Space-dependent Coefficient Research Articles

Inverse problem to identify a space-dependent diffusivity coefficient in a generalized subdiffusion equation from final data

Inverse problem to identify a space-dependent diffusivity coefficient in a generalized subdiffusion equation from final data

Observer-based output feedback control design for a fractional ODE and a fractional PDE cascaded system

This paper considers a new bi-directional cascaded system of a fractional ordinary differential equation (FODE) and a fractional partial differential equation (FPDE) which interacts at an intermediate point. The space-dependent coefficients, interaction between the FODE and FPDE at an intermediate point and the presence of fractional calculus makes the FODE–FPDE cascaded system, representative. In this note, we first apply an invertible integral transformation to convert the system into a FODE–FPDE coupled system, as the target system, which is Mittag–Leffler stable. Using the backstepping method and under some assumptions of the space-dependent coefficients, we work out the kernel functions in the transformation and we design a boundary controller. Also, by the invertibility of the transformation, we show the Mittag–Leffler stability of the closed-loop system via the Lyapunov approach. Second, we propose an observer for which we prove that it can well estimate the original cascaded system. Then, we design an output feedback boundary control law and show that the closed-loop system is Mittag–Leffler stable under the designed output feedback control law. Finally, we present some numerical illustrations to show the correctness of the theoretical results.

Space-dependent heat source determination problem with nonlocal periodic boundary conditions

The purpose of this paper is to identify the space-dependent heat source coefficient numerically, for the first time, in the third-order pseudo-parabolic equation with initial and nonlocal periodic boundary conditions from nonlocal integral observation. This problem emerges significantly in the modelling of numerous phenomena in physics, engineering, mechanics and science. Although, the inverse source problem considered in this article is ill-posed by being sensitive to noise but has a unique solution. For the numerical realization, we apply the finite difference method (FDM) for discretizing the forward problem and the Tikhonov regularization for finding a stable and accurate solution. The resulting nonlinear minimization problem is solved computationally using the MATLAB subroutine lsqnonlin. Numerical results presented for two benchmark test examples with linear and nonlinear source functions show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data. Furthermore, the von Neumann stability analysis is also discussed.

Second-Order and Nonuniform Time-Stepping Schemes for Time Fractional Evolution Equations with Time–Space Dependent Coefficients

The numerical analysis of time fractional evolution equations with the second-order elliptic operator including general time–space dependent variable coefficients is challenging, especially when the classical weak initial singularities are taken into account. In this paper, we introduce a concise technique to construct efficient time-stepping schemes with variable time step sizes for two-dimensional time fractional sub-diffusion and diffusion-wave equations with general time–space dependent variable coefficients. By means of the novel technique, the nonuniform Alikhanov type schemes are constructed and analyzed for the sub-diffusion and diffusion-wave problems. For the diffusion-wave problem, our scheme is constructed by employing the recently established symmetric fractional-order reduction method. The unconditional stability of proposed schemes is rigorously discussed under mild assumptions on variable coefficients and, based on reasonable regularity assumptions and weak time mesh restrictions, the second-order convergence is obtained with respect to discrete \(H^1\)-norm. Numerical experiments are given to demonstrate the theoretical statements.

NON-FICKIAN DIFFUSION IN TIME–SPACE FLUCTUATING DIFFUSIVITY LANDSCAPES: FROM SUPERFAST TO ULTRASLOW

In this study, the local time and space structural derivative diffusion model is introduced to describe non-Fickian diffusion in heterogeneous media from ultraslow to superfast. Its equivalent heterogeneous diffusion process (HDP) model with time and space-dependent diffusion coefficient is also given. The connection and difference between the proposed model and the existing HDP models are compared. Different types of diffusion processes can be captured by selecting proper structural functions in the structural derivative model, which is determined based on the concentration of tracers, mean squared displacement (MSD) and diffusion coefficient of the heterogeneous media. The dynamics of the free cooling granular gases with constant restitution coefficient and velocity-dependent restitution coefficient is investigated to test the proposed model. The results show that the diffusion law of the granular particles is not uniform for the short and long-time scales, but one uniform diffusion law derived from the proposed model can well fit the data in different time scales. The patterns of the time-dependent diffusion coefficient are also simulated to deeply understand the complicated non-Fickian diffusion in complex system.

Zig-zag, bright, short and long solitons formation in inhomogeneous ferromagnetic materials.Kraenkel-Manna-Merle equation with space dependent coefficients

The ferromagnetism induced by an external magnetic field (EMF), in (3+1) dimensions, is governed by Kraenkel–Manna–Merle equation (KMME). A (1+1) dimensionalmodel equation was derived in the literature. Here, this later model equation is considered with space-dependent coefficients, which stands to inhomogeneous ferromagnetism. Exact solutions are obtained by using the extended unified method. The numerical results of the solutions are represented in graphs. They show multiple geometric waves structures, which are zig-zag bright, short and long, solitons in ferromagnetic materials (FMMs). It is found that when the EMF exhibits short (or ultra short), the ferromagnitization exhibits long (or short) solitons. It is also observed that that the ferromagnetism exhibits highly oscillatory waves, with increasing damping parameter. The novelty of this work stems from the fact that the problem of in-homogeneous ferromagentism, studied here is completely new. Further, the zig-zag solitons structures, found here, are so.

A class of integro-differential Fokker–Planck equations with space-dependent coefficients

A class of integro-differential Fokker–Planck equations with different kernels and generic space-dependent coefficients is investigated. Two exact solutions are obtained: one of the exact solutions for space-dependent coefficients is obtained in Laplace space, whereas the second one with different space-dependent coefficients is obtained in terms of the Hermite polynomials. The probability distribution function and n-moment are analyzed for a power-law coefficient and different kernels.

Acoustics of Fractal Porous Material and Fractional Calculus

In this paper, we present a fractal (self-similar) model of acoustic propagation in a porous material with a rigid structure. The fractal medium is modeled as a continuous medium of non-integer spatial dimension. The basic equations of acoustics in a fractal porous material are written. In this model, the fluid space is considered as fractal while the solid matrix is non-fractal. The fluid–structure interactions are described by fractional operators in the time domain. The resulting propagation equation contains fractional derivative terms and space-dependent coefficients. The fractional wave equation is solved analytically in the time domain, and the reflection and transmission operators are calculated for a slab of fractal porous material. Expressions for the responses of the fractal porous medium (reflection and transmission) to an acoustic excitation show that it is possible to deduce these responses from those obtained for a non-fractal porous medium, only by replacing the thickness of the non-fractal material by an effective thickness depending on the fractal dimension of the material. This result shows us that, thanks to the fractal dimension, we can increase (sometimes by a ratio of 50) and decrease the equivalent thickness of the fractal material. The wavefront speed of the fractal porous material depends on the fractal dimension and admits several supersonic values. These results open a scientific challenge for the creation of new acoustic fractal materials, such as metamaterials with very specific acoustic properties.

Transient study of a solar pond under heat extraction from non-convective and lower convective zones considering finite effectiveness of exchangers

The transient performance of a salt gradient solar pond, which experiences heat extraction from both gradient and storage regions has been studied for annual performance via accounting for the overall heat transfer coefficient across heat exchangers. The proposed method after reduced complexities is satisfactorily validated with theory and experiment-based results of pertinent literature. The novelty of the work lies in the fact that, the temperature drop across both non-convective and lower-convective zone exchanger surfaces has been accounted by using a local time and space dependent heat transfer coefficient. Here, it takes into the consideration free convection from the pond to the exchanger surface, and forced convection from the exchanger surface to flowing working fluid. Further, temperature variation of thermo-fluidic parameters of water involving, viscosity, density and thermal conductivity is considered. Detailed numerical investigation reveals that by neglecting this coefficient which is assumed in conventional studies, it can lead to significant errors in the prediction of transient temperature profiles in the pond and the exchangers. Calculations reveal an error of about +69% in the annual extraction efficiency and about −9% in entropy production if the conventional assumption is used as opposed to realistic technique presented herein. Interestingly, the present study reveals that there is an optimum radius of exchanger pipe in each of non-convective and lower convective zones that maximises the annual extraction efficiency. This work presents a useful analysis to assess multi zone extraction from solar ponds under transient state in a more practical and accurate manner.

Active noise-driven particles under space-dependent friction in one dimension.

We study a Langevin equation describing the stochastic motion of a particle in one dimension with coordinate x, which is simultaneously exposed to a space-dependent friction coefficient γ(x), a confining potential U(x) and nonequilibrium (i.e., active) noise. Specifically, we consider frictions γ(x)=γ_{0}+γ_{1}|x|^{p} and potentials U(x)∝|x|^{n} with exponents p=1,2 and n=0,1,2. We provide analytical and numerical results for the particle dynamics for short times and the stationary probability density functions (PDFs) for long times. The short-time behavior displays diffusive and ballistic regimes while the stationary PDFs display unique characteristic features depending on the exponent values (p,n). The PDFs interpolate between Laplacian, Gaussian, and bimodal distributions, whereby a change between these different behaviors can be achieved by a tuning of the friction strengths ratio γ_{0}/γ_{1}. Our model is relevant for molecular motors moving on a one-dimensional track and can also be realized for confined self-propelled colloidal particles.

A class of integro-differential Fokker-Planck equations with space-dependent coefficient

Exact solution for the probability distribution function (PDF) in Laplace space is obtained for a class of integro-differential Fokker-Planck equations with different kernels and generic space-dependent diffusion coefficient. Besides, exact solutions for the PDF and n-moment are also obtained for different kernels and a power-law diffusion coefficient; they are analyzed and compared for different kernels.

Reconstruction of a Space-Dependent Coefficient in a Linear Benjamin–Bona–Mahony Type Equation

Abstract In this paper, we consider the problem of reconstructing a space-dependent coefficient in a linear Benjamin–Bona–Mahony (BBM)-type equation from a single measurement of its solution at a given time. We analyze the well-posedness of the forward initial-boundary value problem and characterize the inverse problem as a constrained optimization one. Our objective consists on reconstructing the variable coefficient in the BBM equation by minimizing an appropriate regularized Tikhonov-type functional constrained by the BBM equation. The well-posedness of the forward problem is studied and approximated numerically by combining a finite-element strategy for spatial discretization using the Python-FEniCS package, together with a second-order implicit scheme for time stepping. The minimization process of the Tikhonov-regularization adopted is performed by using an iterative L-BFGS-B quasi-Newton algorithm as described for instance by Byrd et al. (1995) and Zhu et al. (1997). Numerical simulations are presented to demonstrate the robustness of the proposed method with noisy data. The local stability and uniqueness of the solution to the constrained optimization problem for a fixed value of the regularization parameter are also proved and illustrated numerically.

On the wave equation with multiplicities and space-dependent irregular coefficients

In this paper we study the well-posedness of the Cauchy problem for a wave equation with multiplicities and space-dependent irregular coefficients. As in Garetto and Ruzhansky [Arch. Ration. Mech. Anal. 217 (2015), pp. 113–154], in order to give a meaningful notion of solution, we employ the notion of very weak solution, which construction is based on a parameter dependent regularisation of the coefficients via mollifiers. We prove that, even with distributional coefficients, a very weak solution exists for our Cauchy problem and it converges to the classical one when the coefficients are smooth. The dependence on the mollifiers of very weak solutions is investigated at the end of the paper in some instructive examples.

Boundary Mittag-Leffler stabilization of coupled time fractional order reaction–advection–diffusion systems with non-constant coefficients

Abstract This paper is concerned with boundary control for a class of coupled time fractional order reaction–advection–diffusion (FRAD) systems with non-constant coefficients (space-dependent coefficients) by state feedback. Partial differential equation (PDE) backstepping makes available to stabilize coupled time FRAD systems modeled by fractional PDEs. With boundary controller design and discussion on well-posedness of control kernel equations, the Mittag-Leffler stability of the closed-loop system is analyzed theoretically by the fractional Lyapunov method. A numerical scheme is constructed for coupled FRAD system to simulate numerical examples when the kernel equations have not the explicit solution. Comments on robustness to perturbation parameters in system coefficients are finally stated.

On the generalized nonlinear Camassa–Holm equation

In this paper, a generalized nonlinear Camassa–Holm equation with time- and space-dependent coefficients is considered. We show that the control of the higher order dispersive term is possible by using an adequate weight function to define the energy. The existence and uniqueness of solutions are obtained via a standard Picard iterative method, so that there is no loss of regularity of the solution with respect to the initial condition in some appropriate Sobolev space.

Stability for a Formally Determined Inverse Problem for a Hyperbolic PDE with Space and Time Dependent Coefficients

We prove stability for a formally determined inverse problem for a hyperbolic PDE where the coefficients depend on space and time variables. The hyperbolic operator has constant wave speed and we study the recovery of zeroth order and first order coefficients and the space dimension can be one or higher. We use a modification of the Bukhgeim-Klibanov method to obtain our results.

On the identification of the nonlinearity parameter in the Westervelt equation from boundary measurements

<p style='text-indent:20px;'>We consider an undetermined coefficient inverse problem for a nonlinear partial differential equation occurring in high intensity ultrasound propagation as used in acoustic tomography. In particular, we investigate the recovery of the nonlinearity coefficient commonly labeled as <inline-formula><tex-math id="M1">\begin{document}$ B/A $\end{document}</tex-math></inline-formula> in the literature which is part of a space dependent coefficient <inline-formula><tex-math id="M2">\begin{document}$ \kappa $\end{document}</tex-math></inline-formula> in the Westervelt equation governing nonlinear acoustics. Corresponding to the typical measurement setup, the overposed data consists of time trace measurements on some zero or one dimensional set <inline-formula><tex-math id="M3">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula> representing the receiving transducer array. After an analysis of the map from <inline-formula><tex-math id="M4">\begin{document}$ \kappa $\end{document}</tex-math></inline-formula> to the overposed data, we show injectivity of its linearisation and use this as motivation for several iterative schemes to recover <inline-formula><tex-math id="M5">\begin{document}$ \kappa $\end{document}</tex-math></inline-formula>. Numerical simulations will also be shown to illustrate the efficiency of the methods.</p>

An Analytical Solution of Spatially Dependent One-Dimensional Advection-Dispersion Equation for a Varying Pulse Type Input Source

In present study, solution of advection-dispersion equation is obtained to determine concentration distribution of solute introduced from a varying pulse type point source in one-dimensional heterogeneous semi-infinite porous medium. Heterogeneity of the medium gives rise to space dependent groundwater velocity, dispersion coefficient and retardation factor. Groundwater velocity is some exponent ξ to a linear function of space. The dispersion and retardation factor are also exponents of same linear function with exponents (ξ+1) and (ξ-1), respectively, where ξ takes the value 0 or 1. At one end of the domain, a time dependent varying nature source, which involves step-size increasing function of time, acts along the flow up to a certain time the n eliminated while concentration gradient is considered zero at the other end of the domain. Initially, medium is uniformly polluted. Firstly, the advection-dispersion equation is reduced into constant coefficients by using certain transformations and then Laplace Integral Transformation Technique is utilized to get the solution. The obtained result is illustrated with numerical examples to study the effect of various parameters.

Heterogeneous diffusion processes and nonergodicity with Gaussian colored noise in layered diffusivity landscapes.

Heterogeneous diffusion processes (HDPs) with space-dependent diffusion coefficients D(x) are found in a number of real-world systems, such as for diffusion of macromolecules or submicron tracers in biological cells. Here, we examine HDPs in quenched-disorder systems with Gaussian colored noise (GCN) characterized by a diffusion coefficient with a power-law dependence on the particle position and with a spatially random scaling exponent. Typically, D(x) is considered to be centerd at the origin and the entire x axis is characterized by a single scaling exponent α. In this work we consider a spatially random scenario: in periodic intervals ("layers") in space D(x) is centerd to the midpoint of each interval. In each interval the scaling exponent α is randomly chosen from a Gaussian distribution. The effects of the variation of the scaling exponents, the periodicity of the domains ("layer thickness") of the diffusion coefficient in this stratified system, and the correlation time of the GCN are analyzed numerically in detail. We discuss the regimes of superdiffusion, subdiffusion, and normal diffusion realisable in this system. We observe and quantify the domains where nonergodic and non-Gaussian behaviors emerge in this system. Our results provide new insights into the understanding of weak ergodicity breaking for HDPs driven by colored noise, with potential applications in quenched layered systems, typical model systems for diffusion in biological cells and tissues, as well as for diffusion in geophysical systems.

Observation and stabilisation of coupled time‐fractional reaction–advection–diffusion systems with spatially‐varying coefficients

This study solves the problem of observer-based output feedback stabilisation of a coupled time-fractional reaction–advection–diffusion system with space-dependent coefficients. In light of the backstepping approach, the Mittag–Leffler convergent observer is derived to enable stabilisation by output feedback. By the fractional Lyapunov method, the Mittag–Leffler stability of the estimation error system is then established. Using the separation principle yields a stabilising output feedback controller, with which the closed-loop stability is proved. Finally, the feasibility of the proposed approach is tested by fractional numerical examples when the kernel matrix equation has not the explicit solution.