Linear Initial Boundary Value Problems Research Articles

The Robin problems for the coupled system of reaction–diffusion equations

This article investigates the local well-posedness of Turing-type reaction–diffusion equations with Robin boundary conditions in the Sobolev space. Utilizing the Hadamard norm, we derive estimates for Fokas unified transform solutions for linear initial-boundary value problems subject to external forces. Subsequently, we demonstrate that the iteration map, defined by the unified transform solutions and incorporating nonlinearity instead of external forces, acts as a contraction map within an appropriate solution space. Our conclusive result is established through the application of the contraction mapping theorem.

Numerical treatments of nonlinear Burgers–Fisher equation via a combined approximation technique

A combined spectral matrix collocation strategy is presented to solve the time-dependent nonlinear Burgers–Fisher equation pertaining to various important physical mechanisms such as advection, diffusion, and logistic reaction. The nonlinearity of the model is first tackled by a time-marching scheme based on the Taylor formula. Hence in each time step, we solve a linear initial boundary value problem (IBVP) by using the spectral collocation technique based on novel Boole polynomials. Various numerical computations are carried out to indicate the pertinent features and testify the applicability of the presented combined technique. Comparisons are made between our results and the exact analytical solutions and some available numerical outcomes in the literature to show the validity of the method.

MODELING OF NONLINEAR MASS FLOWS DURING WATER SOFTENING IN A MECHANICAL FILTER

A mathematical model of water softening in a mechanical filter based on the balance relations for the mass components of the thermodynamical system is constructed. The initial-boundary value problem of convective diffusion accompanied by a chemical reaction is formulated for one-dimensional case of spatial coordinates and reduced to the equivalent system of integro-differential equations. The solution of this system is constructed by the method of successive iterations by decomposing the functions of the mass concentrations of impurity in water solution and in the body skeleton into series around the solutions of the homogeneous linear initial-boundary value problem. Based on the expression for the impurity concentration in the aqueous solution the mass flows of impurity are obtained and investigated. The nonlinearity of steady-state regime, which is caused by the transfer of mass simultaneously by diffusion and convection mechanisms, is shown.

Cauchy–Dirichlet Problem to Semilinear Multi-Term Fractional Differential Equations

In this paper, we analyze the well-posedness of the Cauchy–Dirichlet problem to an integro-differential equation on a multidimensional domain Ω⊂Rn in the unknown u=u(x,t), Dtν0(ϱ0u)−Dtν1(ϱ1u)−L1u−∫0tK(t−s)L2u(x,s)ds=f(x,t)+g(u),0<ν1<ν0<1, where Dtνi are the Caputo fractional derivatives, ϱi=ϱi(x,t) with ϱ0≥μ0>0, and Li are uniform elliptic operators with time-dependent smooth coefficients. The principal feature of this equation is related to the integro-differential operator Dtν0(ϱ0u)−Dtν1(ϱ1u), which (under certain assumption on the coefficients) can be rewritten in the form of a generalized fractional derivative with a non-positive kernel. A particular case of this equation describes oxygen delivery through capillaries to tissue. First, under proper requirements on the given data in the linear model and certain relations between ν0 and ν1, we derive a priori estimates of a solution in Sobolev–Slobodeckii spaces that gives rise to providing the Hölder regularity of the solution. Exploiting these estimates and constructing appropriate approximate solutions, we prove the global strong solvability to the corresponding linear initial-boundary value problem. Finally, obtaining a priori estimates in the fractional Hölder classes and assuming additional conditions on the coefficients ϱ0 and ϱ1 and the nonlinearity g(u), the global one-valued classical solvability to the nonlinear model is claimed with the continuation argument method.

Accurate analytical calculation of the rate coefficient for the diffusion-controlled reactions due to hyperbolic diffusion.

Using an approach based on the diffusion analog of the Cattaneo-Vernotte differential model, we find the exact analytical solution to the corresponding time-dependent linear hyperbolic initial boundary value problem, describing irreversible diffusion-controlled reactions under Smoluchowski's boundary condition on a spherical sink. By means of this solution, we extend exact analytical calculations for the time-dependent classical Smoluchowski rate coefficient to the case that includes the so-called inertial effects, occurring in the host media with finite relaxation times. We also present a brief survey of Smoluchowski's theory and its various subsequent refinements, including works devoted to the description of the short-time behavior of Brownian particles. In this paper, we managed to show that a known Rice's formula, commonly recognized earlier as an exact reaction rate coefficient for the case of hyperbolic diffusion, turned out to be only its approximation being a uniform upper bound of the exact value. Here, the obtained formula seems to be of great significance for bridging a known gap between an analytically estimated rate coefficient on the one hand and molecular dynamics simulations together with experimentally observed results for the short times regime on the other hand. A particular emphasis has been placed on the rigorous mathematical treatment and important properties of the relevant initial boundary value problems in parabolic and hyperbolic diffusion theories.

Arbitrary High-Order Unconditionally Stable Methods for Reaction-Diffusion Equations with inhomogeneous Boundary Condition via Deferred Correction

Abstract In this paper, we analyse full discretizations of an initial boundary value problem (IBVP) related to reaction-diffusion equations. To avoid possible order reduction, the IBVP is first transformed into an IBVP with homogeneous boundary conditions (IBVPHBC) via a lifting of inhomogeneous Dirichlet, Neumann or mixed Dirichlet–Neumann boundary conditions. The IBVPHBC is discretized in time via the deferred correction method for the implicit midpoint rule and leads to a time-stepping scheme of order 2 ⁢ p + 2 2p+2 of accuracy at the stage p = 0 , 1 , 2 , … p=0,1,2,\ldots of the correction. Each semi-discretized scheme results in a nonlinear elliptic equation for which the existence of a solution is proven using the Schaefer fixed point theorem. The elliptic equation corresponding to the stage 𝑝 of the correction is discretized by the Galerkin finite element method and gives a full discretization of the IBVPHBC. This fully discretized scheme is unconditionally stable with order 2 ⁢ p + 2 2p+2 of accuracy in time. The order of accuracy in space is equal to the degree of the finite element used when the family of meshes considered is shape-regular, while an increment of one order is proven for a quasi-uniform family of meshes. Numerical tests with a bistable reaction-diffusion equation having a strong stiffness ratio, a Fisher equation, a linear reaction-diffusion equation addressing order reduction and two linear IBVPs in two dimensions are performed and demonstrate the unconditional convergence of the method. The orders 2, 4, 6, 8 and 10 of accuracy in time are achieved. Except for some linear problems, the accuracy of DC methods is better than that of BDF methods of same order.

Pointwise long time behavior for the mixed damped nonlinear wave equation in

In this paper, we investigate the long time behavior of the solution for the nonlinear wave equation with frictional and visco-elastic damping terms in \begin{document}$ \mathbb{R}^n_+ $\end{document} . It is shown that for the long time, the frictional damped effect is dominated. The Green's functions for the linear initial boundary value problem can be described in terms of the fundamental solutions for the full space problem and reflected fundamental solutions coupled with the boundary operator. Using the Duhamel's principle, we get the pointwise long time behavior of the solution \begin{document}$ \partial_{{\bf{x}}}^{\alpha}u $\end{document} for \begin{document}$ |\alpha|\le 1 $\end{document} .

Fokas method for linear boundary value problems involving mixed spatial derivatives.

We obtain solution representation formulae for some linear initial boundary value problems posed on the half space that involve mixed spatial derivative terms via the unified transform method (UTM), also known as the Fokas method. We first implement the method on the second-order parabolic PDEs; in this case one can alternatively eliminate the mixed derivatives by a linear change of variables. Then, we employ the method to biharmonic problems, where it is not possible to eliminate the cross term via a linear change of variables. A basic ingredient of the UTM is the use of certain invariant maps. It is shown here that these maps are well defined provided that certain analyticity issues are appropriately addressed.

Rotationally-axisymmetric Motion of a Binary Mixture with a Flat Free Boundary at Small Marangoni Numbers

Rotationally-axisymmetric motion of a binary mixture with a flat free boundary at small Marangoni numbers is investigated. The problem is reduced to the inverse linear initial-boundary value problem for parabolic equations. Using Laplace transformation properties the exact analytical solution is obtained. It is shown that a stationary solution is the limiting one with the growth of time if there is a certain relationship between the temperature of the solid wall and the external temperature of the gas. If there is no connection, the convergence to the stationary solution is broken. Some examples of numerical reconstruction of the temperature, concentration and velocity fields are given, which confirm the theoretical conclusions

On the Asymptotic Behavior of the Conjugate Problem Describing a Creeping Axisymmetric Thermocapillary Motion

In this paper the conditions for the law of temperature behavior on a solid cylinder wall describes, under which the solution of a linear conjugate inverse initial-boundary value problem describing a two-layer axisymmetric creeping motion of viscous heat-conducting fluids tends to zero exponentially with increases of time

On the connection between solutions of initial boundary-value problems for a some class of integro-differential PDE and a linear hyperbolic equation

We consider the second initial boundary-value problem for a certain class of second-order integro-differential PDE with integral operator. The connection of its solution with the solution of the standard second linear initial boundary-value problem for the hyperbolic equation is shown. Thus, the nonlinear problem is reduced to a standard linear problem, whose numerical solution can be obtained, for example, by the Fourier method or Galerkin method. The article provides examples of five integro-differential equations for various integral operators as particular representatives of the class of integro-differential equations for a better understanding of the problem. The application of the main theorem to these examples is shown. Some simple natural requirement is imposed on the integral operator; so, in four cases out of five the problem’s solution satisfies some phase constraint. The form of these constraints is of particular interest for the further research.

Maximal regularity with weights for parabolic problems with inhomogeneous boundary conditions

In this paper, we establish weighted L^{q}–L^{p}-maximal regularity for linear vector-valued parabolic initial-boundary value problems with inhomogeneous boundary conditions of static type. The weights we consider are power weights in time and in space, and yield flexibility in the optimal regularity of the initial-boundary data and allow to avoid compatibility conditions at the boundary. The novelty of the followed approach is the use of weighted anisotropic mixed-norm Banach space-valued function spaces of Sobolev, Bessel potential, Triebel–Lizorkin and Besov type, whose trace theory is also subject of study.

On (Non) Applicability of a Mode-Truncation of a Damped Traveling String

This study investigates a linear homogeneous initial-boundary value problem for a traveling string under linear viscous damping. The string is assumed to be traveling with constant speed, while it is fixed at both ends. Physically, this problem represents the vertical (lateral) vibrations of damped axially moving materials. The axial belt speed is taken to be positive, constant and small in comparison with a wave speed, and the damping is also considered relatively small. A two timescale perturbation method together with the characteristic coordinate’s method will be employed to establish the approximate analytic solutions. The damped amplitude-response of the system will be computed under specific initial conditions. The obtained results are compared with the finite difference numerical technique for justification. It turned out that the introduced damping has a significant effect on the amplitude-response. Additionally, it is proven that the mode-truncation is applicable for the damped axially traveling string system on a timescale of order ε -1

Gaussian Process Modeling Using the Principle of Superposition

ABSTRACTPartial differential equation (PDE) models of physical systems with initial and boundary conditions are often solved numerically via a computer code called the simulator. To study the dependence of the solution on a functional input, the input is expressed as a linear combination of a finite number of basis functions, and the coefficients of the bases are varied. In such studies, Gaussian process (GP) emulators can be constructed to reduce the amount of simulations required from time-consuming simulators. For linear initial-boundary value problems (IBVPs) with functional inputs as additive terms in the PDE, initial conditions, or boundary conditions, the IBVP solution is theoretically a linear function of the coefficients conditional on all other inputs, which is a result called the principle of superposition. Since numerical errors cause deviation from linearity and nonlinear IBVPs are widely solved in practice, we generalize the result to account for nonlinearity. Based on this generalized result, we propose mean and covariance functions for building GP emulators that capture the approximate conditional linear effect of the coefficients. Numerical simulations demonstrate the substantial improvements in prediction performance achieved with the proposed emulator. Matlab codes for reproducing the results in this article are available in the online supplement.

An improved projection method

Strictly stable high-order accurate finite difference approximations are derived, for linear initial boundary value problems. The boundary closures are based on the diagonal-norm summation-by-parts framework and the boundary conditions are imposed using an improved projection method. The improved projection method removes some of the numerical issues with the previously derived (here referred to as traditional) projection method. The finite difference approximations lead to fully explicit ODE systems. The accuracy and stability properties are demonstrated for linear hyperbolic and parabolic problems in 1D and 2D. In particular we study the problem where initial and boundary data are inconsistent, that causes the traditional projection method to fail. Comparisons are made also with weak (penalty) enforcement of boundary conditions.

Exponential Quadrature Rules Without Order Reduction for Integrating Linear Initial Boundary Value Problems

In this paper a technique is suggested to integrate linear initial boundary value problems with exponential quadrature rules in such a way that the order in time is as high as possible. A thorough error analysis is given both for the classical approach of integrating the problem first in space and then in time and for doing it in the reverse order in a suitable manner. Time-dependent boundary conditions are considered with both approaches and full discretization formulas are given to implement the methods once the quadrature nodes have been chosen for the time integration and a particular (although very general) scheme is selected for the space discretization. Numerical experiments are shown which corroborate that, for example with the suggested technique, order $2s$ is obtained when choosing the $s$ nodes of the Gaussian quadrature rule.

Long time behavior for the visco-elastic damped wave equation in <inline-formula><tex-math id="M1">\begin{document}$\mathbb{R}^n_+$\end{document}</tex-math></inline-formula> and the boundary effect

In this paper, we investigate the existence and long time behavior of the solution for the nonlinear visco-elastic damped wave equation in \begin{document}$\mathbb{R}^n_+$\end{document} , provided that the initial data is sufficiently small. It is shown that for the long time, one can use the convected heat kernel to describe the hyperbolic wave transport structure and damped diffusive mechanism. The Green's function for the linear initial boundary value problem can be described in terms of the fundamental solution (for the full space problem) and reflected fundamental solution coupled with the boundary operator. Using the Duhamel's principle, we get the \begin{document}$ L^p $\end{document} decaying rate for the nonlinear solution \begin{document}$\partial_{{\bf x}}^{\alpha}u$\end{document} for \begin{document}$|\alpha|\le 1$\end{document} .

On Stable Wall Boundary Conditions for the Hermite Discretization of the Linearised Boltzmann Equation

We define certain criteria, using the characteristic decomposition of the boundary conditions and energy estimates, which a set of stable boundary conditions for a linear initial boundary value problem, involving a symmetric hyperbolic system, must satisfy. We first use these stability criteria to show the instability of the Maxwell boundary conditions proposed by Grad (Commun Pure Appl Math 2(4):331–407, 1949). We then recognise a special block structure of the moment equations which arises due to the recursion relations and the orthogonality of the Hermite polynomials; the block structure will help us in formulating stable boundary conditions for an arbitrary order Hermite discretization of the Boltzmann equation. The formulation of stable boundary conditions relies upon an Onsager matrix which will be constructed such that the newly proposed boundary conditions stay close to the Maxwell boundary conditions at least in the lower order moments.

Avoiding order reduction when integrating linear initial boundary value problems with exponential splitting methods

It is well known the order reduction phenomenon which arises when exponential methods are used to integrate time-dependent initial boundary value problems, so that the classical order of these methods is reduced. In particular, this subject has been recently studied for Lie-Trotter and Strang exponential splitting methods, and the order observed in practice has been exactly calculated. In this article, a technique is suggested to avoid that order reduction.We deal directly with nonhomogeneous time-dependent boundary conditions, without having to reduce the problem to the homogeneous ones. We give a thorough error analysis of the full discretization and justify why the computational cost of the technique is negligible in comparison with the rest of the calculations of the method. Some numerical results for dimension splittings are shown, which corroborate that much more accuracy is achieved.

Analysis of order reduction when integrating linear initial boundary value problems with Lawson methods

In this paper, a thorough analysis is given for the order which is observed when integrating evolutionary linear partial differential equations with Lawson methods. The analysis is performed under the general framework of C0-semigroups in Banach spaces and hence it can be applied to the numerical time integration of many initial boundary value problems which are described by linear partial differential equations. Conditions of regularity and annihilation at the boundary of these problems are then stated to justify the precise order which is observed, including fractional order of convergence.