of finding the exact eigenfunctions at the set of functions w , satisfying the boundary condition. Because then the minimization problem (14) is reduced to the approximation theory problem 2 2 i.e. the approximate eigenfunctions of this problem, converge to its exact solutions -eigenfunctions In this case, the number of eigenvalues and eigenfunctions of the boundary value problem in variational form increases indefinitely, and consequently, the finite sum in the limit passes into an infinite series, which for any value 0 > t is an infinite Fourier series. Such a series is the unique solution of problem (1), it follows from Steklov's theorem [11].
The difference between this method for solving initial boundary value problems for regions with curvilinear boundaries and other methods, for example, the finite element method [8], is that in this method, the sequence of finite Fourier series (11) determined by its algorithm converges at each fixed time to the corresponding infinite Fourier series formed on the basis of exact eigenfunctions ) y , x ( ) k ( ϕ and representing an existing exact solution to the problem (1), which cannot be determined. Consequently, these finite Fourier series represent analytical approximate solutions of problem (1) for a region with a curvilinear boundary, which, as the number of grid nodes increases, come close to the exact solution of this problem -an infinite Fourier series, not only in terms of quantitative criteria, but also in their analytical structure. The method gives, in the form of finite Fourier series, arbitrarily accurate approximate analytical solutions to problem (1) for regions with a curvilinear boundary whose structures are similar to the structure of the exact solution, and in this sense the method leads to an exact analytical solution after the required accuracy of the desired solution is set.

Test Problem
The problem (1) and the Sturm-Liouville problem (5) are considered for a region S whose boundary S ∂ is a square with a side . l π = A rectangular uniform grid with steps is used, the nodes of which have coordinates The finite difference equations (10) written for all internal nodes , taking into account the boundary conditions (5), give a homogeneous system of difference equations, the nontrivial solutions of which are the eigenfunctions [12], [13], [ are determined by formulas (12), (13), which take into account the initial conditions (1).
The exact solution of the Sturm-Liouville problem (5) for the case  corresponds to the structure of the partial sum of the infinite series (16), with account of the number of nodes for a concrete calculation. Thus, the formulations of theorems 1 and 2 are confirmed by the results of solving the test problem. These results confirm the conclusions of the theoretical study of the convergence of the method, and that the proposed algorithm of the method of separating variables, which is connected with orthogonal splines, gives approximate analytical solutions to the initial boundary value problem in the form of a finite Fourier series with any given accuracy.

Discussion
Expanding of scopes of application of classical analytical methods of solving initial boundary value problems is an actual problem. One of the directions of development of such methods is to extend the scope of application of the method of separation of variables -the Fourier method, to problems for areas with curvilinear boundaries. Special functions made it possible to use this method when solving problems for areas with curvilinear boundaries. However, the geometry of such boundaries is connected with coordinate surfaces in some curvilinear coordinate systems, which significantly limits the scope of application of methods. Solutions of initial boundary value problems for regions with complex curvilinear boundaries found using numerical methods do not allow direct analogy with analytical exact solutions in the form of an infinite Fourier series. In this article, a method of separating of variables designed to solve initial boundary value problems for regions with curvilinear boundaries in the general case of their geometry is considered. This method gives solutions in the form of finite generalized Fourier series, which are analogous to partial sums of an infinite Fourier series, which is an exact solution of a problem, and converge to an exact solution. This method pulls together numerical methods for solving initial boundary value problems with an analytical method for solving them. The considered modification of the Fourier method related to orthogonal splines can be used for solving other problems of mathematical physics and continuum mechanics.

Conclusions
In this method, the potential capabilities of the variable separation method, orthogonal splines, and the finite difference method are used together, which leads to analytical solutions in the form of finite Fourier series that correspond with high accuracy to partial sums of an infinite Fourier series -the exact solution of the problem.
The article [15] shows the effectiveness of orthogonal splines in the problems of approximation of functions and surfaces. Here the region of application of orthogonal splines is expanded.
The method of modeling temperature data using semiparametric additive generalized linear model is proposed in the article [16]. This method can apparently be used in conjunction with the modified Fourier method proposed here in solving initial boundary value problems for temperature fields in regions with a curved boundary.