Abstract The complex PDEs are a very important and interesting task in nonlinear quantum science. Although there have been extensive studies on the classical models, solving fractional models still has lot of shortcomings, especially for non-homogenous ones. Therefore, present study focuses two-component time-fractional NLS system, our method is to solve prolonged system derived from governed m...

This paper provides both analytical and numerical solutions of (PDEs) involving time-fractional derivatives. We implemented three powerful techniques, including the modified variational iteration technique, Adomian decomposition homotopy analysis to obtain an approximate solution for bounded space variable ν. The Laplace transformation is used in derivative operator enhance proposed methods’ pe...

The present investigation dealing with a hybrid technique coupled new iterative transform method, namely the Elzaki method (IETM), is employed to solve nonlinear fractional Fisher’s model. equation precise mathematical result that arose in population dynamics and genetics, specifically chemistry. Caputo Antagana-Baleanu derivatives sense are used test intricacies of this mechanism numerically. ...

Partial differential equations (PDEs) are concise and understandable representations of domain knowledge, which essential for deepening our understanding physical processes predicting future responses. However, the PDEs many real-world problems uncertain, calls PDE discovery. We propose symbolic genetic algorithm (SGA-PDE) to discover open-form directly from data without prior knowledge about e...

<p style='text-indent:20px;'>In this paper, we study the numerical approximation of a system PDEs which arises from an optimal control problem for time-fractional Fokker-Planck equation with time-dependent drift. The is composed backward Hamilton-Jacobi-Bellman and forward equation. We approximate Caputo derivatives in by means L1 schemes Hamiltonian finite differences. scheme constructed...

In this paper, we are mainly concerned with the physically interesting static Schr\"{o}dinger-Hartree-Maxwell type equations \begin{equation*} (-\Delta)^{s}u(x)=\left(\frac{1}{|x|^{\sigma}}\ast |u|^{p}\right)u^{q}(x) \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\, \mathbb{R}^{n} \end{equation*} involving higher-order or fractional Laplacians, where $n\geq1$, $0<s:=m+\frac{\alpha}{2}<\frac{n}{2}$, $m\...

This paper is devoted to designing fast solvers and efficient preconditioners for the optimal control problems (OCPs) constrained by stochastic fractional elliptic equations. We first prove existence uniqueness of solution then derive optimality system. For numerical approximation, we use Galerkin spectral methods, which apply method discretization random variables employ spectral-Galerkin appr...

I am going to start a series of lectures on nonlinear partial differential equations. Advancements made in the theory of nonlinear PDEs is one of the main achievements in XX century mathematics. Let me first make a remark on linear PDEs. In some sense one can say that there is a complete theory of linear PDEs, and perhaps the best source would be the four volumes of "The Analysis of Linear Part...

Abstract Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the network itself. PINNs nowadays used to solve PDEs, fractional integral-differential and stochastic PDEs. This novel methodology has arisen multi-task learning framework in which NN must fit observed data while reducing PDE resid...