Initial-boundary Value Problem Research Articles

Efficient exponential Rosenbrock methods till order four

In a previous paper, a technique was described to avoid order reduction with exponential Rosenbrock methods when integrating initial boundary value problems with time-dependent boundary conditions. That requires to calculate some information on the boundary from the given data. In the present paper we prove that, under some assumptions on the coefficients of the method which are mainly always satisfied, no numerical differentiation is required to approximate that information in order to achieve order 4 for parabolic problems with Dirichlet boundary conditions. With Robin/Neumann ones, just numerical differentiation in time may be necessary for order 4, but none for order ≤3.Furthermore, as with this technique it is not necessary to impose any stiff order conditions, in search of efficiency, we recommend some methods of classical orders 2, 3 and 4 and we give some comparisons with several methods in the literature, with the corresponding stiff order.

Inverse a time-dependent potential problem of a generalized time-fractional super-diffusion equation with a nonlinear source from a nonlocal integral observation

This paper focus on the problem of reconstructing the time-dependent potential in a class of generalized (including three special cases: the classical/multi-term/distributed order) time-fractional super-diffusion equations with nonlinear sources from a nonlocal integral observation. For such nonlinear equation, we investigate it for both the direct and inverse time-dependent potential problems. For the direct problem, given the time-dependent potential function p(t), we obtain the well-posedness of the corresponding initial–boundary value problem. For the inverse potential problem, by utilizing additional nonlocal measurement data and using the Arzelà–Ascoli theorem and Grönwall’s inequality, we prove the existence and uniqueness of the solution for such nonlinear problem. Meanwhile, we also demonstrate the ill-posedness of the inverse problem. Moreover, to validate the theoretical results, we numerically reconstruct the potential term from Bayesian perspective. Several numerical examples are presented to show the efficiency of the proposed method.

Analysis of fractional Euler-Bernoulli bending beams using Green’s function method

This paper deals with the effect of fractional order on the fractional Euler-Bernoulli beams model. This model is based on the elastic curve that can be approximated using tangent lines and the osculating circle. We introduce a general form of the fractional Green’s function for the highest-order term 1<α≤2, which gives a unique solution for the geometrical fractional bending beams with an inhomogeneous fractional differential equation. The analytical closed-form responses were obtained by inverse problem technique and integration over Green’s function kernel. The theory and properties of fractional Green’s function method are described by generalizing classical theory to the initial and two-point fractional boundary value problems. Static analysis of the numerical examples, such as determinate and indeterminate beams with typical loads and beam property boundary conditions, are presented and discussed. The results were verified using the direct integration method. The fractional conventional ratio indicates the beam's deformation deviation from its classical state. The analysis results reveal that changing the fractional parameter significantly impacts the deflection behavior of flexural beams due to the beam’s characteristics and the fractional order. The behavior of the fractional beam is identical to the classical state when the fractional parameter is an integer number.

Unsteady Magnetohydrodynamics PDE of Monge–Ampère Type: Symmetries, Closed-Form Solutions, and Reductions

The paper studies an unsteady equation with quadratic nonlinearity in second derivatives, that occurs in electron magnetohydrodynamics. In mathematics, such PDEs are referred to as parabolic Monge–Ampère equations. An overview of the Monge–Ampère type equations is given, in which their unusual qualitative features are noted. For the first time, the Lie group analysis of the considered highly nonlinear PDE with three independent variables is carried out. An eleven-parameter transformation is found that preserves the form of the equation. Some one-dimensional reductions allowing to obtain self-similar and other invariant solutions that satisfy ordinary differential equations are described. A large number of new additive, multiplicative, generalized, and functional separable solutions are obtained. Special attention is paid to the construction of exact closed-form solutions, including solutions in elementary functions (in total, more than 30 solutions in elementary functions were obtained). Two-dimensional symmetry and non-symmetry reductions leading to simpler partial differential equations with two independent variables are considered (including stationary Monge–Ampère type equations, linear and nonlinear heat type equations, and nonlinear filtration equations). The obtained results and exact solutions can be used to evaluate the accuracy and analyze the adequacy of numerical methods for solving initial boundary value problems described by highly nonlinear partial differential equations.

Numerical Analysis of Stainless Steel of Crevice Corrosion Initiation Behavior with Varying External Potentials and Initial Crevice Gaps

The potential, current density, pH, and crevice profile distribution within the crevice structure of stainless steel in seawater were numerically analyzed. The crevice corrosion phenomena were mathematically modeled with the initial boundary value problem of diffusion and static electrical field, respectively. This initial boundary value problem was discretized using a finite difference method. The predicts of the crevice corrosion behaviors, set for various external potentials and initial crevice gaps, were classified into three categories: the pitting type, the active type, and no corrosion. The pitting type crevice corrosion was predicted to occur in conditions where the external potentials were noble and the initial crevice gaps were narrow. The active type crevice corrosion was predicted to occur in conditions where the external potentials were less noble and the initial crevice gaps were narrow. The range of external potential conditions where crevice corrosion was predicted not to occur increased in conditions with wide initial crevice gaps. Crevice corrosion was predicted to occur at all external potentials in conditions where the initial crevice gaps were 2 μm or less.

Initial Boundary Value and Inverse Coefficient Problems for One-Dimensional Fractional Diffusion Equation in a Half-Line

Initial Boundary Value and Inverse Coefficient Problems for One-Dimensional Fractional Diffusion Equation in a Half-Line

An analytical solution for vertical infiltration in homogeneous bounded profiles

AbstractIn this study, we derive an analytical solution to address the problem of vertical infiltration within 1D homogeneous bounded profiles. Initially, we consider the Richards equation together with Dirichlet boundary conditions. We assume constant diffusivity and linear dependence between the conductivity and the water content, resulting to a linear partial differential equation of diffusion type. To solve the simplified initial boundary value problem over a finite interval, we apply the unified transform, commonly known as the Fokas method. Through this methodology, we obtain an integral representation of the solution that can be efficiently and directly computed numerically, yielding a convergent scheme. We examine various cases, and we compare our solution with well‐known approximate solutions. This work can be seen as a first step to derive analytical solutions for the far more difficult and complex problem of modelling water flow in heterogeneous layered soils.

Combining Laplace transform and Variational iteration method for solving singular IVPs and BVPs of Lane–Emden type equation

This study uses the combined Laplace transform and variational iteration method to analytically solve initial value problems (IVPs) and boundary value problems (BVPs) of Lane- Emden- equation with singular behaviour at . Initially, this method requires calculating the lagrange multiplayer, and after its value was calculated, it was found to be equal , this method is applying on several problems and the results were improved using Padé approximations, the results were compared with the exact solution by calculating the absolute error value between them.

Version [1.1] - [2D-xtDCR-LTBEM: A Fortran code for numerical solutions to the unsteady anisotropic-DCR equation of spatio-temporal coefficients]

Variable coefficient equations are usually used as governing equations for modeling problems which are associated with Functionally Graded Materials (FGM). Version 1.1 - 2D-xtDCR-LTBEM is a Fortran code that can be used to find numerical solutions of two-dimensional (2D) initial–boundary value problems which are governed by the unsteady anisotropic-diffusion convection reaction (DCR)-type and the anisotropic-diffusion reaction (DR)-type equations with a class of space–time dependent coefficients, by utilizing a combined Laplace Transform-Boundary Element Method (LTBEM). This code may provide valuable help for simulation of problems of FGM.

Inverse Coefficient Problem for the 2D Wave Equation with Initial and Nonlocal Boundary Conditions

In this paper, we consider direct and inverse problems for the two-dimensional wave equation. The direct problem is an initial boundary value problem for this equation with nonlocal boundary conditions. In the inverse problem, it is required to find the time-variable coefficient at the lower term of the equation. The classical solution of the direct problem is presented in the form of a biorthogonal series in eigenvalues and associated functions, and the uniqueness and stability of this solution are proven. For solution to the inverse problem, theorems of existence in local, uniqueness in global, and an estimate of conditional stability are obtained. The problems of determining the right-hand sides and variable coefficients at the lower terms from initial boundary value problems for second-order linear partial differential equations with local boundary conditions have been studied by many authors. Since the nonlinearity is convolutional, the unique solvability theorems in them are proven in a global sense. In the works, the method of separation of variables is used to find the classical solution of the direct problem in the form of a biorthogonal series in terms of eigenfunctions and associated functions. The nonlocal integral condition is used as the overdetermination condition with respect to the solution of the direct problem. The direct problem reduces to equivalent integral equations of the Fourier method. To establish integral inequalities, the generalized Gronwall-Bellman inequality is used. We obtain an a priori estimate of the solution in terms of an unknown coefficient, that are useful for studying the inverse problem.

Error Identities for Parabolic Equations with Monotone Spatial Operators

Abstract The article studies quantitative relations (error identities) that characterize distances between exact solutions of nonlinear evolutionary problems and functions considered as approximations. The restrictions imposed on such a function are minimal and actually come down to the condition that it belongs to the same functional class as the generalized solution of the problem under consideration. Functional identities of this type reflect the most general relations between deviations from exact solutions of parabolic initial boundary value problems and those data that can be observed in a numerical experiment. The identities contain no mesh dependent constants and are valid for any function in the admissible (energy) class regardless of the method by which it was constructed. Therefore, they can serve as basic tools for deriving fully reliable a posteriori estimates of approximation errors as well as for analysis of modeling errors. The corresponding examples are discussed in the paper.

Boundedness of weak solutions to a 3D chemotaxis-Stokes system with slow [formula omitted]Laplacian diffusion and rotation

In this paper, we consider the following chemotaxis-Stokes system with general sensitivity and rotation nt+u⋅∇n=∇⋅(|∇n|p−2∇n)−∇⋅(nS(x,n,c)∇c),x∈Ω,t>0,ct+u⋅∇c=Δc−nc,x∈Ω,t>0,ut+∇P=Δu+n∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0in a smooth bounded domain Ω∈R3 with zero-flux boundary and no-slip boundary condition. We prove the boundedness of the weak solutions to the initial–boundary value problem of the 3D chemotaxis-Stokes system with p−Laplacian diffusion if 43p+α>72−63 and p>2, which improves the results of papers [Chen et al., Nonlinear Anal. Real World Appl., 76 (2024) 103996; Zhuang et al., Nonlinear Anal. Real World Appl., 56 (2020) 103163 and Tao et al., J. Differ. Equ., 268(11) (2020) 6879–6919] and extends the result of the paper [Jin, J. Differ. Equ. 287 (2021) 148–184] to the chemotaxis system with general sensitivity and rotation.

Stability of hybrid time integration scheme for Lord–Shulman thermopiezoelectricity

Based on the existing initial boundary value and variational problems of Lord–Shulman thermopiezoelectricity a transient analysis of the behavior of piezoelectric materials is performed numerically. For space discretization of the variational problem the finite element method is used and for time discretization a hybrid time integration scheme is constructed on the basis of Newmark scheme for hyperbolic equation and generalized trapezoidal rule for parabolic equations. The unconditional stability of the developed time integration scheme is proved for some specific values of the scheme parameters by making use of energy balance law for the obtained time-discretized variational problem. Finally, the applicability of the constructed numerical scheme is demonstrated by the results of the numerical experiment which are compared with the ones available in literature.

The integrability and infinite conservation laws of a variable coefficient higher-order Schrödinger equation

In this paper, a variable coefficient high-order Schrödinger equation is studied to describe the propagation of ultrashort optical pulses in dispersion reducing fibers. It has third-order dispersion, self steepening and stimulated Raman scattering effects. Using the Bell polynomial method, the integrability of the variable coefficient higher-order Schrödinger equation is discussed. And the bilinear Bäcklund transformation, Lax pair, and infinite conservation laws of the equation are derived. It can be obtained that the variable coefficient higher-order Schrödinger equation is integrable in the sense of Lax pair. The soliton solutions are constructed from the obtained bilinear Bäcklund transformation and the bilinear form. The obtained results are analyzed in the initial–boundary value problem. It is worth mentioning that this paper, based on the angle of increasing the order, obtained the bilinear Bäcklund transformation of the third order variable coefficient Schrödinger equation by using the Bell polynomial method. On this basis, relevant properties such as Lax pair and infinite conservation laws are also obtained, and analyzed the application in the initial–boundary value problem. This has not been studied in existing literatures.

Study of self-oscillations of a distributed rotor from a material with hereditarily elastic properties

Self-oscillatory solutions of dynamic equations of transverse vibrations of a distributed rotating ideally balanced rotor (shaft) on an isotropic support made of a material with nonlinear hereditary properties are studied. The derivation of dynamic equations (mathematical model), which are a system of two nonlinear partial differential equations with an infinite (integral) delay of the time argument, is given. A mathematical formulation of the initial–boundary value problem is formulated, its solvability and correctness are proven. The stability of the zero solution (stable rotation), the mechanisms of loss of stability, and the self-oscillatory solutions bifurcating in this case are studied. The possibility of bifurcation of periodic solutions (direct asynchronous precessions) and invariant two-dimensional tori (beat modes) is shown. The stability of self-oscillating solutions is determined by the parameters of the problem under consideration. The method of central manifolds of distributed systems and the theory of bifurcations were used as a research method. Asymptotic formulas are constructed for self-oscillating solutions.

Extremization to fine tune physics informed neural networks for solving boundary value problems

We propose a novel method for fast and accurate training of physics-informed neural networks (PINNs) to find solutions to boundary value problems (BVPs) and initial boundary value problems (IBVPs). By combining the methods of training deep neural networks (DNNs) and Extreme Learning Machines (ELMs), we develop a model which has the expressivity of DNNs with the fine-tuning ability of ELMs. We showcase the superiority of our proposed method by solving several BVPs and IBVPs which include linear and non-linear ordinary differential equations (ODEs), partial differential equations (PDEs) and coupled PDEs. The examples we consider include a stiff coupled ODE system where traditional numerical methods fail, a 3+1D non-linear PDE, Kovasznay flow and Taylor–Green vortex solutions to incompressible Navier–Stokes equations and pure advection solution of 1+1 D compressible Euler equation.The Theory of Functional Connections (TFC) is used to exactly impose initial and boundary conditions (IBCs) of (I)BVPs on PINNs. We propose a modification to the TFC framework named Reduced TFC and show a significant improvement in the training and inference time of PINNs compared to IBCs imposed using TFC. Furthermore, Reduced TFC is shown to be able to generalize to more complex boundary geometries which is not possible with TFC. We also introduce a method of applying boundary conditions at infinity for BVPs and numerically solve the pure advection in 1+1 D Euler equations using these boundary conditions.

On the initial boundary values problem for a mixture of two Cosserat bodies with voids

In this study it is approached a linear model for the mixture of two Cosserat bodies having pores. It is formulated the mixed problem with initial and boundary data in this context. The main goal is to show that the coefficients that realize the coupling of the elastic effect with the one due to voids can vary, without the mixture being essentially affected. In a more precise formulation, this means that a small variation of the coefficients in the constitutive equations of the two continua causes only a small variation of the solutions of the corresponding mixed problems, that is, the continuous dependence of the solutions in relation to these coefficients is ensured. The considered mixture model is consistent because all estimates, specific to continuous dependence, are made based on rigorous mathematical relationships.

DECAY RATES AND INITIAL VALUES FOR TIME-FRACTIONAL DIFFUSION-WAVE EQUATIONS

We consider a solution u(·,t) to an initial boundary value problem for time-fractional diffusion-wave equation with the order α ∈ (0,2)\ {1} where t is a time variable. We first prove that a suitable norm of u(·,t) is bounded by the rate t −α for 0 α 1 and t 1−α for 1 α 2 for all large t &gt; 0. Second, we characterize initial values in the cases where the decay rates are faster than the above critical exponents. Differently from the classical diffusion equation α = 1, the decay rate can keep some local characterization of initial values. The proof is based on the eigenfunction expansions of solutions and the asymptotic expansions of the Mittag-Leffler functions for large time.

Some qualitative results in hyperbolic two-temperature generalized thermoelasticity

This study aims to establish the convolutional-type variational and reciprocity theorems within the framework of the hyperbolic two-temperature generalized thermoelasticity theory for an isotropic thermoelastic material, with the help of alternate formulation of the mixed boundary initial value problem, in which initial conditions are combined with field equations (using the Laplace transform). The convolutional-type variational principle adapts readily to numerical solutions based on the Ritz method and is useful in the finite element method. The reciprocity theorem is helpful in the theoretical development of boundary and finite element methods. The current effort can be valuable for the problem of coupling effects of thermal and mechanical fields, especially in geophysics and mining.

Asymptotics of the Solution of the Initial Boundary Value Problem for the One-Dimensional Klein–Gordon Equation with Variable Coefficients

Asymptotics of the Solution of the Initial Boundary Value Problem for the One-Dimensional Klein–Gordon Equation with Variable Coefficients