Численное решение начально-краевых задач для уравнения теплопроводности методом интегральных уравнений

. Many important questions in the theory of elasticity lead to a variational problem associated with a biharmonic equation and to the corresponding boundary value problems for such an equation. The paper considers the main boundary value problem for the biharmonic equation in the unit circle. This problem leads, for example, to the study of plate deflections in the case of kinematic boundary conditions, when the displacements and their derivatives depend on the circular coordinate. The exact solution of the considered boundary value problem is known. The desired biharmonic function can be represented explicitly in the unit circle by means of the Poisson integral. An approximate solution of this problem is sometimes foundusing difference schemes. To do this, a grid with cells of small diameter is thrown onto the circle, and at each grid node all partial derivatives of the problem are replaced by their finite-difference relations. As a result, a system of linear algebraic equations arises for unknown approximate values of the biharmonic function, from which they are uniquely found. The disadvantage of this method is that the above system is not always easy to solve. In addition, we get the solution not at any point of the circle, but only at the nodes of the grid. For real calculations and numerical analysis of solutions to applied problems, the authors have constructed its unified analytical approximate representation on the basis of the known exact solution of the boundary value problem while using logarithms. The approximate formula has a simple form and can be easily implemented numerically. Uniform error estimates make it possible to perform calculations with a given accuracy. All coefficients of the quadrature formula for the Poisson integral are non-negative, which greatly simplifies the study of the approximate solution. An analysis of the quadrature sum for stability is carried out. An example of solving a boundary value problem is considered.

This thesis: (a) presents the solution of several boundary value problems (BVPs) for the Laplace and the modified Helmholtz equations in the interior of an equilateral triangle; (b) presents the solution of the heat equation in the interior of an equilateral triangle; (c) computes the eigenvalues and eigenfunctions of the Laplace operator in the interior of an equilateral triangle for a variety of boundary conditions; (d) discusses the solution of several BVPs for the non-linear Schrodinger equation on the half line. In 1967 the Inverse Scattering Transform method was introduced; this method can be used for the solution of the initial value problem of certain integrable equations including the celebrated Korteweg-de Vries and nonlinear Schrodinger equations. The extension of this method from initial value problems to BVPs was achieved by Fokas in 1997, when a unified method for solving BVPs for both integrable nonlinear PDEs, as well as linear PDEs was introduced. This thesis applies “the Fokas method” to obtain the results mentioned earlier. For linear PDEs, the new method yields a novel integral representation of the solution in the spectral (transform) space; this representation is not yet effective because it contains certain unknown boundary values. However, the new method also yields a relation, known as “the global relation”, which couples the unknown boundary values and the given boundary conditions. By manipulating the global relation and the integral representation, it is possible to eliminate the unknown boundary values and hence to obtain an effective solution involving only the given boundary conditions. This approach is used to solve several BVPs for elliptic equations in two dimensions, as well as the heat equation in the interior of an equilateral triangle. The implementation of this approach: (a) provides an alternative way for obtaining classical solutions; (b) for problems that can be solved by classical methods, it yields vii novel alternative integral representations which have both analytical and computational advantages over the classical solutions; (c) yields solutions of BVPs that apparently cannot be solved by classical methods. In addition, a novel analysis of the global relation for the Helmholtz equation provides a method for computing the eigenvalues and the eigenfunctions of the Laplace operator in the interior of an equilateral triangle for a variety of boundary conditions. Finally, for the nonlinear Schrodinger on the half line, although the global relation is in general rather complicated, it is still possible to obtain explicit results for certain boundary conditions, known as “linearizable boundary conditions”. Several such explicit results are obtained and their significance regarding the asymptotic behavior of the solution is discussed.

We deal with statistical modeling algorithms for the numerical solution of the first boundary value problem for the heat equation. Unbiased estimators of the solution of a boundary value problem are built on the trajectories of random walks. We consider a random walk on the boundary and a random walk on the cylinders inside the region in which the boundary problem must be solved. The results of computational experiments and some applications are presented. The complexity of algorithms are estimated numerically.

The main goal of this study is to find the solution of initial boundary value problem for the one-dimensional time and space-fractional diffusion equation which is a very intriguing topic for many researchers. With the aim of newly defined inner product, which is the main contribution of this study, the analytic solution of the boundary value problem is obtained. The time and space-fractional derivatives are defined in the Caputo sense which is more suitable than Riemann-Liouville sense. We apply the separation of variables method to reduce the problem to two separate fractional ODEs. The generalized solution is constructed/formed in the form of a Fourier series with respect to the eigenfunctions of a certain eigenvalue problem. In order to obtain the coefficients of the Fourier series for the solution, we define a new inner product which is the key point of study.

The theorem of the existence and uniqueness of the solution of the boundary value problem for the equation in partial derivatives of the fourth order with variable coefficients containing the product of the mixed parabolic-hyperbolic operator and the differential operator of the oscillation string with discontinuous conditions of gluing in the pentagon to the plane is proved. By the method of reducing the order of equations, the solvability of the boundary value problem is reduced to the solution of the Tricomi problem for the mixed parabola-hyperbolic equation with variable coefficients and discontinuous gluing conditions. Solving this problem is reduced to the solution of Fredholm’s integral equation of the second order relative to the trace of the derivative function on y along the line of variation of the equation type. In the hyperbolic part of the domain, the representation of the solution of the problem for the hyperbolic equation with the smallest terms was obtained by using the Riemann function method. In the parabolic part of the domain, the solution of the first boundary value problem for the parabolic equation with the smallest terms is obtained by the method of successive approximations and the Green’s function. As a result, the solution of the problem is realized by the method of solving the Gursa problem and the first boundary value problem for the equation of string oscillation.

A numerical solution of the initial boundary value problem for the heat equation has a great significance as regards the number of applications in engineering sciences. This problem arises also at the solution of the inverse boundary value problems in thermal tomography. The approximate solution can be found with the boundary integral equations method which may be used in the diverse variants. We combine the Cayley transform and the boundary integral equations for the numerical solution of the interior initial boundary value problem for the heat equation.

The nonlinear parabolic equations describing motion of incompressible media are investigated. The rheological equations of most general type are considered. The deviator of the stress tensor is expressed as a nonlinear continuous positive definite operator applied to the rate of strain tensor. The global‐in‐time estimate of solution of initial boundary value problem is obtained. This estimate is valid for systems of equations of any non‐Newtonian fluid. Solvability of initial boundary value problems for such equations is proved under some additional hypothesis. The application of this theory makes it possible to prove the existence of global‐in‐time solutions of two‐dimensional initial boundary value problems for generalized linear viscoelastic liquids, that is, for liquids with linear integral rheological equation, and for third‐grade liquids.

The article deals with a boundary value problem in a rectangular area for a third-order degenerate equation with minor terms. The study of such equations is caused by both a theoretical and applied interest (known as VT (viscous transonic) – the equation can be found in gas dynamics). Imposing some restrictions on the coefficients of lower derivatives and using the method of energy integrals, the unique solvability of the problem is demonstrated. The solution of the problem is sought by separation of variables (Fourier method), thus two one-dimensional boundary value problems for ordinary differential equations are obtained. According to the variable y we have the problem on eigenvalues and eigenfunctions for a second-order degenerate equation. The eigenvalues and eigenfunctions are found. Eigenfunctions are the first-order Bessel functions. In order to obtain some necessary estimates the spectral problem reduces to an integral equation by constructing the Green's function. Hereafter, Bessel inequality is used. The possibility of expansion of boundary functions in the system of eigenfunctions is also shown. In order to obtain the necessary a priori estimates for the solution of one-dimensional boundary value problem with respect to the variable x and its derivatives, the problem reduces to a second-order Fredholm integral equation, with the help of Green's function. The estimates of Green's function and its derivatives are obtained. Fredholm equation is solved by the method of successive approximations, and the necessary estimates for this solution and its derivatives are obtained. The formal solution of the boundary value problem is obtained in the form of an infinite series in eigenfunctions. In order to prove the uniform convergence of the last series composed of the partial derivatives, first using the Cauchy–Bunyakovsky inequality, the series consisting of two variables is decomposed into two one-dimensional series, and then all of the obtained estimates mentioned above and estimates for the Fourier coefficients are used.

This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos–Leroux–Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type: an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L1–error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O(e1/2) in L1–norm; otherwise, as in the initial value problem, the L1–error bound is O(e| ln e|).

We consider the general system of n first order linear ordinary differential equations y'(t)=A(t)y(t)+g(t), a<t< b, subject to "boundary" conditions, or rather linear constraints, of the form Σ^(N)_(ν=1) B_(ν)y(τ_ν)=β Here y(t), g(t) and II are n-vectors and A(t), Bx,..., BN are n × n matrices. The N distinct points {τ_ν} lie in [a,b] and we only require N ≧ 1. Thus as special cases initial value problems, N=1, are included as well as the general 2-point boundary value problem, N=2, with τ_1=a, τ_2=b. (More general linear constraints are also studied, see (5.1) and (5.17).)

Integral equations are equations in which the unknown function appears inside a definite integral. They are closely related to differential equations. Initial value problems and boundary value problems for ordinary and partial differential equations can often be written as integral equations (see [7] for an introduction to the technique), and some integral equations can be written as initial or boundary value problems for differential equations. Problems that can be cast in both forms are generally more familiar as differential equations, owing to the larger collection of analytical procedures for solving differential equations. Many applications are best modeled with integral equations, but most of these problems require a lengthy derivation. A relatively simple example is the model for population dynamics, with birth and death rates that depend on age. This model was first formulated in 1922 by Alfred J. Lotka [8], who is best known for the LotkaVolterra predator-prey population model. A similar model has been used to model the spread of the HIV virus among IV drug users [4]. A simple age-dependent population model will be developed at the end of this article. Integral equations are also important in the theory and numerical analysis of differential equations; this is where the mathematics student is most likely to encounter them. For example, Picard's existence and uniqueness theorem for first-order initial value problems is conveniently proved using integral equations [1]; the proof is constructive and can be used to formulate a method for numerical solution of initial value problems. Systematic study of integral equations is usually undertaken as part of a course in functional analysis (see [6]) or applied mathematics (see [9]). This advanced setting is required for a full appreciation of integral equation theory, but it makes the subject accessible only to the advanced student. By contrast, several of the important results in the theory of integral equations can be demonstrated using nothing more than elementary analysis, and the elaboration of this statement is the goal of the present discussion. In fact, all but one of the results presented here will be derived using nothing more than the material presented in a standard advanced calculus course. This gives less advanced students of mathematics a chance to encounter some integral equation concepts that play an important role in the theory and application of continuous mathematics.

Aim. Purpose is to find exact solutions of the boundary value problem for the biharmonic equation in an infinite 𝑛𝑛-dimensional layer with Navier boundary conditions. Methodology. The paper considers a boundary value problem for a biharmonic equation in an infinite n-dimensional layer. The paper considers a boundary value problem for a biharmonic equation in an infinite n-dimensional layer 𝑥 ∈ Rn, 0 &lt; y &lt; a with Navier boundary conditions. This problem reduces to the sequential solution of two Dirichlet problems for the Poisson equation, the explicit solutions of which were obtained earlier by the authors using the Fourier transform of generalized functions of slow growth. Results. Exact solutions of the Navier boundary value problem are obtained for a biharmonic equation whose right-hand side is a polyharmonic function in 𝑥𝑥, in particular a polynomial. In this case, the solution is also a polyharmonic function in 𝑥, in particular a polynomial. Research implications. They consist in obtaining exact solutions of the Navier boundary value problem for a biharmonic equation in an infinite 𝑛𝑛-dimensional layer.

Numerical solution of two-point boundary value problems

A harmonic equation and the Helmholtz equation are elliptic type equations and describe important physical processes (the first – stationary, the second - stationary and dynamic). Effective solutions of boundary value problems for harmonic equation (in different regions in the plane) are constructed by the methods of the theory of analytic functions of a complex variable. These methods can not be applied directly to solving problems for the Helmholtz equation. In the scientific literature, solutions of boundary value problems for this equation are known only in certain areas that are represented by cumbersome formulas. In the paper, using the solution of the Helmholtz equation in a circle through the functions (not analytical) of complex variable and conformal mapping of a given area on the circle, a general approach to building a solution of the corresponding boundary value problem is formulated. An important prerequisite for presenting this solution as functional series is finding the solution of harmonic equation in a given region that satisfies the given boundary conditions and an analytic function in this region respectively. The solutions of the Helmholtz equation in the plane with an elliptic hole and half-plane are constructed. For effective formulation of boundary value prob­lems and finding analytic functions in these areas, systems of basic functions in the corresponding spaces of analytic functions are found.

The problem of solving the initial value problem of linear differential algebra system and the analytical solution of two-point boundary value problem is studied. In three cases, the expressions of the initial value problem and the analytical solution of the boundary value problem are given respectively. Example illustrates the good results of the proposed method.