Existence of solutions for the semilinear abstract Cauchy problem of fractional order

In this paper we establish the existence of solutions for the nonlinear abstract Cauchy problem of order α ∈ (1, 2), where the fractional derivative is considered in the sense of Caputo. The autonomous and nonautonomous cases are studied. We assume the existence of an α-resolvent family for the homogeneous linear problem. By using this α-resolvent family and appropriate conditions on the forcing function, we study the existence of classical solutions of the nonhomogeneus semilinear problem. The non-autonomous problem is discussed as a perturbation of the autonomous case. We establish a variation of the constants formula for the nonautonomous and nonhomogeneous equation.

How to solve model equation of hierarchical diffusion using some matrix algebra

Problems of a random walk on a binary tree have been reformulated on any homogeneous tree. Cauchy problem of the random walk for homogeneous and nonhomogeneous equation having a Parisi matrix as a coefficient is formulated and solved with help of a special commutative ring of matrices. The ring containing the Parisi matrix is constructed. The method can be generalized on multidimensional case, for differential equations in non-Archimedean time, and for difference equations.

Dynamic Responses for Rails and Panels of Electromagnetic Rail Launcher in the Electromagnetic Forces

The mechanical model for rails and panels of rectangle electromagnetic rail launcher under working condition were simplified as dynamic responses of bi-layer elastic foundation beams. The electromagnetic forces make rails and panels vibrate, which directly affects the launching accuracy. The mechanical equilibrium equation of bi-layer elastic foundations beams was established in this article, and the distributed loads related to armature movement were expressed by Heaviside function. The modes of vibration were given by the boundary conditions. Based on the above results, the dynamic displacement and stress analytical solutions for rails and panels were given by solving homogeneous and nonhomogeneous equation. Dynamic responses of rails and panels were calculated and analyzed against the due kinetic and structure parameters, and the analytical solutions were verified by the comparison with ANSYS results. The research results can be some reference to strength and rigidity design for rails and panels of rectangle electromagnetic rail launcher.

Semiconservative Systems of Integral Equations with Two Kernels

The solvability and the properties of solutions of nonhomogeneous and homogeneous vector integral equation , where , are matrix valued functions, , with nonnegative integrable elements, are considered in one semiconservative (singular) case, where the matrix is stochastic one and the matrix is substochastic one. It is shown that in certain conditions the nonhomogeneous equation simultaneously with the corresponding homogeneous one possesses positive solutions.