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For the final outcomes of the combination scheme with their fixed probability distribution, an iterative process is developed for their complete repetition-free enumeration, which is represented by an inhomogeneous Markov chain. To this aim we determe the probabilities of its initial states and the transition probabilities (iterative transitions of the process). To calculate this process, we analyze the procedure for compiling a system of equations for them and a general method for solving it (for finding the probabilities of iterative transitions when listing the outcomes of any combinatorial schemes that lead to their desired final probability distribution) with an illustration on an example of a combination scheme with an equiprobable distribution on the set of its outcomes. Based on the process thus constructed, we propose methods for modeling the outcome of the combination scheme both from the result of solving the direct numbering problem and for sequential elementwise modeling of the outcome of the scheme, as the state of a Markov chain. [ABSTRACT FROM AUTHOR]

In the present paper we study a Darboux-Goursat problem for a hyperbolic equation with a singular coefficient, where a third-type boundary condition on the non-characteristic segment is imposed. This problem is related to a four-dimensional Protter's problem for the wave equation and it has an unique generalized solution with possible singularity at a boundary point. We find an integral representation of this solution via Riemann-Hadamard function. The singularity of the generalized solution depends on orthogonality condition on the right-hand side of the equation in respect to nontrivial classical solutions of the corresponding adjoint homogeneous problem. Here we find an explicit formula for such nontrivial classical solutions. [ABSTRACT FROM AUTHOR]

We apply Pohozhaev type identity to establish nonexistence of nontrivial regular solution of a problem of mixed type. The result is obtained in both supercritical and critical case with brief discussion on the derivation of Sobolev embedding constant. Since boundary conditions are imposed on elliptic boundary and part of hyperbolic boundaries we apply Hardy-Sobolev inequality to ensure that integrals on the boundaries where no boundary conditions are imposed have appropriate sign. [ABSTRACT FROM AUTHOR]

The main purpose of this article is to present an approximation method for solving two-dimensional Hammerstein- Volterra fuzzy integral equations with partial integrals (2D-HVFIEPI). The proposed approach is based on homotopy analysis method (HAM). We transform this equation to Hammerstein-Volterra fuzzy integral equations with partial integrals to crisp case. The convergence analysis for the proposed methods is proved and error estimate between the exact and the approximate solution is find. We give numerical application to show efficiency and accuracy of the method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new method for solving fuzzy partial differential equation (FPDE) using double fuzzy Natural transform (DFNT). We provide fundamental results of the double fuzzy Natural transform for fuzzy partial derivatives of the m-th order. By using them we construct the solution of fuzzy partial differential equation. We show that DFNT method is a simple and reliable approach for solving such equation analytically. Finally, we give illustrative example to show the ability of the proposed method. [ABSTRACT FROM AUTHOR]

In the present work we obtain existence of positive solutions for a class of third order equations with parameter depen- dence. The exact expression of the related Green's function is given and some of its properties are proven. Although the Green's function is not positive, the obtained solutions are still positive. The results are based on the classical Krasosel'skii's fixed point theorem in cones. [ABSTRACT FROM AUTHOR]

The existence of infinitely many weak solutions for the fourth-order differential equation (φ p ( u ″ (t) )) ″ ​ ′ + w (φ p ( u ′ (t) )) ′ + a (t) φ p (u (t)) + b (t) φ q (u (t)) − c (t) φ r (u (t)) = 0 , for t ∈ [0, T] is studied. Here φ p (t) = | t | p − 2 t , 1 < p < q < r , w is a constant, and a, b, c are positive continuous functions on the interval [0, T]. The equation is considered with Dirichlet boundary conditions u (0) = u (T) = u ′ (0) = u ′ (T) = 0. An existence result is proved using the variational method and the symmetric mountain-pass theorem. Some earlier results of [L. Saavedra & S. Tersian, Nonlinear Analysis: Real World Applications 34, (2017), 507–519] and [S. Tersian, Proceedings of 48th Spring Conference of the Union of Mathematicians in Bulgaria, 2019, 27–34] are extended. [ABSTRACT FROM AUTHOR]

In the present paper we show that the ground state standing waves eiωt u(x) for the generalized Klein–Gordon–Hartree equation ψtt−Δψ+ψ−(|x|−(n−α)*|ψ|p)|ψ|p−2ψ=0,α∈(0,n),n≥3 in ℝn, are orbitally unstable in the domain of all frequencies 0 ≤ ω < 1 for the supercritical case and for frequencies, strictly less than a critical frequency ωc=√p−1/n+α+1−p(n−1), in the subcritical case. [ABSTRACT FROM AUTHOR]

We prove new type of conservation laws and unconditional well-posedness in the energy space for the solution to the Ginzburg-Landau equation with complex pure power nonlinearity. The main tools used are an approach unfolding the conservation of mass and energy as basically independent from the regularity of the solution in combination with classical Strichartz estimates and a new extended Strichartz estimates. [ABSTRACT FROM AUTHOR]