Initial-boundary Value Problem Research Articles

Inverse Problem for the Moisture Transfer Equation: Development of a Method for Finding the Unknown Parameter and Proof of the Convergence of the Iterative Process

This study develops a mathematical model for soil moisture diffusion, addressing the inverse problem of determining both the diffusion coefficient and the variation coefficient in a nonlinear moisture transfer equation. The model incorporates specific boundary and initial conditions and utilizes experimentally measured moisture values at a boundary point as input data. An iterative method, based on an explicit gradient scheme, is introduced to estimate the soil parameters. The initial boundary value problem is discretized, leading to a difference analog and the formulation of a conjugate difference problem. Iterative formulas for calculating the unknown parameters are derived, with a priori estimates ensuring the convergence of the iterative process. Additionally, the research establishes the convergence of the numerical model itself, providing a rigorous foundation for the proposed approach. The study also emphasizes symmetry in moisture calculations, ensuring consistency regardless of the calculation direction (from right to left or left to right) and confirming that moisture distribution remains symmetric within specified intervals. This preservation of symmetry enhances the model’s robustness and accuracy in parameter estimation. The numerical simulations were successfully conducted over a 7-day period, demonstrating the model’s reliability. The discrepancy between the numerical predictions and experimental observations remained within the margin of measurement error, confirming the model’s accuracy.

Nonlinear evolution of vortical disturbances entrained in the entrance region of a circular pipe

The nonlinear evolution of free-stream vortical disturbances entrained in the entrance region of a circular pipe is investigated using asymptotic and numerical methods. Attention is focused on the low-frequency disturbances that induce streamwise elongated structures. A pair of vortical modes with opposite azimuthal wavenumbers is used to model the free-stream disturbances. Their amplitude is assumed to be intense enough for nonlinear interactions to occur inside the pipe. The formation and evolution of the perturbation flow are described by the nonlinear unsteady boundary-region equations in the cylindrical coordinate system, derived and solved herein for the first time. Matched asymptotic expansions are employed to construct appropriate initial conditions and the initial–boundary value problem is solved numerically by a marching procedure in the streamwise direction. Numerical results show the stabilising effect of nonlinearity on the intense algebraic growth of the disturbances and an increase of the wall-shear stress due to the nonlinear interactions. A parametric study is carried out to evince the effect of the Reynolds number, the streamwise and azimuthal wavelengths, and the radial length scale of the inlet disturbance on the nonlinear flow evolution. Elongated pipe-entrance nonlinear structures (EPENS) occupying the whole pipe cross-section are discovered. EPENS with $h$ -fold rotational symmetry comprise $h$ high-speed streaks positioned near the wall, and $h$ low-speed streaks centred around the pipe core. These distinct structures display a striking resemblance to nonlinear travelling waves found numerically and observed experimentally in fully developed pipe flow. Good agreement of our mean-flow and root mean square data with experimental measurements is obtained.

ON WELL-POSED BOUNDARY CONDITIONS AND ENERGY STABLE FINITE-VOLUME METHOD FOR THE LINEAR SHALLOW WATER WAVE EQUATION

Abstract We derive and analyse well-posed boundary conditions for the linear shallow water wave equation. The analysis is based on the energy method and it identifies the number, location and form of the boundary conditions so that the initial boundary value problem is well-posed. A finite-volume method is developed based on the summation-by-parts framework with the boundary conditions implemented weakly using penalties. Stability is proven by deriving a discrete energy estimate analogous to the continuous estimate. The continuous and discrete analysis covers all flow regimes. Numerical experiments are presented verifying the analysis.

Blow Up of Solutions to Wave Equations with Combined Logarithmic and Power-Type Nonlinearities

In this paper, we study the initial boundary value problem for wave equations with combined logarithmic and power-type nonlinearities. For arbitrary initial energy, we prove a necessary and sufficient condition for blow up at infinity of the global weak solutions. In addition, we derive a growth estimate for the blowing up global solutions.

Thermostatted Kinetic Theory Structures in Biophysics: Generalizations and Perspectives

The mathematical modeling of multicellular systems is an important branch of biophysics, which focuses on how the system properties emerge from the elementary interaction between the constituent elements. Recently, mathematical structures have been proposed within the thermostatted kinetic theory for the modeling of complex living systems and have been profitably employed for the modeling of various complex biological systems at the cellular scale. This paper deals with a class of generalized thermostatted kinetic theory frameworks that can stand in as background paradigms for the derivation of specific models in biophysics. Specifically, the fundamental homogeneous thermostatted kinetic theory structures of the recent literature are recovered and generalized in order to take into consideration further phenomena in biology. The generalizations concern the conservative, the nonconservative, and the mutative interactions between the inner system and the outer environment. In order to sustain the strength of the new structures, some specific models of the literature are reset into the style of the new frameworks of the thermostatted kinetic theory. The selected models deal with breast cancer, genetic mutations, immune system response, and skin fibrosis. Future research directions from the theoretical and modeling viewpoints are discussed in the whole paper and are mainly devoted to the well-posedness in the Hadamard sense of the related initial boundary value problems, to the spatial–velocity dynamics and to the derivation of macroscopic-scale dynamics.

Downwind and upwind approximations for primal and dual problems of elasto-plasticity with Prandtl–Reuss type material laws

Modern adaptive finite element (FE) algorithms for solution of initial-boundary-value-problems (IBVP) employ goal-oriented error measures in order to assess the quality of computational results for the physical event under investigation. However, traditional time-stepping algorithms for solution of the corresponding dual problem run backwards-in-time, which due to additional storage requirements might become a serious drawback when an extensive number of time steps for the FEM simulation arises. In this paper, we take advantage of an end-boundary-value-problem (EBVP) associated to IBVP, with corresponding dual-problem running forwards in time. In order to obtain a unified framework for numerical approximation of primal and dual weak forms for both, IBVP and EBVP, respectively, we apply the concept of downwind and upwind approximations not only to the trial functions but also to the test functions. This results into eight different integration schemes. On this basis, as a main result of this contribution, a time-stepping algorithm is obtained, which runs forwards-in-time for the dual problem and therefore avoids the additional storage requirements of the traditional backwards-in-time stepping procedures. The presented algorithm is numerically tested and validated for a CT-specimen for elastic and elasto-plastic behavior, where the constitutive equations are written in Prandtl–Reuss type format.

Solvability of initial boundary value problems for a quasihydrodynamic system of equations in the case of a weakly compressible fluid

Subject of research: solvability of the first initial-boundary value problem for a quasi-hydrodynamic system of equations in the case of a weakly compressible fluid. Method of research: the proof is based on the Galerkin method using a priori estimates. Main results of research: the existence theorem for generalized solutions is proven.

Heat equation for Sturm–Liouville operator with singular propagation and potential

Abstract This article considers the initial boundary value problem for the heat equation with the time-dependent Sturm–Liouville operator with singular potentials. To obtain a solution by the method of separation of variables, the problem is reduced to the problem of eigenvalues of the Sturm–Liouville operator. Further on, the solution to the initial boundary value problem is constructed in the form of a Fourier series expansion. A heterogeneous case is also considered. Finally, we establish the well-posedness of the equation in the case when the potential and initial data are distributions, also for singular time-dependent coefficients.

Nonlinear mechanics of phase-change-induced accretion

In this paper, we formulate a continuum theory of solidification within the context of finite-strain coupled thermoelasticity. We aim to fill a gap in the existing literature, as the existing studies on solidification typically decouple the thermal problem (the classical Stefan’s problem) from the elasticity problem, and often limit themselves to linear elasticity with small strains. Treating solidification as an accretion problem, with the growth velocity correlated with the jump in the heat flux across the boundary, it presents an initial boundary-value problem (IBVP) over a domain whose boundary location is a priori unknown. This IBVP is solved numerically for the specific example of radially inward solidification in a spherical container. Several parametric studies are conducted to compare the numerical results with the rigid cases in the literature and gain insights into the role of elastic deformations in solidification.

Chaos of the initial and boundary value problems for the reaction-diffusion equations

This paper investigates an initial and boundary value problem for the reaction-diffusion equations, which can be considered as a linearized form of the advective Fisher-KPP equations. It is demonstrated that the initial and boundary value problem is chaotic when the three parameters of the reaction-diffusion equation vary above a specific surface. However, stable solutions are obtained both on and below this surface within a particular subset of initial values. The chaos and stability of the nonhomogeneous initial boundary value problem are further studied. Finally, some numerical examples are provided to illustrate the validity of the obtained results.

On the quadratic closeness of eigenfunctions of «unperturbed» and «perturbed» differentiation operators on an interval

The article considers the eigenvalue problem of the differentiation operator, when the spectral param- eter is also present in the boundary condition with an integral perturbation, where the integrand has the property of limited variation and has a value of unity at the ends of the segment [-1, 1]. The derivatives with respect to the time variable included in the boundary condition naturally arise when solving (by the Fourier method) initial boundary value problems for evolution equations. The conjugate operator is constructed. It is shown that the spectral questions of the conjugate operator have a similar structure. The characteristic determinant of the original direct spectral problem with an integral perturbation of the boundary condition and for the eigenvalue problem of a loaded first-order differential equation on a segment with a periodic boundary condition, which is an entire analytical function of the spectral parameter, is constructed. Based on the formula of the characteristic determinant, conclusions are drawn about the asymptotic behavior of the spectrum of the original «perturbed» differentiation operator and the loaded first-order differential equation on the segment. A special feature of the operator under consideration is the non-self-adjointness of the operator in L2(–1,1). The quadratic proximity of the systems of eigen functions of the «unperturbed» and «perturbed» differentiation operators and the Riesz basis property of these systems are proved. In this case, the system of eigenfunctions of the «perturbed» operator is not orthonormal.

Дослідження процесів самозаймання у насипі з прямокутним перерізом методами Роте та двобічних наближень

Self-ignition of a stockpile of material (coal, peat, grain) occurs as a result of the accumulation of heat released by an exothermic reaction of oxidation, which makes it possible to consider the stockpile as a body with an internal heat source. The research of self-ignition processes using mathematical modeling methods leads to the need to find a solution to the initial boundary value problem for a two-dimensional semilinear heat conduction equation. This cannot always be done analytically, so it makes sense to use numerical analysis methods. The aim of this article is to apply Rothe’s method in combination with the method of two-sided approximations based on the use of the Green's function to find the solution of the initial boundary value problem for the two-dimensional semilinear heat conduction equation that arises during the mathematical modeling of self-ignition processes of an stockpile of bulk material of cylindrical shape with a rectangular base. To achieve the goal, after the discretization of the heat conduction equation by the time variable, a sequence of boundary value problems was obtained by the Rothe’s method, each of which was reduced to the Hammerstein equation. For this nonlinear operator equation, an iterative process of the two-way method was constructed with its stopping condition obtained through a posteriori error estimation. The power of the internal heat source was approximated using an exponential dependence. As a result of the computational experiment, a sequence of approximate solutions was obtained. The graphs of heat maps constructed for them made it possible to examine over time the course of the self-ignition process in the cross-section of a stockpile of cylindrical shape with a rectangular base and to identify areas of heat accumulation

Non-homogeneous initial boundary value problems and small data scattering of the Hirota equations posed on the half line

Non-homogeneous initial boundary value problems and small data scattering of the Hirota equations posed on the half line

Inverse coefficient problem for a partial differential equation with multi-term orders fractional Riemann–Liouville derivatives

This work studies direct initial boundary value and inverse coefficient determination problems for a one-dimensional partial differential equation with multi-term orders fractional Riemann–Liouville derivatives. The unique solvability of the direct problem is investigated and a priori estimates for its solution are obtained in weighted spaces, which will be used for studying the inverse problem. Then, the inverse problem is equivalently reduced to a nonlinear integral equation. The fixed-point principle is used to prove the unique solvability of this equation.

A Two-Dimensional Nonlocal Fractional Parabolic Initial Boundary Value Problem

In this paper, we investigate a two-dimensional singular fractional-order parabolic partial differential equation in the Caputo sense. The partial differential equation is supplemented with Dirichlet and weighted integral boundary conditions. By employing a functional analysis method based on operator theory techniques, we prove the existence and uniqueness of the solution to the posed nonlocal initial boundary value problem. More precisely, we establish an a priori bound for the solution from which we deduce the uniqueness of the solution. For proof of its existence, we use various density arguments.

Simulation of gas mixture dynamics in a pipeline network using explicit staggered-grid discretization

We develop an explicit staggered finite difference discretization scheme for simulating the transport of highly heterogeneous gas mixtures through pipeline networks. This study is motivated by the proposed blending of hydrogen into natural gas pipelines to reduce end use carbon emissions while using existing pipeline systems throughout their planned lifetimes. Our computational method accommodates an arbitrary number of constituent gases with very different physical properties that may be injected into a network with significant spatiotemporal variation. In this setting, the gas flow physics are highly location- and time- dependent, so that local composition and nodal mixing must be accounted for. The resulting conservation laws are formulated in terms of pressure, partial densities and flows, and volumetric and mass fractions of the constituents. We include non-ideal equations of state that employ linear approximations of gas compressibility factors, so that the pressure dynamics propagate locally according to a variable wave speed that depends on mixture composition and density. We derive compatibility relationships for network edge boundary values that are more complex than for a homogeneous gas. The simulation method is evaluated on initial boundary value problems for a single pipe and a small network, is cross-validated with a lumped element simulation, and used to demonstrate a local monitoring and control policy for maintaining allowable concentration levels.

Existence of global weak solutions and simulations to a Dirichlet problem for a generalized Swift–Hohenberg equation

In this paper, we shall investigate an initial–boundary value problem of a generalized Swift–Hohenberg model subject to homogeneous Dirichlet boundary conditions in two spatial dimensions. The model consists of a nonlinear term of the form ψ2Δ2ψ2 in the free energy functional, which is used to model the stability of fronts between hexagons and squares in pinning effect. We first prove the global-in-time existence and uniqueness of weak solutions to this initial–boundary value problem in the case with the parameter β<0, where we employ the energy method and make use of various techniques to derive delicate a priori estimates. At the end, a few numerical experiments of the model are also performed to study the competition between hexagons and squares.

General boundary conditions for a Boussinesq model with varying bathymetry

AbstractThis paper is devoted to the theoretical and numerical investigation of the initial boundary value problem for a system of equations used for the description of waves in coastal areas, namely, the Boussinesq–Abbott system in the presence of topography. We propose a procedure that allows one to handle very general linear or nonlinear boundary conditions. It consists in reducing the problem to a system of conservation laws with nonlocal fluxes and coupled to an ordinary differential equation. This reformulation is used to propose two hybrid finite volumes/finite differences schemes of first and second order, respectively. The possibility to use many kinds of boundary conditions is used to investigate numerically the asymptotic stability of the boundary conditions, which is an issue of practical relevance in coastal oceanography since asymptotically stable boundary conditions would allow one to reconstruct a wave field from the knowledge of the boundary data only, even if the initial data are not known.

On One Point Singular Nonlinear Initial Boundary Value Problem for a Fractional Integro-Differential Equation via Fixed Point Theory

In this article, we focus on examining the existence, uniqueness, and continuous dependence of solutions on initial data for a specific initial boundary value problem which mainly arises from one-dimensional quasi-static contact problems in nonlinear thermo-elasticity. This problem concerns a fractional nonlinear singular integro-differential equation of order θ∈[0,1]. The primary methodology involves the application of a fixed point theorem coupled with certain a priori bounds. The feasibility of solving this problem is established under the context of data related to a weighted Sobolev space. Furthermore, an additional result related to the regularity of the solution for the formulated problem is also presented.

Error analysis of an L2-type method on graded meshes for semilinear subdiffusion equations

A semilinear initial–boundary value problem with a Caputo time derivative of fractional order α∈(0,1) is considered, solutions of which typically exhibit a singular behaviour at an initial time. For an L2-type discretization of order 3−α, we give sharp pointwise-in-time error bounds on graded temporal meshes with arbitrary degree of grading.