Time-dependent Partial Differential Equations Research Articles

Numerical Analysis of the Transient and Non-Isothermal Channel Flow of a Third-Grade Fluid with Convective Cooling

We investigate the unsteady, non-isothermal, pressure driven channel flow of a third grade liquid subject to exothermic reactions. We assume temperature dependent fluid viscosity and also that the flow is subjected to convective cooling at the channel walls. The exothermic reactions are modelled via Arrhenius kinetics and the convective heat exchange with the ambient at the channel walls follows Newton’s law of cooling. The time-dependent, coupled, and nonlinear partial differential equations governing the flow and heat transfer problem are solved numerically using efficient, semi-implicit finite difference algorithms. The sensitivity of the fluid flow and heat transfer system to the various embedded parameters is explored.

Linkages of calcium-induced calcium release in a cardiomyocyte simulated by a system of seven coupled partial differential equations

Cardiac arrhythmias affect millions of adults in the U.S. each year. This irregularity in the beating of the heart is often caused by dysregulation of calcium in cardiomyocytes, the cardiac muscle cell. Cardiomyocytes function through the interplay between electrical excitation, calcium signaling, and mechanical contraction, an overall process known as calcium-induced calcium release (CICR). A system of seven coupled nonlinear time-dependent partial differential equations (PDEs), which model physiological variables in a cardiac cell, link the processes of cardiomyocytes. Through parameter studies for each component system at a time, we create a set of values for critical parameters that connect the calcium store in the sarcoplasmic reticulum, the effect of electrical excitation, and mechanical contraction in a physiologically reasonable manner. This paper shows the design process of this set of parameters and then shows the possibility to study the influence of a particular problem parameter using the overall model.

Determination of the time-dependent thermal grooving coefficient

Changes in morphology of a polycrystalline material may occur through interface motion under the action of a driving force. An important special case that is considered in this paper is the thermal grooving that occurs when a grain boundary intersects the flat surface of a recently solidified metal slab giving rise to the formation of a thin symmetric groove. In case the transient surface diffusion is the main forming mechanism this yields a fourth-order time-dependent partial differential equation with unknown time-dependent surface diffusivity. In order to determine it, the profile of the free grooving surface at a fixed location is recorded in time. The grooving boundaries are supported by self-adjoint boundary conditions. We provide sufficient conditions on the input data for which the resulting coefficient identification problem is proved to be well-posed. Furthermore, we develop a predictor–corrector finite-difference spline method for obtaining an accurate and stable numerical solution to the nonlinear coefficient identification problem. Numerical results illustrate the performance of the inversion of both exact and noisy data.

APPROXIMATING THE SOLUTIONS OF HIGHER ORDER LINEAR TIME-DEPENDENT PARTIAL DIFFERENTIAL EQUATIONS USING RADIAL BASIS FUNCTION METHOD OF LINES (RBF-MOLS)

APPROXIMATING THE SOLUTIONS OF HIGHER ORDER LINEAR TIME-DEPENDENT PARTIAL DIFFERENTIAL EQUATIONS USING RADIAL BASIS FUNCTION METHOD OF LINES (RBF-MOLS)

BoSSS: A package for multigrid extended discontinuous Galerkin methods

The software package BoSSS serves the discretization of (steady-state or time-dependent) partial differential equations with discontinuous coefficients and/or time-dependent domains by means of an eXtended Discontinuous Galerkin (XDG, resp. DG) method, aka. cut-cell DG or unfitted DG. This work consists of two major parts: First, the XDG method is introduced and a formal notation is developed, which captures important numerical details such as cell-agglomeration and a multigrid framework. In the second part, iterative solvers for extended DG systems are presented and their performance is evaluated.

Analytical and Numerical Study on Cross Diffusion Effects on Magneto-Convection of a Chemically Reacting Fluid with Suction/Injection and Convective Boundary Condition

The purpose of this paper is to investigate the Soret and Dufour effects on unsteady mixed convective boundary layer flow of a viscous fluid over a stretching surface in a porous medium in the presence of magnetic field with heat generation/absorption, chemical reaction, suction/injection and convective boundary condition. The governing time-dependent partial differential equations are transformed into non-linear ordinarydifferential equations using similarity transformations. These equations subject to the appropriate boundary conditions are solved analytically by homotopy analysis method (HAM) and numerically by Runge-Kutta fourth order method and shooting technique.The numerical solution is compared with analytical solution. The influence of the different parameters on velocity, temperature and concentration profiles are discussed in graphical as well as in tabular form. It is observed that the fluid velocity and temperature increase on increasing the buoyancy ratio parameter and heat generation/absorption parameter. Also found that the surface heat and mass transfer rates increase on increasingthe suction/injection and heat generation/absorption parameters.

Numerical solution for the Kawahara equation using local RBF-FD meshless method

When the number of nodes increases more than thousands, the arising system of global radial basis functions (RBFs) method becomes dense and ill-conditioned. To solve this difficulty, local RBFs generated finite difference method (RBF-FD) were introduced. RBF-FD method is based on local stencil nodes and so it has a sparsity system. The main goal in this work is to develop the RBF-FD method in order to obtain numerical solution for the Kawahara equation as a time dependent partial differential equation that appears in the shallow water and acoustic waves in plasma. For this purpose, we have discretized the temporal and spatial derivatives with a finite difference, θ-weighted, and RBF-FD methods. Then by applying the collocation technique at the grid nodes, a system of linear equations is obtained which gives the numerical solution of the Kawahara equation. The stability analysis is given. The efficiency and accuracy of the proposed approach are tested by four examples. In addition, a comparison between proposed method and the methods, RBFs, differential quadrature (DQM), cosine expansion based differential quadrature (CDQ) and modified cubic B-Spline differential quadrature (MCBC-DQM) is shown.

An optimal-order error estimate of the lowest-order ELLAM-MFEM approximation to miscible displacement in three space dimensions

An optimal-order error estimate for the lowest-order ELLAM-MFEM time-stepping procedure for the coupled system of time-dependent nonlinear partial differential equations modeling miscible displacement that moves through the three-dimensional subsurface porous media is proved, under a practically reasonable time-step constraint of Δt=O(h). In the procedure, the Eulerian–Lagrangian localized adjoint method (ELLAM) and the mixed finite element method (MFEM) are used to solve the transport equation and the pressure equation, respectively. This mathematically justifies the numerical advantages of the ELLAM-MFEM solution procedure.

A meshless technique based on generalized moving least squares combined with the second-order semi-implicit backward differential formula for numerically solving time-dependent phase field models on the spheres

In the current research paper, the generalized moving least squares technique is considered to approximate the spatial variables of two time-dependent phase field partial differential equations on the spheres in Cartesian coordinate. This is known as a direct approximation (it is the standard technique for generalized finite difference scheme [69,77]), and it can be applied for scattered points on each local sub-domain. The main advantage of this approach is to approximate the Laplace-Beltrami operator on the spheres using different types of distribution points simply, in which the studied mathematical models are involved. In fact, this scheme permits us to solve a given partial differential equation on the sphere directly without changing the original problem to a problem on a narrow band domain with pseudo–Neumann boundary conditions. A second-order semi-implicit backward differential formula (by adding a stabilized term to the chemical potential that is the second-order Douglas-Dupont-type regularization) is applied to approximate the temporal variable. We show that the time discretization considered here guarantees the mass conservation and energy stability. Besides, the convergence analysis of the proposed time discretization is given. The resulting fully discrete scheme of each partial differential equation is a linear system of algebraic equations per time step that is solved via an iterative method, namely biconjugate gradient stabilized algorithm. Some numerical experiments are presented to simulate the phase field Cahn-Hilliard, nonlocal Cahn-Hilliard (for diblock copolymers as microphase separation patterns) and crystal equations on the two-dimensional spheres.

Simulation-based Anomaly Detection and Damage Localization: An application to Structural Health Monitoring

We propose a simulation-based decision strategy for the proactive maintenance of complex structures with a particular application to structural health monitoring (SHM). The strategy is based on a data-driven approach which exploits an offline–online decomposition. A synthetic dataset is constructed offline by solving a parametric time-dependent partial differential equation for multiple input parameters, sampled from their probability distributions of natural variation. The collected time-signals, extracted at sensor locations, are used to train classifiers at such sensor locations, thus constructing multiple databases of healthy configurations. These datasets are then used to train one class Support Vector Machines (OC-SVMs) to detect anomalies. During the online stage, a new measurement, possibly obtained from a damaged configuration, is evaluated using the classifiers. Information on damage is provided in a hierarchical manner: first, using a binary feedback, the entire structure response is either classified as inlier (healthy) or outlier (damaged). Then, for the outliers, we exploit the outputs of multiple classifiers to retrieve information both on the severity and the spatial location of the damages. Because of the large number of signals needed to construct the datasets offline, a model order reduction strategy is implemented to reduce the computational burden. We apply this strategy to both 2D and 3D problems to mimic the vibrational behavior of complex structures under the effect of an active source and show the effectiveness of the approach for detecting and localizing cracks.

Data-driven deep learning of partial differential equations in modal space

We present a framework for recovering/approximating unknown time-dependent partial differential equation (PDE) using its solution data. Instead of identifying the terms in the underlying PDE, we seek to approximate the evolution operator of the underlying PDE numerically. The evolution operator of the PDE, defined in infinite-dimensional space, maps the solution from a current time to a future time and completely characterizes the solution evolution of the underlying unknown PDE. Our recovery strategy relies on approximation of the evolution operator in a properly defined modal space, i.e., generalized Fourier space, in order to reduce the problem to finite dimensions. The finite dimensional approximation is then accomplished by training a deep neural network structure, which is based on residual network (ResNet), using the given data. Error analysis is provided to illustrate the predictive accuracy of the proposed method. A set of examples of different types of PDEs, including inviscid Burgers' equation that develops discontinuity in its solution, are presented to demonstrate the effectiveness of the proposed method.

Fast Fourier Solvers for the Tensor Product High-Order FEM for a Poisson Type Equation

Logarithmically optimal in theory and fast in practice, direct algorithms for implementing a tensor product finite element method (FEM) based on tensor products of 1D high-order FEM spaces on multi-dimensional rectangular parallelepipeds are proposed for solving the N-dimensional Poisson-type equation $$ - \Delta u + \alpha u = f$$ ($$N \geqslant 2$$) with Dirichlet boundary conditions. The algorithms are based on well-known Fourier approaches. The key new points are a detailed description of the eigenpairs of the 1D eigenvalue problems for the high-order FEM, as well as fast direct and inverse eigenvector expansion algorithms that simultaneously employ several versions of the fast Fourier transform. Results of numerical experiments in the 2D and 3D cases are presented. The algorithms can be used in numerous applications, in particular, to implement tensor product high-order finite element methods for various time-dependent partial differential equations, including the multidimensional heat, wave, and Schrodinger ones.

Discontinuous Galerkin approximation for excitatory-inhibitory networks with delay and refractory periods

In this study, we consider the network of noisy leaky integrate-and-fire (NNLIF) model, which governs by a second-order nonlinear time-dependent partial differential equation (PDE). This equation uses the probability density approach to describe the behavior of neurons with refractory states and the transmission delays. A numerical approximation based on the discontinuous Galerkin (DG) method is used for the spatial discretization with the analysis of stability. The strong stability-preserving explicit Runge–Kutta (SSPERK) method is performed for the temporal discretization. Finally, some test examples and numerical simulations are given to examine the behavior of the solution. The execution of the constructed scheme is measured by the quantitative comparison with the existing finite difference technique, namely weighted essentially nonoscillatory (WENO) scheme.

Numerical Analysis of Transient Heat Transfer in Radial Porous Moving Fin with Temperature Dependent Thermal Properties

In this article, a time dependent partial differential equation is used to model the nonlinear boundary value problem describing heat transfer through a radial porous moving fin with rectangular profile. The study is performed by applying a numerical solver in MATLAB (pdepe), which is a centered finite difference scheme. The thermal conductivity and fin surface emissivity are linearly dependent on temperature while the heat transfer coefficient is given by power law function of temperature. The effects of thermo-physical parameters, such as the Peclet number, surface emissivity coefficient, power index of heat transfer coefficient, convective-conductive parameter, radiative-conductive parameter and non-dimensional ambient temperature on temperature are studied.

Two-scale homogenization of abstract linear time-dependent PDEs

Many time-dependent linear partial differential equations of mathematical physics and continuum mechanics can be phrased in the form of an abstract evolutionary system defined on a Hilbert space. In this paper we discuss a general framework for homogenization (periodic and stochastic) of such systems. The method combines a unified Hilbert space approach to evolutionary systems with an operator theoretic reformulation of the well-established periodic unfolding method in homogenization. Regarding the latter, we introduce a well-structured family of unitary operators on a Hilbert space that allows to describe and analyze differential operators with rapidly oscillating (possibly random) coefficients. We illustrate the approach by establishing periodic and stochastic homogenization results for elliptic partial differential equations, Maxwell’s equations, and the wave equation.

Approximate solution of nonlinear Black–Scholes equation via a fully discretized fourth-order method

In this work, a new fourth-order finite difference (FD) approximation (for both structured and unstructured grid of nodes) is contributed and equipped with the fourth-order Runge–Kutta scheme to tackle the financial nonlinear partial differential equation (PDE) of Black–Scholes. This timedependent PDE problem is converted to a set of ordinary differential equations (ODEs). It is proved that under several criteria the procedure is time stable. Computational illustrations presented here, show that our approach is fast and accurate. The proposed technique reduces the computational cost, when more accurate results are requested.

Physics Informed Extreme Learning Machine (PIELM)–A rapid method for the numerical solution of partial differential equations

There has been rapid progress recently on the application of deep networks to the solution of partial differential equations, collectively labeled as Physics Informed Neural Networks (PINNs). In this paper, We develop Physics Informed Extreme Learning Machine (PIELM), a rapid version of PINNs which can be applied to stationary and time-dependent linear partial differential equations. We demonstrate that PIELM matches or exceeds the accuracy of PINNs on a range of problems. We also discuss the limitations of neural network-based approaches, including our PIELM, in the solution of PDEs on large domains and suggest an extension, a distributed version of our algorithm -- DPIELM. We show that DPIELM produces excellent results comparable to conventional numerical techniques in the solution of time-dependent problems. Collectively, this work contributes towards making the use of neural networks in the solution of partial differential equations in complex domains as a competitive alternative to conventional discretization techniques.

Space-time kernel based numerical method for generalized Black-Scholes equation

In approximating time-dependent partial differential equations, major error always occurs in the time derivatives as compared to the spatial derivatives. In the present work the time and the spatial derivatives are both approximated using time-space radial kernels. The proposed numerical scheme avoids the time stepping procedures and produced sparse differentiation matrices. The stability and accuracy of the proposed numerical scheme is tested for the generalized Black-Scholes equation.

On an improved computational solution for the 3D HCIR PDE in finance

Abstract The aim of this work is to tackle the three–dimensional (3D) Heston– Cox–Ingersoll–Ross (HCIR) time–dependent partial differential equation (PDE) computationally by employing a non–uniform discretization and gathering the finite difference (FD) weighting coe cients into differentiation matrices. In fact, a non–uniform discretization of the 3D computational domain is employed to achieve the second–order of accuracy for all the spatial variables. It is contributed that under what conditions the proposed procedure is stable. This stability bound is novel in literature for solving this model. Several financial experiments are worked out along with computation of the hedging quantities Delta and Gamma.

Dynamically orthogonal tensor methods for high-dimensional nonlinear PDEs

We develop new dynamically orthogonal tensor methods to approximate multivariate functions and the solution of high-dimensional time-dependent nonlinear partial differential equations (PDEs). The key idea relies on a hierarchical decomposition of the approximation space obtained by splitting the independent variables of the problem into disjoint subsets. This process, which can be conveniently visualized in terms of binary trees, yields series expansions analogous to the classical Tensor-Train and Hierarchical Tucker tensor formats. By enforcing dynamic orthogonality conditions at each level of the binary tree, we obtain coupled evolution equations for the modes spanning each subspace within the hierarchical decomposition. This allows us to effectively compute the solution to high-dimensional time-dependent nonlinear PDEs on tensor manifolds of constant rank, with no need for rank reduction methods. We also propose new algorithms for dynamic addition and removal of modes within each subspace. Numerical examples are presented and discussed for high-dimensional hyperbolic and parabolic PDEs in bounded domains.