We propose an analytically tractable class of models for the dynamics a limit order book, described through stochastic partial differential equation with multiplicative noise book centered at mid-price, along mid-price which is consistent flow dynamics. provide conditions under model admits finite-dimensional realization driven by (low-dimensional) Markov process, leading to efficient methods e...

In order to investigate the effects of non-linear springs in vibrating behavior of vehicles, the independent suspension of conventional vehicles could be modeled as a non-linear single degree of freedom system. The equation of motion for the system would be a non-linear third order ordinary differential equation, when considering the elasticity of rubber bushings in joints of shock absorber. It...

In this paper we consider analytical and numerical solutions to the Dirichlet boundary value problem for the biharmonic partial differential equation, on a disk of finite radius in the plane. The physical interpretation of these solutions is that of the harmonic oscillations of a thin, clamped plate. For the linear, fourth-order, biharmonic partial differential equation in the plane, it is well...

the numerical methods are of great importance for approximating the solutions of nonlinear ordinary or partial differential equations, especially when the nonlinear differential equation under consideration faces difficulties in obtaining its exact solution. in this latter case, we usually resort to one of the efficient numerical methods. in this paper, the chebyshev collocation method is sugge...

Symmetries play an important role in solving partial differential equations. In this paper, the determining equations for a class of nonlinear partial differential equations with arbitrary order are considered. It is shown that the determining equations for the nonclassical reduction can be obtained by requiring the compatibility between the original equation and the invariant surface condition...

in this paper, we investigate some problems which can be reduced to the goursat problem for afourth order equation. some results and theorems are given concerning the existence and uniqence for thesolution of the suggested problem.