Numerical Solution with Non-Polynomial Cubic Spline Technique of Order Four Homogeneous Parabolic Partial Differential Equations

An innovative technique of NPCS are being used in engineering, computer sciences and natural sciences field to solve PDEs and ODEs Problems. There are many problems not having exact solution or not much stable and convergent exact solution, to solve such problem one apply different approximation, iterative and many other methods. The developed technique is one of them and implemented on some homogeneous parabolic PDEs of different dimensions and getting results will compare with exact solution and one other existing method, by tabular and graphically as well. Graphs and Mathematical result are found by using MATHEMATICA. Copyright(c) The Authors

Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning

In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are potentially free of the curse of dimensionality for many different applications and have been proven to be so in the case of some nonlinear Monte Carlo methods for nonlinear parabolic PDEs. In this paper, we review these numerical and theoretical advances. In addition to algorithms based on stochastic reformulations of the original problem, such as the multilevel Picard iteration and the deep backward stochastic differential equations method, we also discuss algorithms based on the more traditional Ritz, Galerkin, and least square formulations. We hope to demonstrate to the reader that studying PDEs as well as control and variational problems in very high dimensions might very well be among the most promising new directions in mathematics and scientific computing in the near future.

Uniformly Convergent Numerical Scheme for Singularly Perturbed Parabolic PDEs with Shift Parameters

In this article, singularly perturbed parabolic differential difference equations are considered. The solution of the equations exhibits a boundary layer on the right side of the spatial domain. The terms containing the advance and delay parameters are approximated using Taylor series approximation. The resulting singularly perturbed parabolic PDEs are solved using the Crank–Nicolson method in the temporal discretization and nonstandard finite difference method in the spatial discretization. The existence of a unique discrete solution is guaranteed using the discrete maximum principle. The uniform stability of the scheme is investigated using solution bound. The uniform convergence of the scheme is discussed and proved. The scheme converges uniformly with the order of convergence O N − 1 + Δ t 2 , where N is number of subintervals in spatial discretization and Δ t is mesh length in temporal discretization. Two test numerical examples are considered to validate the theoretical findings of the scheme.

Time Adaptivity in Model Predictive Control

The core of the Model Predictive Control (MPC) method in every step of the algorithm consists in solving a time-dependent optimization problem on the prediction horizon of the MPC algorithm, and then to apply a portion of the optimal control over the application horizon to obtain the new state. To solve this problem efficiently, we propose a time-adaptive residual based a-posteriori error control concept based on the optimality system of this optimal control problem. This approach not only delivers an adaptive time discretization of the prediction horizon, but also suggests an adaptive time discretization of the application horizon, whose length could be either adaptive or fixed. We apply this concept for systems governed by linear parabolic PDEs and present several numerical examples which demonstrate the performance and the robustness of our adaptive MPC control concept.