Delay Partial Differential Equations Research Articles

A Robust Numerical Scheme via Grid Equidistribution for Singularly Perturbed Delay Partial Differential Equations Arising in Control Theory

A Robust Numerical Scheme via Grid Equidistribution for Singularly Perturbed Delay Partial Differential Equations Arising in Control Theory

Crank–Nicolson method for solving time-fractional singularly perturbed delay partial differential equations

Crank–Nicolson method for solving time-fractional singularly perturbed delay partial differential equations

Machine learning for a class of partial differential equations with multi-delays based on numerical Gaussian processes

Delay partial differential equations (PDEs) are widely utilized in many fields, such as climate prediction and epidemiology. But observation data in real world is often noisy and discrete. And in order to expand the applications of delay PDEs, we consider numerical Gaussian processes to solve these models. In this paper, numerical Gaussian processes for predicting the latent solution of a type of delay PDEs with multi-delays are investigated, and various delay PDEs are studied, including problems governed by variable-order fractional order operators and nonlinear operators, so as to adapt to the needs of practical applications. Numerical Gaussian processes are very good at fitting latent solution of PDEs, when all observation data is noisy and discontinuous. And the methodology can clearly quantify the uncertainty of the predicted solution. For complex boundaries controlled by ODEs, we consider mixed boundary conditions of delay PDEs in this paper. And we also apply Runge-Kutta methods to enhance the prediction accuracy of these problems. Finally, we design seven numerical examples to investigate the efficiency of NGPs and how the noisy data influences the solution of our studied problems.

Averaging principles for two-time-scale neutral stochastic delay partial differential equations driven by fractional Brownian motions

We prove the validity of averaging principles for two-time-scale neutral stochastic delay partial differential equations (NSDPDEs) driven by fractional Brownian motions (fBms) under two-time-scale formulation. Firstly, in the sense of mean-square convergence, we obtain not only the averaging principles for NSDPDEs involving two-time-scale Markov switching with a single weakly recurrent class but also for the case of two-time-scale Markov switching with multiple weakly irreducible classes. Secondly, averaging principles for NSDPDEs driven by fBms with random delay modulated by two-time-scale Markovian switching are established. We proved that there is a limit process in which the fast changing noise is averaged out. The limit process is substantially simpler than that of the original full fast–slow system.

Generalized neural closure models with interpretability

Improving the predictive capability and computational cost of dynamical models is often at the heart of augmenting computational physics with machine learning (ML). However, most learning results are limited in interpretability and generalization over different computational grid resolutions, initial and boundary conditions, domain geometries, and physical or problem-specific parameters. In the present study, we simultaneously address all these challenges by developing the novel and versatile methodology of unified neural partial delay differential equations. We augment existing/low-fidelity dynamical models directly in their partial differential equation (PDE) forms with both Markovian and non-Markovian neural network (NN) closure parameterizations. The melding of the existing models with NNs in the continuous spatiotemporal space followed by numerical discretization automatically allows for the desired generalizability. The Markovian term is designed to enable extraction of its analytical form and thus provides interpretability. The non-Markovian terms allow accounting for inherently missing time delays needed to represent the real world. Our flexible modeling framework provides full autonomy for the design of the unknown closure terms such as using any linear-, shallow-, or deep-NN architectures, selecting the span of the input function libraries, and using either or both Markovian and non-Markovian closure terms, all in accord with prior knowledge. We obtain adjoint PDEs in the continuous form, thus enabling direct implementation across differentiable and non-differentiable computational physics codes, different ML frameworks, and treatment of nonuniformly-spaced spatiotemporal training data. We demonstrate the new generalized neural closure models (gnCMs) framework using four sets of experiments based on advecting nonlinear waves, shocks, and ocean acidification models. Our learned gnCMs discover missing physics, find leading numerical error terms, discriminate among candidate functional forms in an interpretable fashion, achieve generalization, and compensate for the lack of complexity in simpler models. Finally, we analyze the computational advantages of our new framework.

Integral Quadratic Constraints with Infinite-Dimensional Channels.

Modern control theory provides us with a spectrum of methods for studying the interconnection of dynamic systems using input-output properties of the interconnected subsystems. Perhaps the most advanced framework for such input-output analysis is the use of Integral Quadratic Constraints (IQCs), which considers the interconnection of a nominal linear system with an unmodelled nonlinear or uncertain subsystem with known input-output properties. Although these methods are widely used for Ordinary Differential Equations (ODEs), there have been fewer attempts to extend IQCs to infinite-dimensional systems. In this paper, we present an IQC-based framework for Partial Differential Equations (PDEs) and Delay Differential Equations (DDEs). First, we introduce infinite-dimensional signal spaces, operators, and feedback interconnections. Next, in the main result, we propose a formulation of hard IQC-based input-output stability conditions, allowing for infinite-dimensional multipliers. We then show how to test hard IQC conditions with infinite-dimensional multipliers on a nominal linear PDE or DDE system via the Partial Integral Equation (PIE) state-space representation using a sufficient version of the Kalman-Yakubovich-Popov lemma (KYP). The results are then illustrated using four example problems with uncertainty and nonlinearity.

Robust spectral treatment for time-fractional delay partial differential equations

Fractional delay differential equations (FDDEs) and time-fractional delay partial differential equations (TFDPDEs) are the focus of the present research. The FDDEs is converted into a system of algebraic equations utilizing a novel numerical approach based on the spectral Galerkin (SG) technique. The suggested numerical technique is likewise utilized for TFDPDEs. In terms of shifted Jacobi polynomials, suitable trial functions are developed to fulfill the initial-boundary conditions of the main problems. According to the authors, this is the first time utilizing the SG technique to solve TFDPDEs. The approximate solution of five numerical examples is provided and compared with those of other approaches and with the analytic solutions to test the superiority of the proposed method.

Homotopy perturbation method for solving time-fractional nonlinear Variable-Order Delay Partial Differential Equations

The homotopy perturbation method is extend to derive the approximate solution of the variable order fractional partial differential equations with time delay. The variable order fractional derivative is taken in the Caputo sense. An approximation formula of the Caputo derivative of fractional variable order is presented in terms of standard (integer order) derivatives only. Then the original problem will be transformed into a systems of partial differential equations with delay. By employing the homotopy perturbation method the explicit approximate solutions are found. The error and convergence analysis of the homotopy perturbation method has been discussed for the applicability of the method. The absolute errors and the approximate solutions are presented graphically and by tables at the values of various variable fractional order. From the results of the illustrated examples, we can Judge that the homotopy perturbation method is very effective, and simple accelerates the rapid convergence of the solution.

A Systematic Review on the Solution Methodology of Singularly Perturbed Differential Difference Equations

This review paper contains computational methods or solution methodologies for singularly perturbed differential difference equations with negative and/or positive shifts in a spatial variable. This survey limits its coverage to singular perturbation equations arising in the modeling of neuronal activity and the methods developed by numerous researchers between 2012 and 2022. The review covered singularly perturbed ordinary delay differential equations with small or large negative shift(s), singularly perturbed ordinary differential–differential equations with mixed shift(s), singularly perturbed delay partial differential equations with small or large negative shift(s) and singularly perturbed partial differential–difference equations of the mixed type. The main aim of this review is to find out what numerical and asymptotic methods were developed in the last ten years to solve such problems. Further, it aims to stimulate researchers to develop new robust methods for solving families of the problems under consideration.

Blowing-up solutions of differential equations with shifts: A survey

<p style='text-indent:20px;'>We present a survey on delay ordinary differential equations, delay fractional differential equations and delay partial differential equations with blowing-up or growing-up solutions. Moreover, we indicate the techniques used for the obtained results.</p>

On the Solution of Multi-Term Nonlinear Partial Derivative Delay Differential Equations

In our paper, we used the Adomian decomposition method to solve multi-term nonlinear delay differential equations of partial derivative order, these types of equations are studied. When we used this method, the being of an exclusive solution will be provided, approximate analytics of this method applied to these kinds of equations will be disputed, and the maximum absolute briefed error of the Adomians series solution will be rated. A digital example is made ready to make clear the effectiveness of the offered method.

A generalized Gegenbauer wavelet collocation method for solving p-type fractional neutral delay differential and delay partial differential equations

A generalized Gegenbauer wavelet collocation method for solving p-type fractional neutral delay differential and delay partial differential equations

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.

Stability of a Cross-Flow Heat-Exchanger Tube With Asymmetric Supports

Abstract Heat-exchanger tubes subjected to cross-flow experience a fluid elastic instability (FEI) at a critical flow velocity. Baffle plates are often used to provide additional stiffness to long tubes. A cladding is provided between the baffle plates and the tubes to allow for thermal expansion, and the cladding acts like a soft spring. In this paper, we study the contact between the cladding and the tube when the stiffness of the cladding is asymmetric. The tube is modeled as a cantilever Euler–Bernoulli beam, and the fluid forces are represented using an added mass, added damping and a time-delayed displacement term. This results in a partial delay differential equation model for the tube vibration. Using the Galerkin approximation and considering only a single mode, a delay differential equation (DDE) model is developed. However, due to the presence of the delay, the single mode model is also infinite-dimensional. This model contains sign function nonlinearity due to the asymmetry in the contact stiffness of the cladding. The DDE model is essentially nonlinear—that is, no linear approximation exists. However, its solutions are scalable: if q(t) is a solution, then α q(t) is also a solution for any α>0. By exploiting this scalability property of solutions, we use Lyapunov-like exponents to probe the stability of the nonlinear DDE. Further, by numerical continuation, we numerically demonstrate the existence of periodic solutions on the stability boundary. Our stability charts will be beneficial to the designers of heat-exchangers.

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.

A Hybrid Finite Difference Method for Singularly Perturbed Delay Partial Differential Equations with Discontinuous Coefficient and Source

The article presents a hybrid finite difference scheme to solve a singularly perturbed parabolic functional differential equation with discontinuous coefficient and source. The simultaneous presence of deviating argument with a discontinuous source and coefficient makes the problem stiff. The solution of the problem exhibits turning point behaviour across discontinuity as tends to zero. The hybrid scheme presented is a composition of a central difference scheme and a midpoint upwind scheme on a specially generated mesh. At the same time, an implicit finite difference method is used to discretize the time variable. Consistency, stability, and convergence of the presented numerical approach have been investigated. The presented method converges uniformly independent of the perturbation parameter. Numerical results have been presented for two test examples that verify the effectiveness of the scheme.

Transmission Dynamics of Tuberculosis with Age-specific Disease Progression.

Demographic structure and latent phenomenon are two essential factors determining the rate of tuberculosis transmission. However, only a few mathematical models considered age structure coupling with disease stages of infectious individuals. This paper develops a system of delay partial differential equations to model tuberculosis transmission in a heterogeneous population. The system considers demographic structure coupling with the continuous development of disease stage, which is crucial for studying how aging affects tuberculosis dynamics and disease progression. Here, we determine the basic reproduction number, and several numerical simulations are used to investigate the influence of various progression rates on tuberculosis dynamics. Our results support that the aging effect on the disease progression rate contributes to tuberculosis permanence.

Two-dimensional Haar Wavelet Method for Numerical Solution of Delay Partial Differential Equations

In this paper, a two-dimensional Haar wavelet collocation method is applied to obtain the numerical solution of delay and neutral delay partial differential equations. Both linear and nonlinear problems can be solved using this method. Some benchmark test problems are given to verify the efficiency and accuracy of the aforesaid method. The results are compared with the exact solution and performance of the two-dimensional Haar collocation technique is measured by calculating the maximum absolute and root mean square errors for different numbers of grid points. The results are also compared with finite difference technique and one-dimensional Haar wavelet technique. The numerical results show that the two-dimensional Haar method is simply applicable, accurate and efficient.

Exponential Multistep Methods for Stiff Delay Differential Equations

Stiff delay differential equations are frequently utilized in practice, but their numerical simulations are difficult due to the complicated interaction between the stiff and delay terms. At the moment, only a few low-order algorithms offer acceptable convergent and stable features. Exponential integrators are a type of efficient numerical approach for stiff problems that can eliminate the influence of stiffness on the scheme by precisely dealing with the stiff term. This study is concerned with two exponential multistep methods of Adams type for stiff delay differential equations. For semilinear delay differential equations, applying the linear multistep method directly to the integral form of the equation yields the exponential multistep method. It is shown that the proposed k-step method is stiffly convergent of order k. On the other hand, we can follow the strategy of the Rosenbrock method to linearize the equation along the numerical solution in each step. The so-called exponential Rosenbrock multistep method is constructed by applying the exponential multistep method to the transformed form of the semilinear delay differential equation. This method can be easily extended to nonlinear delay differential equations. The main contribution of this study is that we show that the k-step exponential Rosenbrock multistep method is stiffly convergent of order k+1 within the framework of a strongly continuous semigroup on Banach space. As a result, the methods developed in this study may be utilized to solve abstract stiff delay differential equations and can be served as time matching methods for delay partial differential equations. Numerical experiments are presented to demonstrate the theoretical results.

Existence and Hyers–Ulam Stability Results for Partial Fractional-Order Delay Differential Equations

Existence and Hyers–Ulam Stability Results for Partial Fractional-Order Delay Differential Equations