A new approach using optimization technique for constructing low-dimensional dynamical systems of nonlinear partial differential equations (PDEs) is presented. After the spatial basis functions of the nonlinear PDEs are chosen, spatial basis functions expansions combined with weighted residual methods are used for time/space separation and truncation to obtain a high-dimensional dynamical syste...

The selection of the spatial basis functions is very important for model reduction of the nonlinear partial differential equations (PDEs) under time/space separation framework, which will significantly affect the accuracy and efficiency of the modeling. Using the spatial basis functions expansions and the Galerkin method, the finite-dimensional ordinary differential equation (ODE) systems can b...

In this paper we develop a dressing method for constructing and solving some classes of matrix quasi-linear Partial Differential Equations (PDEs) in arbitrary dimensions. This method is based on a homogeneous integral equation with a nontrivial kernel, which allows one to reduce the nonlinear PDEs to systems of non-differential (algebraic or transcendental) equations for the unknown fields. In ...

Within the nonlinear theory of generalized functions introduced earlier by the author a number of existence and regularity results have been obtained. One of them has been the first global version of the Cauchy-Kovalevskaia theorem, which proves the existence of generalized solutions on the whole of the domain of analyticity of arbitrary analytic nonlinear PDEs. These generalized solutions are ...

High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonli...

COURSE DESCRIPTION The course is an introduction to the study of partial differential equations (PDEs) using functional analysis and energy methods. Questions of existence, uniqueness and regularity for weak solutions to linear elliptic and parabolic PDEs will be emphasized. Various nonlinear PDEs will also be studied, using a variety of different approaches, like variational and monotonicity m...

Travelling wave solutions are important in nonlinear science. These solutions describe phenomena such as vibrations, solitons and propagation with finite speed. In recent years, direct search for exact solutions of nonlinear partial differential equations (PDEs) has become more and more attractive, partly due to the availability of computer systems like Maple or Mathematica, which allow to perf...

In this work, we have applied Elzaki transform and He's homotopy perturbation method to solvepartial dierential equation (PDEs) with time-fractional derivative. With help He's homotopy per-turbation, we can handle the nonlinear terms. Further, we have applied this suggested He's homotopyperturbation method in order to reformulate initial value problem. Some illustrative examples aregiven in ord...