The backward Euler algorithm for the multidimensional nonhomogeneous heat equation is analyzed, based on the finite element method. The existence and uniqueness of the numerical solution is investigated. Also, the convergence of the numerical solutions is studied.

We investigate the existence of solutions for the Dirichlet problem including the generalized balance of a membrane equation. We present a duality theory and variational principle for this problem. As one of the consequences of the duality we obtain some numerical results which give a measure of a duality gap between the primal and dual functional for approximate solutions.

The existence of nontrivial solutions is considered for the fractional Schrödinger-Poisson system with double quasi-linear terms: $$\left\{\begin{array}{cc}{(-\Delta )}^{s}u+V\left(x\right)u+\phi u-\frac{1}{2}u{(-\Delta )}^s{u}^2=f(x,u),\hfill & x\in {ℝ}^3,\hfill \\ {(-\Delta )}^t\phi =u^2,\hfill & x\in {ℝ}^3,\hfill \end{array}\right.$$ where ${(-\Delta )}^{\alpha }$ is the fractional Laplacian for $\alpha =s$, $t\in (0,1]$ with $s<t$ and $2t+4s>3$. Under assumptions on $V$ and $f$, we prove the existence of positive solutions and negative solutions for the above system by using perturbation method and the mountain pass theorem.

We describe qualitative behaviour of solutions of the Gross-Pitaevskii equation in 2D in terms of motion of vortices and radiation. To this end we introduce the notion of the intervortex energy. We develop a rather general adiabatic theory of motion of well separated vortices and present the method of effective action which gives a fairly straightforward justification of this theory. Finally we mention briefly two special situations where we are able to obtain rather detailed picture of the vortex...

A mathematical model of the equilibrium problem of elastic sandwich plates is established. Using the theory of inequalities of Korn's type for a general class of elliptic systems the existence and uniqueness of a variational solution is proved.

In this paper, we study the spectrum for the following eigenvalue problem with the p-biharmonic operator involving the Hardy term: ${\Delta \left(\right|\Delta u|}^{p-2}{\Delta u)=\lambda (\left|u\right|}^{p-2}u)/\left(\delta {\left(x\right)}^{2p}\right)$ in Ω, $u\in W{₀}^{2,p}\left(\Omega \right)$. By using the variational technique and the Hardy-Rellich inequality, we prove that the above problem has at least one increasing sequence of positive eigenvalues.