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The necessary and sufficient terms of solution existence of nonlinear autonomous Noetherian boundary-value problem are found in special critical case. The characteristic feature of the set problems is impossibility of direct application of traditional research schematic representation and construction of solutions of critical boundary-value problems, which was created in works of I.G. Malkin, A.M. Samoilenko, E.A. Grebenikov, Yu.A. Ryabov and A.A. Boichuk. For the solution construction of Noetherian boundary-value problem in special critical case an iterative procedure is recommended, it is constructed according to the scheme of least-squares method. Efficiency of the offered technique is shown on the example of analysis for periodic problems for Hill equation.

In the current paper, we study a Galerkin–Petrov method for a parabolic equations of higher order in domain with a moving boundary. Asymptotic estimates for the convergence rate of approximate solutions are obtained.

We find sufficient conditions for the solvability and construction of the generalized Green’s operator for linear Noether boundary value problem for linear differential-algebraic system.

We find sufficient conditions for the solvability and construction of the generalized Green’s operator for linear Noether boundary value problem for degenerate linear differential-algebraic system with interface conditions.

Currently, different nonlinear numerical schemes of the spatial approximation are used in numerical simulation of boundary value problems for hyperbolic systems of partial differential equations (e. g. gas dynamics equations, MHD, deformable rigid body, etc.). This is due to the need to improve the order of accuracy and perform simulation of discontinuous solutions that are often occurring in such systems. The need for non-linear schemes is followed from the barrier theorem of S. K. Godunov that states the impossibility of constructing a linear scheme for monotone approximation of such equations with approximation order two or greater. One of the most accurate non-linear type schemes are ENO (essentially non oscillating) and their modifications, including WENO (weighted, essentially non oscillating) scemes. The last received the most widespread, since the same stencil width has a higher order of approximation than the ENO scheme. The benefit of ENO and WENO schemes is the ability to maintain a high-order approximation to the areas of non-monotonic solutions. The main difficulty of the analysis of such schemes comes from the fact that they themselves are nonlinear and are used to approximate the nonlinear equations. In particular, the linear stability condition was obtained earlier only for WENO5 scheme (fifth-order approximation on smooth solutions) and it is a numerical one. In this paper we consider the problem of construction and stability for WENO5, WENO7, WENO9, WENO11, and WENO13 finite volume schemes for the Hopf equation. In the first part of this article we discuss WENO methods in general, and give the explicit expressions for the coefficients of the polynomial weights and linear combinations required to build these schemes. We prove a series of assertions that can make conclusions about the order of approximation depending on the type of local solutions. Stability analysis is carried out on the basis of the principle of frozen coefficients. The cases of a smooth and discontinuous behavior of solutions in the field of linearization with frozen coefficients on the faces of the final volume and spectra of the schemes are analyzed for these cases. We prove the linear stability conditions for a variety of Runge-Kutta methods applied to WENO schemes. As a result, our research provides guidance on choosing the best possible stability parameter, which has the smallest effect on the nonlinear properties of the schemes. The convergence of the schemes is followed from the analysis.

The second part of paper is devoted to final study of three classic partial differential equations (Laplace, Diffusion and Wave) solution using simple numerical methods in terms of Cellular Automata. Specificity of this solution has been shown by different examples, which are related to the hexagonal grid. Also the next statements that are mentioned in the first part have been proved: the matter conservation law and the offensive effect of excessive hexagonal symmetry.

From the point of CA view diffusion equation is the most important. While solving of diffusion equation at the infinite time interval we can find solution of boundary value problem of Laplace equation and if we introduce vector-variable we will solve wave equation (at least, for scalar). The critical requirement for the sampling of the boundary conditions for CA-cells has been shown during the solving of problem of circular membrane vibrations with Neumann boundary conditions. CA-calculations using the simple scheme and Margolus rotary-block mechanism were compared for the quasione-dimensional problem “diffusion in the half-space”. During the solving of mixed task of circular membrane vibration with the fixed ends in a classical case it has been shown that the simultaneous application of the Crank–Nicholson method and taking into account of the second-order terms is allowed to avoid the effect of excessive hexagonal symmetry that was studied for a simple scheme.

By the example of the centrally symmetric Neumann problem a new method of spatial derivatives introducing into the postfix CA procedure, which is reflecting the time derivatives (on the base of the continuity equation) was demonstrated. The value of the constant that is related to these derivatives has been empirically found in the case of central symmetry. The low rate of convergence and accuracy that limited within the boundaries of the sample, in contrary to the formal precision of the method (4-th order), prevents the using of the CAmethods for such problems. We recommend using multigrid method. During the solving of the quasi-diffusion equations (two-dimensional CA) it was showing that the rotary-block mechanism of CA (Margolus mechanism) is more effective than simple CA.

The paper presents a numerical solution to the heat wave motion problem for a degenerate second-order nonlinear parabolic equation with a source term. The nonlinearity is conditioned by the power dependence of the heat conduction coefficient on temperature. The problem for the case of two spatial variables is considered with the boundary condition specifying the heat wave motion law. A new solution algorithm based on an expansion in radial basis functions and the boundary element method is proposed. The solution is constructed stepwise in time with finite difference time approximation. At each time step, a boundary value problem for the Poisson equation corresponding to the original equation at a fixed time is solved. The solution to this problem is constructed iteratively as the sum of a particular solution to the nonhomogeneous equation and a solution to the corresponding homogeneous equation satisfying the boundary conditions. The homogeneous equation is solved by the boundary element method. The particular solution is sought by the collocation method using inhomogeneity expansion in radial basis functions. The calculation algorithm is optimized by parallelizing the computations. The algorithm is implemented as a program written in the C++ language. The parallel computations are organized by using the OpenCL standard, and this allows one to run the same parallel code either on multi-core CPUs or on graphic CPUs. Test cases are solved to evaluate the effectiveness of the proposed solution method and the correctness of the developed computational technique. The calculation results are compared with known exact solutions, as well as with the results we obtained earlier. The accuracy of the solutions and the calculation time are estimated. The effectiveness of using various systems of radial basis functions to solve the problems under study is analyzed. The most suitable system of functions is selected. The implemented complex computational experiment shows higher calculation accuracy of the proposed new algorithm than that of the previously developed one.

Recently, to describe various mathematical models of physical processes, fractional differential calculus has been widely used. In this regard, much attention is paid to partial differential equations of fractional order, which are a generalization of partial differential equations of integer order. In this case, various settings are possible.

Loaded differential equations in the literature are called equations containing values of a solution or its derivatives on manifolds of lower dimension than the dimension of the definitional domain of the desired function. Currently, numerical methods for solving loaded partial differential equations of integer and fractional orders are widely used, since analytical solving methods for solving are impossible. A fairly effective method for solving this kind of problem is the finite difference method, or the grid method.

We studied the initial-boundary value problem in the rectangle $\overline{D}=\{(x,\,t)\colon 0\leqslant x\leqslant l,\;0\leqslant t\leqslant T\}$ for the loaded differential heat equation with composition fractional derivative of Riemann – Liouville and Caputo – Gerasimov and with boundary conditions of the first and third kind. We have gotten an a priori assessment in differential and difference interpretations. The obtained inequalities mean the uniqueness of the solution and the continuous dependence of the solution on the input data of the problem. A difference analogue of the composition fractional derivative of Riemann – Liouville and Caputo –Gerasimov order $(2-\beta )$ is obtained and a difference scheme is constructed that approximates the original problem with the order $O\left(\tau +h^{2-\beta } \right)$. The convergence of the approximate solution to the exact one is proven at a rate equal to the order of approximation of the difference scheme.

A boundary value problem for partial differential equation with nonlocal boundary relations of special type is resolved by means of a slight modification of the separation of variables method. Ordinal differential operator of the second order subject to boundary conditions of the main problem is not self-adjoint. The system of eigenfunctions generated by the operator has no basis property in L₂[0,1] space. A special system of functions is proposed to expand the solution of the boundary value problem.

We construct necessary and sufficient conditions for the existence of solution of seminonlinear matrix boundary value problem for a parametric excitation system of ordinary differential equations. The convergent iteration algorithms for the construction of the solutions of the semi-nonlinear matrix boundary value problem for a parametric excitation system differential equations in the critical case have been found. Using the convergent iteration algorithms we expand solution of seminonlinear periodical boundary value problem for a parametric excitation Riccati type equation in the neighborhood of the generating solution. Estimates for the value of residual of the solutions of the seminonlinear periodical boundary value problem for a parametric excitation Riccati type equation are found.