Problems For Partial Differential Equations Research Articles

Bent-half space model problem for Lame equation with surface tension

The study of fluid flow is a very fascinating area of fluid dynamics. Fluid motion has received more and more attention in recent years and numerous researchers have looked into this topic. However, they rarely used a mathematical analysis approach to analyse fluid motion; instead, they used numerical analysis. This serves as a significant justification for the researcher's decision to study fluid flow from the perspective of mathematical analysis. In this paper, we consider the ${\mathcal R}$-boundedness of the solution operator families of the Lam\'e equation with surface tension in bent half-space model problem by taking into account the surface tension in a bounded domain of {\it N}-dimensional Euclidean space ($N \geq 2$). The motion of the model problem can be described by linearizing an equation system of a model problem. This research is a continuation of [13]. They investigated the ${\mathcal R}$-boundedness of the solution operator families in the half-space case for the model problem of the Lam\'e equation with surface tension. First of all, by using Laplace transformation we consider the resolvent of the model problem, then treat the problem in bent half-space case. By using Weis's operator-valued Fourier multiplier theorem, we know that ${\mathcal R}$-boundedness implies the maximal $L_p$-$L_q$ regularity for the initial boundary value. This regularity is an essential tool for the partial differential equation problem.

Schwarz Problem for Model Partial Differential Equations with One Complex Variable

This paper investigates the Schwarz problem. Initially, the focus lies on analyzing the problem for the first, second orders. Subsequently, attention shifts towards studying the same problem for equations of higher order. In the realm of second-order equations, the Schwarz problem is specifically examined for some operators; Laplace, Bitsadze and its complex conjugate. The findings demonstrate that the Schwarz problem for an n-order equation, when equipped with solely one boundary condition, exhibits an infinite number of solutions. However, by incorporating additional boundary conditions, it becomes feasible to obtain a unique solution for problem concerning n-order equations, effectively rendering it a well-posed problem.

An Application-Driven Method for Assembling Numerical Schemes for the Solution of Complex Multiphysics Problems

Within recent years, considerable progress has been made regarding high-performance solvers for partial differential equations (PDEs), yielding potential gains in efficiency compared to industry standard tools. However, the latter largely remains the status quo for scientists and engineers focusing on applying simulation tools to specific problems in practice. We attribute this growing technical gap to the increasing complexity and knowledge required to pick and assemble state-of-the-art methods. Thus, with this work, we initiate an effort to build a common taxonomy for the most popular grid-based approximation schemes to draw comparisons regarding accuracy and computational efficiency. We then build upon this foundation and introduce a method to systematically guide an application expert through classifying a given PDE problem setting and identifying a suitable numerical scheme. Great care is taken to ensure that making a choice this way is unambiguous, i.e., the goal is to obtain a clear and reproducible recommendation. Our method not only helps to identify and assemble suitable schemes but enables the unique combination of multiple methods on a per-field basis. We demonstrate this process and its effectiveness using different model problems, each comparing the resulting numerical scheme from our method with the next best choice. For both the Allen–Cahn and advection equations, we show that substantial computational gains can be attained for the recommended numerical methods regarding accuracy and efficiency. Lastly, we outline how one can systematically analyze and classify a coupled multiphysics problem of considerable complexity with six different unknown quantities, yielding an efficient, mixed discretization that in configuration compares well to high-performance implementations from the literature.

Numerical analysis of a nonsmooth quasilinear elliptic control problem: I. Explicit second-order optimality conditions

In this paper, we derive explicit second-order necessary and sufficient optimality conditions of a local minimizer to an optimal control problem for a quasilinear second-order partial differential equation with a piecewise smooth but not differentiable nonlinearity in the leading term. The key argument rests on the analysis of level sets of the state. Specifically, we show that if a function vanishes on the boundary and its the gradient is different from zero on a level set, then this set decomposes into finitely many closed simple curves. Moreover, the level sets depend continuously on the functions defining these sets. We also prove the continuity of the integrals on the level sets. In particular, Green's first identity is shown to be applicable on an open set determined by two functions with nonvanishing gradients. In the second part to this paper, the explicit sufficient second-order conditions will be used to derive error estimates for a finite-element discretization of the control problem.

Well-posedness and Tikhonov regularization of an inverse source problem for a parabolic equation with an integral constraint

Abstract We investigate a time-dependent inverse source problem for a parabolic partial differential equation with an integral constraint and subject to Neumann boundary conditions in a domain of R d \mathbb{R}^{d} , d ≥ 1 d\geq 1 . We prove the well-posedness as well as higher regularity of solutions in Hölder spaces. We then develop and implement an algorithm that we use to approximate solutions of the inverse problem by means of a finite element discretization in space. Due to instability in inverse problems, we apply Tikhonov regularization combined with the discrepancy principle for selecting the regularization parameter in order to obtain a stable reconstruction. Our numerical results show that the proposed scheme is an accurate technique for approximating solutions of this inverse problem.

A boundary value problem for the fourth-order degenerate equation of the mixed type

Many problems in mechanics, physics, and geophysics lead to solving partial differential equations that are not included in the known classes of elliptic, parabolic or hyperbolic equations. Such equations, as a rule, began to be called non-classical equations of mathematical physics. The theory of degenerate equations is one of the central branches of the modern theory of partial differential equations. This is primarily due to the identification of a variety of applied problems, the mathematical modeling of which serves the study of various types of degenerate equations. The study of boundary value problems for mixed type’s equations of the fourth-order with power-law degeneration remains relevant. In this work, a boundary value problem in a rectangular domain for a degenerate equation of the fourth-order mixed-type is posed and investigated. Well-posedness of the boundary value problem for a fourth-order partial differential equation is established by proving the existence and uniqueness of the solution. Under sufficient conditions, a solution to the problem under consideration was explicitly found by the variable separation method.

Solvability of an Inverse Problem for an Elliptic-Type Equation

In this study, we consider an inverse problem of determining an unknown source function in the right-hand side of an elliptic equation which is ill-posed in the Hadamard sense. To investigate the solvability of the problem, we reduce it to a Dirichlet problem for a third-order partial differential equation with homogeneous boundary condition. Since the problem is linear, the proof of the uniqueness theorem is based on the Fredholm Alternative Theorem. We prove the existence of the solution to the problem by using the Galerkin method.

A multi-fidelity transfer learning strategy based on multi-channel fusion

Multi-fidelity strategies leverage a large amount of low-fidelity data combined with a smaller set of high-fidelity data, thereby achieving satisfactory results at a reasonable cost. In our research, we introduce an innovative multi-fidelity strategy that integrates the concepts of multi-fidelity data fusion and transfer learning. In the proposed framework, we incorporate auto-encoders and a multi-channel transfer learning strategy, enabling the network model to comprehend the relationship between the low-fidelity and high-fidelity models in both explicit and implicit manners. This approach not only enhances prediction accuracy but also mitigates issues such as overfitting and negative transfer, which may arise in scenarios with sparse samples. Additionally, Bayesian optimization is employed for effective hyperparameter selection. To evaluate and analyze the performance of our proposed method, we present a series of benchmark test cases. Furthermore, we also show the application of the proposed method to engineering problems. Firstly, we consider a parametrized partial differential equation problem, where high-fidelity and low-fidelity data are obtained using exact methods and simplified algorithms, respectively. Subsequently, we extend this strategy to convolutional neural network architectures, specifically addressing a pressure Poisson equation problem. We also explore the effect of the reliability of the low-fidelity data and the number of high-fidelity data on the results. The results show that the proposed method exhibits low requirements in terms of both the reliability of the low-fidelity data and the number of high-fidelity data while maintaining satisfactory accuracy metrics.

A Study for Solving Pseudo-Parabolic Viscous Diffusion, Telegraph, Poisson and Helmholtz PDE using Legendre-Collocation Method

Legendre collocation approach is implemented employing the operative-differential array escorted by the shifted Legendre polynomials. The significance of the proposed approach is reducing the intended partial differential equation problem into an algebraic system of equations by the aid of Legendre polynomials as orthogonal basis. Using mesh collocation points, the domain of the resulting algebraic system solution domain is discretized, and it is then solved to obtain an approximation solution. To demonstrate the accuracy of the procedure, the maximum absolute error and the estimate error are calculated. Also, Illustrative numerical examples are demonstrated to display the veracity of the potential approach in particular pseudo-parabolic viscous diffusion, telegraph, Poisson and Helmholtz partial differential equations. The generated results are contrasted with the exact solutions/other methods to highlight the efficiency of the proposed approach.

On Some Problems for Partial Differential Equations with a Small Parameter in the Principal Part

On Some Problems for Partial Differential Equations with a Small Parameter in the Principal Part

Potential of physics-informed neural networks for solving fluid flow problems with parametric boundary conditions

Fluid flows are present in various fields of science and engineering, so their mathematical description and modeling is of high practical importance. However, utilizing classical numerical methods to model fluid flows is often time consuming and a new simulation is needed for each modification of the domain, boundary conditions, or fluid properties. As a result, these methods have limited utility when it comes to conducting extensive parameter studies or optimizing fluid systems. By utilizing recently proposed physics-informed neural networks (PINNs), these limitations can be addressed. PINNs approximate the solution of a single or system of partial differential equations (PDEs) by artificial neural networks (ANNs). The residuals of the PDEs are used as the loss function of the ANN, while the boundary condition is imposed in a supervised manner. Hence, PDEs are solved by performing a nonconvex optimization during the training of the ANN instead of solving a system of equations. Although this relatively new method cannot yet compete with classical numerical methods in terms of accuracy for complex problems, this approach shows promising potential as it is mesh-free and suitable for parametric solution of PDE problems. This is achieved without relying on simulation data or measurement information. This study focuses on the impact of parametric boundary conditions, specifically a variable inlet velocity profile, on the flow calculations. For the first time, a physics-based penalty term to avoid the suboptimal solution along with an efficient way of imposing parametric boundary conditions within PINNs is presented.

Polynomial Fuzzy Observer-Based Feedback Control for Nonlinear Hyperbolic PDEs Systems.

This article explores the observer-based feedback control problem for a nonlinear hyperbolic partial differential equations (PDEs) system. Initially, the polynomial fuzzy hyperbolic PDEs (PFHPDEs) model is established through the utilization of the fuzzy identification approach, derived from the nonlinear hyperbolic PDEs model. Various types of state estimation and controller design problems for the polynomial fuzzy PDEs system are discussed concerning the state estimation problem. To investigate the relaxed stability problem, Euler's homogeneous theorem, Lyapunov-Krasovskii functional with polynomial matrices (LKFPM), and the sum-of-squares (SOSs) approach are adopted. The exponential stabilization condition is formulated in terms of the spatial-derivative-SOSs (SD-SOSs). Additionally, a segmental algorithm is developed to find the feasible solution for the SD-SOS condition. Finally, a hyperbolic PDEs system and several numerical examples are provided to illustrate the validity and effectiveness of the proposed results.

Mathematical Modeling of Cell Growth via Inverse Problem and Computational Approach

A simple cell population growth model is proposed, where cells are assumed to have a physiological structure (e.g., a model describing cancer cell maturation, where cells are structured by maturation stage, size, or mass). The main question is whether we can guarantee, using the death rate as a control mechanism, that the total number of cells or the total cell biomass has prescribed dynamics, which may be applied to modeling the effect of chemotherapeutic agents on malignant cells. Such types of models are usually described by partial differential equations (PDE). The population dynamics are modeled by an inverse problem for PDE in our paper. The main idea is to reduce this model to a simplified integral equation that can be more easily studied by various analytical and numerical methods. Our results were obtained using the characteristics method.

To the Question of the Solvability of the Ionkin Problem for Partial Differential Equations

We study the solvability of the Ionkin problem for some differential equations with one space variable. These equations include parabolic and quasiparabolic, hyperbolic and quasihyperbolic, pseudoparabolic and pseudohyperbolic, elliptic and quasielliptic equations and equations of many other types. For the above equations, the following theorems are proved with the use of the splitting method: the existence of regular solutions—solutions that all have weak derivatives in the sense of S. L. Sobolev and occur in the corresponding equation.

Sensor Fault Detection and Diagnosis of Linear Parabolic PDE Systems With Unknown Inputs

This paper investigates the sensor bias fault detection and diagnosis problem for linear parabolic partial differential equation (PDE) systems under the existence of unknown input signals. A variation of Wirtinger's inequality is used to design a Luenberger-type PDE observer and radial basis function (RBF) neural networks are applied to approximate the unknown inputs, guaranteeing the boundedness of the state estimation error. Taking the measurement error as the fault indication signal, a residual evaluation logic is developed to achieve fault detection. An adaptive diagnostic law is constructed to estimate the values of sensor bias faults once they are detected. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sigma$</tex-math></inline-formula> -modification RBF neural networks are designed to compensate for the unknown inputs in the presence of sensor faults, ensuring that both state and sensor bias faults estimation errors converge to some adjustable sets. The effectiveness and applicability of the developed fault diagnosis strategy are demonstrated by simulation results on a heat diffusion process.

Multi-fidelity reduced-order surrogate modelling

High-fidelity numerical simulations of partial differential equations (PDEs) given a restricted computational budget can significantly limit the number of parameter configurations considered and/or time window evaluated. Multi-fidelity surrogate modelling aims to leverage less accurate, lower-fidelity models that are computationally inexpensive in order to enhance predictive accuracy when high-fidelity data are scarce. However, low-fidelity models, while often displaying the qualitative solution behaviour, fail to accurately capture fine spatio-temporal and dynamic features of high-fidelity models. To address this shortcoming, we present a data-driven strategy that combines dimensionality reduction with multi-fidelity neural network surrogates. The key idea is to generate a spatial basis by applying proper orthogonal decomposition (POD) to high-fidelity solution snapshots, and approximate the dynamics of the reduced states—time-parameter-dependent expansion coefficients of the POD basis—using a multi-fidelity long short-term memory network. By mapping low-fidelity reduced states to their high-fidelity counterpart, the proposed reduced-order surrogate model enables the efficient recovery of full solution fields over time and parameter variations in a non-intrusive manner. The generality of this method is demonstrated by a collection of PDE problems where the low-fidelity model can be defined by coarser meshes and/or time stepping, as well as by misspecified physical features.

Robust PRESB Preconditioning of a 3-Dimensional Space-Time Finite Element Method for Parabolic Problems

Abstract We present a recently developed preconditioning of square block matrices (PRESB) to be used within a parallel method to solve linear systems of equations arising from tensor-product discretizations of initial boundary-value problems for parabolic partial differential equations. We consider weak formulations in Bochner–Sobolev spaces and tensor-product finite element approximations for the heat and eddy current equations. The fast diagonalization method is employed to decouple the arising linear system of equations into auxiliary spatial complex-valued linear systems that can be solved concurrently. We prove that the real part of the system matrix is positive definite, which allows us to accelerate the flexible generalized minimal residual method (FGMRES) by the PRESB preconditioner. The action of PRESB on a vector includes two solutions with positive definite matrices. The spectrum of the preconditioned system lies between 1/2 and 1. Finally, we combine the PRESB-FGMRES method with multigrid-CG iterations and illustrate the numerical efficiency and the robustness for spatial discretizations up to 12 millions degrees of freedom.

On a reaction diffusion problem with a moving impulse on boundary

Abstract We study an asymptotic problem of a semilinear partial differential equation (PDE) with Neumann boundary condition, periodic coefficients and highly oscillating drift and nonlinear terms. Our analysis focuses on the double limiting behavior of the PDE-solution perturbed by ε (viscosity parameter) and δ (scaling coefficient) both tending to zero. To do so, we state basic properties of the large deviations principle (LDP) and we express the logarithmic asymptotic of the PDE-solution. Particularly, we provide it for the case when ε converges more quickly than δ.

Inverse problems for stochastic partial differential equations: Some progresses and open problems

The aim of this paper is to survey some recent works on inverse problems for stochastic partial differential equations (SPDEs for short), which were solved by means of Carleman estimate. For simplicity, we focus on two typical SPDEs, i.e, stochastic hyperbolic equations and stochastic parabolic equations. Our particular concerns are those inverse problems which are genuinely stochastic and therefore cannot be reduced to any deterministic ones. Also, some open problems are presented.

Sequence-to-Sequence Stacked Gate Recurrent Unit Networks for Approximating the Forward Problem of Partial Differential Equations

Sequence-to-Sequence Stacked Gate Recurrent Unit Networks for Approximating the Forward Problem of Partial Differential Equations