Studied here is the large-time behavior and eventual periodicity of solutions of initial-boundary-value problems for the BBM equation and the KdV equation, with and without a Burgers-type dissipation appended. It is shown that the total energy of a solution of these problems grows at an algebraic rate which is in fact sharp for solutions of the associated linear equations. We also establish tha...

with the condition γ+ δ+ ǫ = α+β+1 [12]. It has four singularities {0, 1, t,∞} and they are all regular. Heun’s equation is known to be a standard form of the secondorder Fuchsian differential equation with four singularities. The parameter q is not determined by the local monodromy, and is called an accessory parameter. Heun’s differential equation frequently appears in Physics, i.e. black hol...

Abstract In this paper, other types of exact solution of a first-order nonlinear ordinary differential equation is further investigated. By using the solutions of this equation, and a new general ansatz, we give some types of solutions of a class of nonlinear partial differential equations and MKdV equation. These solutions includes solitary wave solutions, singulary solitary solitary solutions...

Questions of reachability for continuous and hybrid systems can be formulated as optimal control or game theory problems, whose solution can be characterized using variants of the Hamilton-Jacobi-Bellman or Isaacs partial differential equations. The formal link between the solution to the partial differential equation and the reachability problem is usually established in the framework of visco...

Uncertain partial differential equation is a type of partial differential equation driven by Liu processes. This paper first provides two methods for solving the first order linear uncertain partial differential equation and the inverse uncertainty distribution of solution is discussed. Moreover, on the basis of the study of uncertainty theory and the propagation process of internet public opin...

In this article, we will show the complex inversion formula for the inversion of the L2-transform and also some applications of the L2, and Post Widder transforms for solving singular integral equation with trigonometric kernel. Finally, we obtained analytic solution for a partial differential equation with non-constant coefficients.

In this paper, we obtain a suitable mathematical model for the seismic response of dams. By using the shear beam model (SB model), we give a mathematical formulation that it is a partial differential equation and transform it to the Sturm-Liouville equation.

Nonstandard finite difference schemes for the Black-Scholes partial differential equation preserving the positivity property are proposed. Computationally simple schemes are derived by using a nonlocal approximation in the reaction term of the Black-Scholes equation. Unlike the standard methods, the solutions of new proposed schemes are positive and free of the spurious oscillations.

We prove a general theorem that the L2ρ(R ;R) ⊗ L2ρ(R ;R) valued solution of an infinite horizon backward doubly stochastic differential equation, if exists, gives the stationary solution of the corresponding stochastic partial differential equation. We prove the existence and uniqueness of the L2ρ(R ;R)⊗Lρ(R ;R) valued solutions for backward doubly stochastic differential equations on finite a...