Problems For Partial Differential Equations Research Articles

Darboux problem for proportional partial fractional differential equations

In this paper, we introduce and study a novel fractional derivative called the generalized proportional Caputo fractional derivative. Under adequate assumptions on the source term, we prove existence and uniqueness aspects of the Darboux problems involving the generalized proportional fractional operator. A numerical scheme based on finite differences approximation is also investigated, and results of convergence are established. Several numerical tests are performed in illustration.

Mathematical modelling and simulation of three phase lag bio-heat transfer model during cancer treatment

We have developed a two-dimensional mathematical model that described the study of bio-heat transfer. Our model is an initial boundary value problem of partial differential equation. The solution consists of the three step procedure- (i) transformation of problem in dimensionless form (ii) by using finite differences, the problem converted into ordinary matrix differential equation (iii) applying Legendre wavelet Galerkin method, the problem is transferred into the generalised system of Sylvester equations which are solved by applying Bartels-Stewart Algorithm of generalised inverse. We have used this method to determine the temperature profile in three different boundary conditions. The consequence of boundary conditions on temperature profile are discussed in detail. The effect of phase lag due to heat flux, phase lag due to temperature gradient and phase lag due to thermal displacement have been observed. And, we have seen that temperature profile increases when the phase lags decreases. We have also observed the effect of blood perfusion rate and metabolic heat generation in specific. Results are validated with exact results in particular case.

Solution of a Semi-Boundary-Value Problem for a First-Order Degenerate Partial Differential Equation

A first-order partial differential equation with constant irreversible coefficients in a Banach space is considered. In the particular case of a finite-dimensional space, the initial-boundary-value problem with irreversible matrix coefficients has no solution; hence, we pose Showalter-type conditions. Due to the regularity of the operator pencil, the equation splits into differential equations in subspaces and given conditions lead to initial conditions in subspaces. A solution to the problem is constructed and an example is provided.

ИССЛЕДОВАНИЕ РАЗРЕШИМОСТИ ЗАДАЧ ВОССТАНОВЛЕНИЯ ВНЕШНЕГО ВОЗДЕЙСТВИЯ В ГИПЕРБОЛИЧЕСКОМ УРАВНЕНИИ ПЕРВОГО ПОРЯДКА

We study the solvability in Sobolev spaces of the problem of recovering the coefficients of the right-hand side, or the external influence, in the first order hyperbolic differential equations. Such problems belong to the class of linear inverse problems for partial differential equations. For the problems under study,  we prove the existence and uniqueness theorems for regular solutions (having all generalized in Sobolev’s sense derivatives entering the equation).

Optimal Location of Exit Doors for Efficient Evacuation of Crowds at Gathering Places

This work deals with the optimal design for the location of the exit doors at meeting places (such as sports centers, public squares, street markets, transport stations, etc.) to guarantee a safer emergency evacuation in events of a sporting, social, entertainment or religious type. This problem is stated as an optimal control problem of nonlinear partial differential equations, where the state system is a reformulation of the Hughes model (coupling the eikonal equation for a density-weighted walking velocity of pedestrians and the continuity equation for conservation of the pedestrian density), the control is the location of the exit doors at the domain boundary (subject to several geometric constraints), and the cost function is related to the evacuation rate. We provide a full numerical algorithm for solving the problem (a finite element technique for the discretization and a gradient-free procedure for the optimization), and show several numerical results for a realistic case.

A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks

Physics-informed neural networks (PINNs) have shown to be effective tools for solving both forward and inverse problems of partial differential equations (PDEs). PINNs embed the PDEs into the loss of the neural network using automatic differentiation, and this PDE loss is evaluated at a set of scattered spatio-temporal points (called residual points). The location and distribution of these residual points are highly important to the performance of PINNs. However, in the existing studies on PINNs, only a few simple residual point sampling methods have mainly been used. Here, we present a comprehensive study of two categories of sampling for PINNs: non-adaptive uniform sampling and adaptive nonuniform sampling. We consider six uniform sampling methods, including (1) equispaced uniform grid, (2) uniformly random sampling, (3) Latin hypercube sampling, (4) Halton sequence, (5) Hammersley sequence, and (6) Sobol sequence. We also consider a resampling strategy for uniform sampling. To improve the sampling efficiency and the accuracy of PINNs, we propose two new residual-based adaptive sampling methods: residual-based adaptive distribution (RAD) and residual-based adaptive refinement with distribution (RAR-D), which dynamically improve the distribution of residual points based on the PDE residuals during training. Hence, we have considered a total of 10 different sampling methods, including six non-adaptive uniform sampling, uniform sampling with resampling, two proposed adaptive sampling, and an existing adaptive sampling. We extensively tested the performance of these sampling methods for four forward problems and two inverse problems in many setups. Our numerical results presented in this study are summarized from more than 6000 simulations of PINNs. We show that the proposed adaptive sampling methods of RAD and RAR-D significantly improve the accuracy of PINNs with fewer residual points for both forward and inverse problems. The results obtained in this study can also be used as a practical guideline in choosing sampling methods.

Numerical Investigation of a Class of Nonlinear Time-Dependent Delay PDEs Based on Gaussian Process Regression

Probabilistic machine learning and data-driven methods gradually show their high efficiency in solving the forward and inverse problems of partial differential equations (PDEs). This paper will focus on investigating the forward problem of solving time-dependent nonlinear delay PDEs with multi-delays based on multi-prior numerical Gaussian processes (MP-NGPs), which are constructed by us to solve complex PDEs that may involve fractional operators, multi-delays and different types of boundary conditions. We also quantify the uncertainty of the prediction solution by the posterior distribution of the predicted solution. The core of MP-NGPs is to discretize time firstly, then a Gaussian process regression based on multi-priors is considered at each time step to obtain the solution of the next time step, and this procedure is repeated until the last time step. Different types of boundary conditions are studied in this paper, which include Dirichlet, Neumann and mixed boundary conditions. Several numerical tests are provided to show that the methods considered in this paper work well in solving nonlinear time-dependent PDEs with delay, where delay partial differential equations, delay partial integro-differential equations and delay fractional partial differential equations are considered. Furthermore, in order to improve the accuracy of the algorithm, we construct Runge–Kutta methods under the frame of multi-prior numerical Gaussian processes. The results of the numerical experiments prove that the prediction accuracy of the algorithm is obviously improved when the Runge–Kutta methods are employed.

Discrete gradient flow approximations of high dimensional evolution partial differential equations via deep neural networks

We consider the approximation of initial/boundary value problems involving, possibly high-dimensional, dissipative evolution partial differential equations (PDEs) using a deep neural network framework. More specifically, we first propose discrete gradient flow approximations based on non-standard Dirichlet energies for problems involving essential boundary conditions posed on bounded spatial domains. The imposition of the boundary conditions is realized weakly via non-standard functionals; the latter classically arise in the construction of Galerkin-type numerical methods and are often referred to as “Nitsche-type” methods. Moreover, inspired by the seminal work of Jordan, Kinderlehrer, and Otto (JKO) Jordan et al. (1998), we consider the second class of discrete gradient flows for special classes of dissipative evolution PDE problems with non-essential boundary conditions. These JKO-type gradient flows are solved via deep neural network approximations. A key, distinct aspect of the proposed methods is that the discretization is constructed via a sequence of residual-type deep neural networks (DNN) corresponding to implicit time-stepping. As a result, a DNN represents the PDE problem solution at each time node. This approach offers several advantages in the training of each DNN. We present a series of numerical experiments which showcase the good performance of Dirichlet-type energy approximations for lower space dimensions and the excellent performance of the JKO-type energies for higher spatial dimensions.

Solving the2Dand3Dnonlinear inverse source problems of elliptic type partial differential equations by a homogenization function method

AbstractIn this article, the inverse source problems of 2D and 3D elliptic type nonlinear partial differential equations are resolved. For this purpose, a family of single‐parameter homogenization functions that automatically meet the given boundary conditions are deduced and employed as the bases to expand the solution. We solve a linear algebraic equations system which satisfies the over‐specified Neumann boundary condition to obtain the unspecified coefficients, and then the solution in the entire domain is permitted. Taking the solution into the governing equation, the unknown source function can be determined quickly. The present novel method is verified to be an accurate, effective, and robust scheme which is without solving nonlinear equations and iterations, and the additional data used are quite economical.

Remarks on the Initial and Terminal Value Problem for Time and Space Fractional Diffusion Equation

The fractional problem for partial differential equation has many applications in science and technology. The main objective of the paper is to investigate the convergence of the mild solution of the diffusion equation with time and space fractional. We consider the problem in two cases which are forward problem and inverse problem. We use new techniques to overcome some of the complex assessments.

How to lighten parametric inverse computations if the misfit is non-convex and the forward solver needs stabilization

Solving parametric inverse problems (IPs) for partial differential equations (PDEs) plays an important role in medical diagnosis, resource investigation, nondestructive testing and many other human activities. Real-world IPs formulated as misfit functional minimization tasks are frequently ill-conditioned. The origins of this ill-posedness are the misfit multimodality and insensitivity as well as the instability of the utilized numerical PDE solver. We propose a complex multi-population memetic strategy HMS combined with the Petrov–Galerkin method stabilized by the Demkowicz operator to overcome these obstacles. The paper delivers a rigorous mathematical formulation of the strategy as well as a theoretical motivation for common inverse/forward error scaling, which significantly reduces the computational cost of the strategy. The presented theory is illustrated with two examples. The first one shows the analytical construction of the Demkowicz operator. The second one is an application of HMS in solving an IP utilizing the approximate stabilization with a forward solver that uses the isogeometric residual minimization (iGRM) method.

Adaptive Event-Triggered Control of Networked Fuzzy PDE Systems Under Hybrid Cyber Attacks

This study focuses on the security control problem of networked fuzzy partial differential equation (PDE) systems under hybrid cyber attacks. First, the nonlinear parabolic PDE system is exactly represented by a Takagi–Sugeno fuzzy PDE model. Second, with taking the insecure transmission network into consideration, a more practical stochastic hybrid cyber attacks model for PDE-based networked control systems is introduced. This model contains both denial-of-service attacks and deception attacks in a unified framework. Then, based on spatially pointwise nonuniform sampled-data measurements, an adaptive event-triggered control scheme with an adjustable threshold is newly designed to counter the hybrid cyber attacks’ impact while saving communication resources. Moreover, by constructing a novel Lyapunov–Krasovskii functional and employing some inequality techniques, stability conditions with less conservatism are derived. Finally, the developed method is applied to the Fisher equation to demonstrate its viability and superiority.

Lp -approximations for solutions of parabolic differential equations on manifolds

The paper considers the Cauchy problem for a parabolic partial differential equation in a Riemannian manifold of bounded geometry. A formula is given that expresses arbitrarily accurate (in the Lp-norm) approximations to the solution of the Cauchy problem in terms of parameters - the coefficients of the equation and the initial condition. The manifold is not assumed to be compact, which creates significant technical difficulties - for example, integrals over the manifold become improper in the case when the manifold has an infinite volume. The presented approximation method is based on Chernoff theorem on approximation of operator semigroups.

Stability of the time-dependent identification problem for delay hyperbolic equations

Time-dependent and space-dependent source identification problems for partial differential and difference equations take an important place in applied sciences and engineering, and have been studied by several authors. Moreover, the delay appears in complicated systems with logical and computing devices, where certain time for information processing is needed. In the present paper, the time-dependent identification problem for delay hyperbolic equation is investigated. The theorems on the stability estimates for the solution of the time-dependent identification problem for the one dimensional delay hyperbolic differential equation are established. The proofs of these theorems are based on the Dalambert’s formula for the hyperbolic differential equation and integral inequality.

When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?

Physics-informed neural networks (PINNs) have become a popular choice for solving high-dimensional partial differential equations (PDEs) due to their excellent approximation power and generalization ability. Recently, Extended PINNs (XPINNs) based on domain decomposition methods have attracted considerable attention due to their effectiveness in modeling multiscale and multiphysics problems and their parallelization. However, theoretical understanding on their convergence and generalization properties remains unexplored. In this study, we take an initial step towards understanding how and when XPINNs outperform PINNs. Specifically, for general multi-layer PINNs and XPINNs, we first provide a prior generalization bound via the complexity of the target functions in the PDE problem, and a posterior generalization bound via the posterior matrix norms of the networks after optimization. Moreover, based on our bounds, we analyze the conditions under which XPINNs improve generalization. Concretely, our theory shows that the key building block of XPINN, namely the domain decomposition, introduces a tradeoff for generalization. On the one hand, XPINNs decompose the complex PDE solution into several simple parts, which decreases the complexity needed to learn each part and boosts generalization. On the other hand, decomposition leads to less training data being available in each subdomain, and hence such model is typically prone to overfitting and may become less generalizable. Empirically, we choose five PDEs to show when XPINNs perform better than, similar to, or worse than PINNs, hence demonstrating and justifying our new theory.

A stochastic regularized second-order iterative scheme for optimal control and inverse problems in stochastic partial differential equations.

Numerous applied models used in the study of optimal control problems, inverse problems, shape optimization, machine learning, fractional programming, neural networks, image registration and so on lead to stochastic optimization problems in Hilbert spaces. Under a suitable convexity assumption on the objective function, a necessary and sufficient optimality condition for stochastic optimization problems is a stochastic variational inequality. This article presents a new stochastic regularized second-order iterative scheme for solving a variational inequality in a stochastic environment where the primary operator is accessed by employing sampling techniques. The proposed iterative scheme, which fits within the general framework of the stochastic approximation approach, has its almost-sure convergence analysis given in a Hilbert space. We test the feasibility and the efficacy of the proposed stochastic approximation approach for a stochastic optimal control problem and a stochastic inverse problem, both associated with a second-order stochastic partial differential equation. This article is part of the theme issue 'Non-smooth variational problems and applications'.

Control of Partial Differential Equations via Physics-Informed Neural Networks

This paper addresses the numerical resolution of controllability problems for partial differential equations (PDEs) by using physics-informed neural networks. Error estimates for the generalization error for both state and control are derived from classical observability inequalities and energy estimates for the considered PDE. These error bounds, that apply to any exact controllable linear system of PDEs and in any dimension, provide a rigorous justification for the use of neural networks in this field. Preliminary numerical simulation results for three different types of PDEs are carried out to illustrate the performance of the proposed methodology.

Convergence of a spectral regularization of a time-reversed reaction-diffusion problem with high-order Sobolev-Gevrey smoothness

The present paper analyzes a spectral regularization of a time-reversed reaction-diffusion problem with globally and locally Lipschitz nonlinearities. This type of inverse and ill-posed problems arises in a variety of real-world applications concerning heat conduction and tumour source localization. In accordance with the weak solvability result for the forward problem, we focus on the inverse problem with the high-order Sobolev-Gevrey smoothness and with Sobolev measurements. As expected from the well-known results for the linear case, we prove that this nonlinear spectral regularization possesses a logarithmic rate of convergence in the high-order Sobolev norm. The proof can be done by the verification of variational source condition; this way interestingly validates such a fine strategy in the framework of inverse problems for nonlinear partial differential equations. Ultimately, we design an iteration-based version of the proposed method for a class of reaction-diffusion problems with non-degenerate nonlinearity, taking into account the conventional time discretization. The convergence of this iterative scheme is also investigated.

Solvability to an initial-periodic problem for delay partial differential equations of Sobolev type

The initial-boundary value problem for higher-order delay partial differential equations is considered. We study the existence of its solution and also propose a method for finding approximate solutions. We are established a sufficient conditions for the existence and uniqueness of the solution to the problem under consideration. Introducing new unknown functions, we reduce the considered problem to an equivalent problem consisting of a family of problems for delay differential equation with functional parameters and integral relations. An algorithm for finding an approximate solution to the problem under study is proposed and its convergence is proved. Sufficient conditions for the existence of a unique solution to an equivalent problem with parameters are established. The conditions for the unique solvability of the initial-boundary value problem for delay partial differential equations are obtained in terms of the initial data. Unique solvability to the initial-boundary value problem for higher-order delay partial differential equations is interconnected with unique solvability to the family of problems with parameters for delay differential equations.

An inverse problem of fourth-order partial differential equation with nonlocal integral condition

In this study, the time-dependent potential coefficient in a higher-order PDE with initial and boundary conditions is numerically constructed for the first time from a nonlocal integral condition. Even though the inverse identification problem investigated in this study is ill-posed, it has a unique solution. For discretizing the direct problem and finding stable and accurate solutions, we employ the Quintic B-spline (QBS) collocation and Tikhonov regularization methods, respectively. The following nonlinear minimization problem is solved using MATLAB. The collected findings demonstrate that accurate and stable solutions can be found.