Real-time full-field estimation of transient responses in time-dependent partial differential equations using causal physics-informed neural networks with sparse measurements

CUDA programs for solving the time-dependent dipolar Gross–Pitaevskii equation in an anisotropic trap

This paper deals with an analytical solution of an initial value system of time dependent linear and nonlinear partial differential equations by implementing reduced differential transform (RDT) method. The effectiveness and the convergence of RDT method are tested by means of five test problems, which indicates the validity and great potential of the reduced differential transform method for solving system of partial differential equations.

Bivariate spline solution of time dependent nonlinear PDE for a population density over irregular domains

We introduce the concept of numerical Gaussian processes, which we define as Gaussian processes with covariance functions resulting from temporal discretization of time-dependent partial differential equations. Numerical Gaussian processes, by construction, are designed to deal with cases where (a) all we observe are noisy data on black-box initial conditions, and (b) we are interested in quantifying the uncertainty associated with such noisy data in our solutions to time-dependent partial differential equations. Our method circumvents the need for spatial discretization of the differential operators by proper placement of Gaussian process priors. This is an attempt to construct structured and data-efficient learning machines, which are explicitly informed by the underlying physics that possibly generated the observed data. The effectiveness of the proposed approach is demonstrated through several benchmark problems involving linear and nonlinear time-dependent operators. In all examples, we are able to r...

Pseudospectral method and Darvishi’s preconditioning for solving system of time dependent partial differential equations

OpenMP Fortran and C programs for solving the time-dependent Gross–Pitaevskii equation in an anisotropic trap

: The authors discuss a moving finite element method for solving vector systems of time dependent partial differential equations in one space dimension. The mesh is moved so as to equidistribute the spatial component of the discretization error in H1. They present a method of estimating this error by using p-hierarchic finite elements. The error estimate is also used in an adaptive mesh refinement procedure to give an algorithm that combines mesh movement and refinement. The authors discretize the partial differential equations in space using a Galerkin procedure with piecewise linear elements to approximate the solution and quadratic elements to estimate the error. A system of ordinary differential equations for mesh velocities are used to control element motions. The authors use existing software for stiff ordinary differential equations for the temporal integration of the solution, the error estimate, and the mesh motion. Computational results using a code based on this method are presented for several examples.

In this study, we analyze the performance of a numerical scheme based on 3-scale Haar wavelets for dynamic Euler-Bernoulli equation, which is a fourth order time dependent partial differential equation. This type of equations governs the behaviour of a vibrating beam and have many applications in elasticity. For its solution, we first rewrite the fourth order time dependent partial differential equation as a system of partial differential equations by introducing a new variable, and then use finite difference approximations to discretize in time, as well as 3-scale Haar wavelets to discretize in space. By doing so, we obtain a system of algebraic equations whose solution gives wavelet coefficients for constructing the numerical solution of the partial differential equation. To test the accuracy and reliability of the numerical scheme based on 3-scale Haar wavelets, we apply it to five test problems including variable and constant coefficient, as well as homogeneous and non-homogeneous partial differential equations. The obtained results are compared wherever possible with those from previous studies. Numerical results are tabulated and depicted graphically. In the applications of the proposed method, we achieve high accuracy even with small number of collocation points.

In this paper, we solve system of time dependent partial differential equations (PDEs) by using pseudospectral method. Firstly, theory of application of spectral collocation method on system of time dependent partial differential equations presented. This method yields a system of ordinary differential equations (ODEs). Secondly, we use forth-order Runge-Kutta formula for the numerical integration of the system of ODE. we consider some examples to illustrate the performance of the method described.

In this paper, we develop a new discontinuous Galerkin (DG) finite element method for solving time dependent partial differential equations (PDEs) with higher order spatial derivatives. Unlike the traditional local discontinuous Galerkin (LDG) method, the method in this paper can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system. Stability is ensured by a careful choice of interface numerical fluxes. The method can be designed for quite general nonlinear PDEs and we prove stability and give error estimates for a few representative classes of PDEs up to fifth order. Numerical examples show that our scheme attains the optimal $(k+1)$-th order of accuracy when using piecewise $k$-th degree polynomials, under the condition that $k+1$ is greater than or equal to the order of the equation.

A moving collocation method for solving time dependent partial differential equations

Reprint of: Residual equilibrium schemes for time dependent partial differential equations

Residual equilibrium schemes for time dependent partial differential equations

The MOL solution of time dependent partial differential equations

In this study, viscoelastic damped dynamic behaviors of laminated composite shells (LCS) under different dynamic loads were investigated. In order to obtain better numerical stability, the truncated series method was used in the formation of equations governing the system. While deriving the equations governing the system, the z/R terms are usually neglected, whereas only 3 and higher order terms are truncated here. The method of truncated equations has been used for the first time as suggested here to obtain viscoelastic damped behavior of dynamically loaded LCS. The governing equation of composite shells was obtained with the help of Hamilton’s principle. Afterwards, time dependent partial differential equations were obtained by applying Navier solution method to these valid equations. These equations were transformed into Laplace space in order to solve time dependent partial differential equations. The transformation of the resulting calculations from Laplace domain into the time domain was conducted with the help of Modified Durbin algorithm. In addition, one of the goals of this research is to highlight the importance of including the (1+z/R) part in the equations to account for the curvature effect of the shells. The addition of these effects and truncated series method to the equations and the investigation of their effects are the originalities of the present study. The present work obtained values were compared with the results obtained with Newmark’s approach and ANSYS finite element methods. The numerical results showed that the proposed approach is a highly effective and efficient solution that can be easily applied to laminated viscoelastic shell problems.