Locally one-dimensional difference scheme for a fractional tracer transport equation

A locally one-dimensional scheme for a fractional tracer transport equation of order is considered. An a priori estimate is obtained for the solution of the scheme, and its convergence is proved in the uniform metric.     

Locally one-dimensional difference scheme for the convective diffusion equation

We consider the mixed boundary-value problem for the nonstationary convective diffusion equation in a rectangular region. The summation approximation method is applied to construct a locally homogeneous difference scheme with O(t ^1/2 + h^3/2) rate of convergence in the L₂ grid metric.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 74, pp. 64–67, 1992;     

Locally one-dimensional scheme for a loaded heat equation with Robin boundary conditions

The third boundary value problem for a loaded heat equation in a p-dimensional parallelepiped is considered. An a priori estimate for the solution to a locally one-dimensional scheme is derived, and the convergence of the scheme is proved.     

Locally one-dimensional difference schemes for the fractional order diffusion equation

Locally-one-dimensional difference schemes for the fractional diffusion equation in multidimensional domains are considered. Stability and convergence of locally one-dimensional schemes for this equation are proved.     

Locally one-dimensional scheme for fractional diffusion equations with robin boundary conditions

For a fractional diffusion equation with Robin boundary conditions, locally one-dimensional difference schemes are considered and their stability and convergence are proved.     

A variational difference scheme for the one-dimensional diffusion equation

In this paper we construct a homogeneous variational difference scheme for the diffusion equation assuming its coefficients to be bounded and measurable; the order of convergence of the scheme is O(h²). We consider the boundary value problem (1) $$\frac{d}{{dx}}\left( {K(x)\frac{{du}}{{dx}}} \right) - g(x)u = - \frac{{dF}}{{dx}},0< x< X$$ subject to the boundary conditions (2) $$u(0) = a,u(X) = b$$ .     

Nonlocal boundary problems for a third-order one-dimensional nonlinear pseudoparabolic equation

We consider initial boundary value problems for a third-order nonlinear pseudoparabolic equation with one space dimension. The boundary condition is given by an integral; the function involved could exhibit singularities, which distinguishes this nonlocal condition from its Dirichlet or Neumann counterparts. By means of appropriate elliptic estimates we are able to seek solutions not only in the weighted spaces but also in the usual Sobolev spaces. The procedure is carried out in a unified way. Our results characterize a regularity of the pseudoparabolic operator that is different from that of the parabolic operator.     

A nonlinear nonlocal mixed problem for a second order pseudoparabolic equation

We study a nonlocal mixed problem for a nonlinear pseudoparabolic equation, which can, for example, model the heat conduction involving a certain thermodynamic temperature and a conductive temperature. We prove the existence, uniqueness and continuous dependence of a strong solution of the posed problem. We first establish for the associated linear problem a priori estimate and prove that the range of the operator generated by the considered problem is dense. The technique of deriving the a priori estimate is based on constructing a suitable multiplicator. From the resulted energy estimate, it is possible to establish the solvability of the linear problem. Then, by applying an iterative process based on the obtained results for the linear problem, we establish the existence, uniqueness and continuous dependence of the weak solution of the nonlinear problem.     

A moment problem for one-dimensional nonlinear pseudoparabolic equation

We are concerned with a moment problem for a nonlinear pseudoparabolic equation with one space dimension on an interval. The boundary conditions are imposed in terms of the zero-order moment and the first-order moment. Based on an elliptic estimate and an iteration method we established the well-posedness of solutions in the usual Sobolev space. We are able to get regularity of the solution so that both solution and its derivative with respect to the time variable belong to the same Sobolev space with respect to the space variable. This feature is different from problems with parabolic equations, where the regularity order of solution is higher than that of the time derivative with respect to the space variable. Previous results reflected only this parabolic nature for the pseudoparabolic equation.     

Inverse problem for a pseudoparabolic equation with integral overdetermination conditions

We consider inverse problems of finding an unknown coefficient in the leading term of a linear pseudoparabolic equation of filtration type on the basis of integral data over the entire boundary or its part under the assumption that the unknown coefficient depends on time. We derive conditions for the time-global solvability and uniqueness of the solution of the inverse problem.     

Finite-difference method for a nonlocal boundary value problem for a third-order pseudoparabolic equation

We consider a nonlocal boundary value problem for a third-order pseudoparabolic equation with variable coefficients. For its solution, in the differential and finite-difference settings, we derive a priori estimates that imply the stability of the solution with respect to the initial data and the right-hand side on a layer as well as the convergence of the solution of the difference problem to that of the differential problem.     

On the numerical solution of a nonlocal boundary value problem for a degenerating pseudoparabolic equation

We study a nonlocal boundary value problem for a degenerating pseudoparabolic third-order equation of the general form. For the solution of the problem, we obtain a priori estimates in differential and difference form, which imply the stability of the solution with respect to the initial data and right-hand side on a layer as well as the convergence of the solution of the difference problem to the solution of the differential problem.     

Finite difference scheme for time-space fractional diffusion equation with fractional boundary conditions

In this paper, the finite difference scheme is developed for the time-space fractional diffusion equation with Dirichlet and fractional boundary conditions. The time and space fractional derivatives are considered in the senses of Caputo and Riemann-Liouville, respectively. The stability and convergence of the proposed numerical scheme are strictly proved, and the convergence order is O(τ^2−α+h²). Numerical experiments are performed to confirm the accuracy and efficiency of our scheme.     

A problem with nonlocal initial data for one-dimensional hyperbolic equation

In this paper we consider a boundary-value problem for one-dimensional hyperbolic equation with nonlocal initial data in integral form. We prove the existence and uniqueness of the generalized solution.     

On a nonlocal problem for nonlinear pseudoparabolic equations

For a nonlinear pseudoparabolic equation with one space dimension we consider its initial boundary value problem on an interval. The boundary condition on the left end is of Dirichlet type, the right end condition is replaced by a nonlocal one. Because it is given by an integral, the function involved could exhibit singularities, which distinguishes this nonlocal condition from its Dirichlet counterpart. Based on an elliptic estimate and an iteration method we established the well-posedness of solutions in a weighted Sobolev space.     

On the stability of an explicit difference scheme for hyperbolic equations with nonlocal boundary conditions

We consider the stability of an explicit finite-difference scheme for a linear hyperbolic equation with nonlocal integral boundary conditions. By studying the spectrum of the transition matrix of the explicit three-layer difference scheme, we obtain a sufficient condition for stability in a special norm.     

A parabolic equation with nonlocal conditions

For a linear parabolic equation with the principal part in divergence form, a boundary-value problem with nonlocal (irregular) conditions of integral type is considered. Sufficient conditions of the unique solvability are found for the above-mentioned problem. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 17, Differential and Functional Differential Equations. Part 3, 2006.     

Solving nonlinear pseudoparabolic equations with nonlocal boundary conditions in reproducing kernel space

In this paper, a function space is constructed, in which an arbitrary function satisfies the nonlocal boundary conditions of a nonlinear pseudoparabolic equation. A very simple numerical algorithm for the approximations of the nonlinear pseudoparabolic equation with nonlocal boundary conditions based on the function space is provided. A numerical example is given to illustrate the applicability and efficiency of the algorithm.     

Stable difference scheme for a nonlinear Klein-Gordon equation

For a nonlinear Klein-Gordon equation, we obtain a stable difference scheme for large time intervals. We prove that this scheme has the sixth order of accuracy.