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We consider an initial-boundary value problem for the heat equation with an inhomogeneous initial condition and boundary conditions. One of the boundary conditions contains a mixed derivative. When solving this problem by the method of separation of variables, a spectral problem arises. A system of eigenfunctions of this spectral problem and a biorthogonally conjugate system, are constructed explicitly. Also we obtain an asymptotic formula for the eigenvalues. In this paper we formulate theorems on the properties of the system of eigenfunctions of the spectral problem and a theorem about representing the solution of the initial initial-boundary value problem in the form of a Fourier series in the system of eigenfunctions. Thus, the existence of a solution is shown if the initial condition belongs to the Holder class. However, it has been shown that the solution is not unique. We show that additional condition guarantees the uniqueness of the solution. The unique solution of this problem is also obtained in the article.

In this paper, we study the third boundary value problem for the convection-diffusion equation with a time fractional derivative and variable coefficients in a multidimensional domain. For an approximate solution in a rectangular parallelepiped of the problem set, in the same area, the third boundary value problem for a differential equation with a small parameter is considered. An a priori estimate is obtained from which it follows the convergence of the solution of the differential problem with a small parameter to the solution of the original problem for small values of the parameter. For a problem with a small parameter, a locally one-dimensional difference scheme by A. A. Samarski is constructed. Using the maximum principle for solving the difference problem, an a priori estimate is obtained in the grid norm $$C$$ , which expresses the stability of the locally one-dimensional difference scheme. The uniform convergence of the locally one-dimensional scheme is proved for $$0

In this paper, we consider the questions of the unique solvability of a boundary value problem for a third-order partial integro-differential equation with a degenerate kernel and multiple characteristics. An explicit solution of the boundary value problem is constructed. In this case, a combination of three methods was used: the method for constructing Green’s function, the method of Fourier series and the Fredholm method for the degenerate kernel.

In this paper, we study the existence, uniqueness and boundary behavior of positive boundary blow-up solutions to the quasilinear system $$\Delta_{\infty}u=a(x)u^{p}v^{q}$$ , $$\Delta_{\infty}v=b(x)u^{r}v^{s}$$ in a smooth bounded domain $$\Omega\subset R^{N}$$ , with the explosive boundary condition $$u=v=+\infty$$ on $$\partial\Omega$$ , where the operator $$\Delta_{\infty}$$ is the $$\infty$$ -Laplacian, the positive weight functions $$a(x)$$ , $$b(x)$$ are Holder continuous in $$\Omega$$ , and the exponents verify $$p$$ , $$s > 3$$ , $$q$$ , $$r>0$$ , and $$(p-3)(s-3) > qr$$ .

The purpose of this article is to demonstrate with a simple example the complexities that arise when considering boundary-value problems for differential-difference equations. In this paper we discuss in detail smoothness issues of solutions to the Dirichlet problem for linear differential-difference equation with integer shifts. After analysis of this example we consider more general statement of the problem and certain results concerning the solvability and smoothness of solutions.

In this paper, we investigate a problem on small motions of a system of bodies partially filled with ideal fluids under the action of an elastic damping device. The initial boundary value problem is reduced to the Cauchy problem for a first-order differential operator equation on a Hilbert space. Properties of the resulting operator matrices, which are coefficients of the equation, are studied. Theorems on solvability of the Cauchy problem and the initial boundary value problem are proven.

A class of quadratic irregular pencils of the second order ordinary differential operators with constant coefficients is considered in this paper. It is assumed that the boundary conditions are splitting and the characteristic equation has positive roots. Both amounts of two-fold expansions in the series in terms of eigenfunctions of such pencils and necessary and sufficient conditions for convergence of these expansions to the decomposed vector-valued function are found. For this class of irregular pencils, the question of a two-fold expansion in terms of eigenfunctions was not considered earlier.

In this paper, we investigate the global structure of positive solutions of $$\begin{cases} u''''(x)=\lambda h(x)f(u(x)), & 0 0$$ is a parameter, $$h\in C[0,1]$$ , $$f\in C[0,\infty)$$ and $$f(s)>0$$ for $$s>0$$ . We show that the problem has three positive solutions suggesting suitable conditions on the nonlinearity. Furthermore, we also establish the existence of infinitely many positive solutions. The proof is based on the bifurcation method.