Nonhomogeneous Diffusion Equation Research Articles

The Stefan Problem With Internal Heat Generation in Spherical Coordinates

Abstract A weakly time-dependent equation for the evolution of the solid–liquid interface in spherical coordinates, driven by internal heat generation, is derived for constant surface temperature boundary conditions. The derivation comes by making an assumption that the interface moves slowly compared to the changes in the temperature so that the technique of separation of variables may be applied for Stefan numbers less than one. Under this approximation, we can separate the nonhomogeneous heat diffusion equation into transient and steady-state terms, and then integrate to get the temperature relations. With the temperature equations in hand, the derivatives are inserted into the interface equation giving a first-order differential equation for the location of the solid–liquid interface as a function of time. The results are compared to a previously derived quasi-static solution and a numerical simulation generated using the method of catching of the front. This method allows for direct tracking of a moving boundary via the calculation of the time it takes to move from node to node in a discretized grid characteristic of classical finite difference methods. Plots of the interface evolution show excellent agreement between the three methods, especially for lower Stefan numbers. The quality of the approximation decreases as the Stefan number increases, but the model is more accurate than the previously studied quasi-static model. For the Stefan numbers St = 1.0 and 10.0, the weakly time-dependent solutions are in better agreement with the numerical results than the quasi-static solutions.

Regularization for a Sideways Problem of the Non-Homogeneous Fractional Diffusion Equation

In this article, we investigate a sideways problem of the non-homogeneous time-fractional diffusion equation, which is highly ill-posed. Such a model is obtained from the classical non-homogeneous sideways heat equation by replacing the first-order time derivative by the Caputo fractional derivative. We achieve the result of conditional stability under an a priori assumption. Two regularization strategies, based on the truncation of high frequency components, are constructed for solving the inverse problem in the presence of noisy data, and the corresponding error estimates are proved.

Regularization Error Analysis for a Sideways Problem of the 2D Nonhomogeneous Time-Fractional Diffusion Equation

The inverse and ill-posed problem of determining a solute concentration for the two-dimensional nonhomogeneous fractional diffusion equation is investigated. This model is much worse than its homogeneous counterpart as the source term appears. We propose a modified kernel regularization technique for the stable numerical reconstruction of the solution. The convergence estimates under both a priori and a posteriori parameter choice rules are proven.

Generalized Analytical Solutions of The Advection-Dispersion Equation with Variable Flow and Transport Coefficients

Demand has increased for analytical solutions to determine the velocities and dispersion coefficients that describe solute transport with spatial, temporal, or spatiotemporal variations encountered in the field. However, few analytical solutions have considered spatially, temporally, or spatiotemporally dependent dispersion coefficients and velocities. The proposed solutions consider eight cases of dispersion coefficients and velocities: both spatially dependent, both spatiotemporally dependent, both temporally dependent, spatiotemporally dependent dispersion coefficient with spatially dependent velocity, temporally dependent dispersion coefficient with constant velocity, both constant, spatially dependent dispersion coefficient with spatiotemporally dependent velocity, and constant dispersion coefficient with temporally dependent velocity. The spatial dependence is linear, while the temporal dependence may be exponential, asymptotical, or sinusoidal. An advection–dispersion equation with these variable coefficients was reduced to a non-homogeneous diffusion equation using the pertinent coordinate transform method. Then, solutions were obtained in an infinite medium using Green’s function. The proposed analytical solutions were validated against existing analytical solutions or against numerical solutions when analytical solutions were unavailable. In this study, we showed that the proposed analytical solutions could be applied for various spatiotemporal patterns of both velocity and the dispersion coefficient, shedding light on feasibility of the proposed solution under highly transient flow in heterogeneous porous medium.

The fractional Landweber method for identifying the space source term problem for time-space fractional diffusion equation

This paper is devoted to solve an inverse problem for identifying the source term of a time-fractional nonhomogeneous diffusion equation with a fractional Laplacian in a non-local boundary. Based on the expression of the solution for the direct problem, the inverse problem for searching the space source term is converted into solving the first kind of Fredholm integral equation. The conditional stability for the inverse source problem is investigated. The fractional Landweber method is used to deal with this inverse problem and the regularized solution is also obtained. Furthermore, the convergence rates for the regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori parameter choice rule. Several numerical examples are given to show the proposed method is efficient and stable.

Equivalence and finite time blow-up of solutions and interfaces for two nonlinear diffusion equations

In this work, we construct a transformation between the solutions to the following reaction-convection-diffusion equation∂tu=(um)xx+a(x)(um)x+b(x)um, posed for x∈R, t≥0 and m>1, where a, b are two continuous real functions, and the solutions to the nonhomogeneous diffusion equation of porous medium typef(y)∂τθ=(θm)yy, posed in the half-line y∈[0,∞) with τ≥0, m>1 and suitable density functions f(y). We apply this correspondence to the case of constant coefficients a(x)=1 and b(x)=K>0. For this case, we prove that compactly supported solutions to the first equation blow up in finite time, together with their interfaces, as x→−∞. We then establish the large time behavior of solutions to a homogeneous Dirichlet problem associated to the first equation on a bounded interval. We also prove a finite time blow-up of the interfaces for compactly supported solutions to the second equation when f(y)=y−γ with γ>2.

Error analysis of sinc-Galerkin method for time-dependent partial differential equations

In this paper, we apply the Galerkin method with sinc bases for solving the boundary-value problems involving nonhomogeneous heat, convection diffusion, wave, and telegraph equations. The accuracy of the method for equation is $ \mathrm {O} (({\Delta } t)+e^{-k\sqrt {N} })$ . Error analysis is included. Several examples are given to illustrate the efficiency and implementation of the sinc-Galerkin method. Comparisons are made to confirm the reliability of the method.

ECUACIÓN FRACCIONARIA DE BURGERS NO HOMOGÉNEA

En este artículo se estudian diferentes tipos de soluciones de la ecuación fraccionaria unidimensional no lineal de Burgers con un término no homogéneo asociado a fuerzas externas. Esta ecuación es una generalización de la ecuación de difusión no homogénea en la que se incluye una derivada fraccionaria de Caputo que describe una no linealidad no local. Por medio de la transformación de Cole-Hopf generalizada, la ecuación de Burgers fraccionaria no homogénea se convierte en una ecuación diferencial lineal en derivadas parciales, lo que permite obtener soluciones analíticas. Se exploran los efectos asociados al término no homogéneo y al orden de la derivada fraccionaria.

High‐order compact finite difference and laplace transform method for the solution of time‐fractional heat equations with dirchlet and neumann boundary conditions

The work presents a novel coupling of the Laplace Transform and the compact fourth‐order finite‐difference discretization scheme for the efficient and accurate solution of linear time‐fractional nonhomogeneous diffusion equations subject to both Dirichlet and Neumann boundary conditions. A translational transformation of the dependent variable ensures the Caputo derivative is aligned with the Riemann‐Louiville fractional derivative. The resulting scheme is computationally efficient and shown to be uniquely solvable in all cases, accurate and convergent to in the spatial domain. The convergence rates in the temporal domain are contour dependent but exhibit geometric convergence. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1184–1199, 2016

Radial Equivalence of Nonhomogeneous Nonlinear Diffusion Equations

We establish one-to-one transformations and self-maps between nonlinear diffusion equations in nonhomogeneous media, where the density function is given by a power. We use these transformations to deduce new interesting self-similar, radially symmetric solutions of the equations. In particular, Barenblatt, dipole and focusing Aronson-Graveleau type solutions are deduced, and some equations with singular potentials are studied. The new solutions are example of interesting or unexpected mathematical features of these equations, providing also natural candidates for the asymptotic behavior.

The method of fundamental solutions with eigenfunctions expansion method for 3D nonhomogeneous diffusion equations

AbstractAfter the successful applications of the combination of the method of fundamental solutions (MFS), the method of particular solutions (MPS), and the eigenfunctions expansion method (EEM) to solve 2D homogeneous and nonhomogeneous diffusion equations by Young et al. (Young et al., Numer Meth Part Differ Equat 22 (2006), 1173), this article intends to extend the same fundamental concepts to calculate more challenging 3D nonhomogeneous diffusion equations. The nonhomogeneous diffusion equations with time‐independent source terms and boundary conditions are analyzed by the proposed meshless MFS‐MPS‐EEM model. Nonhomogeneous diffusion equation in any complex domains can be decomposed into a Poisson equation and a homogeneous diffusion equation by the principle of linear superposition. This approach is proved to be far better off than solutions by using classic method of separation of variables with inefficient multisummation of very sophisticated series expansion from special functions, which can only limit to treat very simple 3D geometries such as cube, cylinder, or sphere. Poisson equation is solved by using the MPS‐MFS model, in which the source term in the Poisson equation is first handled by the MPS based on the compactly‐supported radial basis functions and the Laplace equation is solved by the MFS. On the other hand, by utilizing the EEM, the homogeneous diffusion equation is first transformed into a Helmholtz equation, which is then solved by the MFS together with the technique of singular value decomposition (SVD) to acquire the eigenvalues and eigenfunctions. After the eigenfunctions are obtained, we can synthesize the diffusion solutions like the orthogonal Fourier series expansions but with only one summation for the series even for multidimensional problems. Numerical results for four case studies of 3D homogeneous and nonhomogeneous diffusion problems show good agreement with the analytical and other numerical solutions, such as finite element method (FEM). Thus, the present numerical scheme has provided a promising meshfree numerical approach to solve 3D nonhomogeneous diffusion equations with time‐independent source terms and boundary conditions for very irregular domains. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

FDMFS for Diffusion Equation with Unsteady Forcing Function

In this paper, a novel numerical scheme called (FDMFS), which combines the fi- nite difference method (FDM) and the method of fundamental solutions (MFS), is proposed to simulate the nonhomogeneous diffusion problem with an unsteady forcing function. Most mesh- less methods are confined to the investigations of nonhomogeneous diffusion equations with steady forcing functions due to the difficulty to find an unsteady particular solution. Therefore, we pro- posed a FDM with Cartesian grid to handle the unsteady nonhomogeneous term of the equations. The numerical solution in FDMFS is decomposed into a particular solution and a homogeneous so- lution. The particular solution is constructed us- ing the FDM in an artificial regular domain which contains the real irregular domain without bound- ary conditions, and the homogeneous solution can be obtained by the time-space unification MFS in the irregular domain with boundary conditions. Besides, the Cartesian grid for particular solution is very simple to generate automatically. Our pa- per is the first time to propose an algorithm to solve nonhomogeneous diffusion equations with unsteady forcing functions using MFS to solve homogeneous solutions and FDM to calculate the particular solutions. Numerical experiments are presented for 2D problems in regular and irregu- lar domains to show the high performance of this proposed scheme. Moreover, the stabilities of ex- plicit and implicit FDM for particular solution are analyzed. Numerical studies suggest that the pro- posed FDMFS can speed up the simulation and save the CPU time and memory storage substan- tially.

The method of fundamental solutions with eigenfunction expansion method for nonhomogeneous diffusion equation

AbstractIn this article we describe a numerical method to solve a nonhomogeneous diffusion equation with arbitrary geometry by combining the method of fundamental solutions (MFS), the method of particular solutions (MPS), and the eigenfunction expansion method (EEM). This forms a meshless numerical scheme of the MFS‐MPS‐EEM model to solve nonhomogeneous diffusion equations with time‐independent source terms and boundary conditions for any time and any shape. Nonhomogeneous diffusion equation with complex domain can be separated into a Poisson equation and a homogeneous diffusion equation using this model. The Poisson equation is solved by the MFS‐MPS model, in which the compactly supported radial basis functions are adopted for the MPS. On the other hand, utilizing the EEM the diffusion equation is first translated to a Helmholtz equation, which is then solved by the MFS together with the technique of the singular value decomposition (SVD). Since the present meshless method does not need mesh generation, nodal connectivity, or numerical integration, the computational effort and memory storage required are minimal as compared with other numerical schemes. Test results for two 2D diffusion problems show good comparability with the analytical solutions. The proposed algorithm is then extended to solve a problem with irregular domain and the results compare very well with solutions of a finite element scheme. Therefore, the present scheme has been proved to be very promising as a meshfree numerical method to solve nonhomogeneous diffusion equations with time‐independent source terms of any time frame, and for any arbitrary geometry. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006

A new analytic approach for dealing with hysteretic materials

We present analytic formulations for studying the energetic behavior of hysteretic magnetic materials. One formulation reduces the full nonlinear diffusion problem to a linear problem through an optimization procedure. A second formulation attempts to approximate the magnetic permeability tensor by a complete set of functions. By means of scalar product defined in the function space, we obtain a series of linear nonhomogeneous diffusion equations. We analyze for the vector case qualitatively and give solutions for a one-dimensional field configuration. For the scalar case, we investigate two different magnetic materials and, for simplicity, we approximate the relevant hysteresis cycles by a closed polygonal. A scalar Preisach model, numerically treated, is used as a benchmark.

Modeling wound healing in bone

A previous model of wound healing in bone [1] is further analyzed numerically to derive the solution for given parameters. As in the previous model, the time-independent nonhomogeneous diffusion equation is solved for the equilibrium distribution of wound-induced growth factor over a hemispherical surface. The solution is then coded and implemented to produce graphical results.

The effect of surface curvature on wound healing in bone

The time-independent nonhomogeneous diffusion equation is solved for the equilibrium distribution of wound-induced growth factor over a hemispherical surface. The growth factor is produced at the inner edge of a circular wound and stimulates healing in regions where the concentration exceeds a certain threshold value. An implicit analytic criterion is derived for complete healing of the wound.

The Generalized Quasi-linearization Method for Reaction Diffusion Equations on an Unbounded Domain

The method of generalized quasi-linearization has been well developed for ordinary differential equations. In this paper, we extend the method of generalized quasi-linearization to reaction diffusion equations on an unbounded domain. The iterates, which are solutions of linear equations starting from lower and upper solutions, converge uniformly and monotonically to the unique solution of the nonlinear reaction diffusion equation in an unbounded domain. Initially an existence theorem for the linear nonhomogeneous reaction diffusion equation in an unbounded domain has been proved under improved conditions. The quadratic convergence has been proved by using a comparison theorem of reaction diffusion equations with ordinary differential equations. This avoids the computational complexity of the quasi-linearization method, since the computation of Green's function at each stage of the iterates is avoided.

Knowledge-based tensor anisotropic diffusion of cardiac magnetic resonance images

We present a general formulation for a new knowledge-based approach to anisotropic diffusion of multi-valued and multi-dimensional images, with an illustrative application for the enhancement and segmentation of cardiac magnetic resonance (MR) images. In the proposed method all available information is incorporated through a new definition of the conductance function which differs from previous approaches in two aspects. First, we model the conductance as an explicit function of time and position, and not only of the differential structure of the image data. Inherent properties of the system (such as geometrical features or non-homogeneous data sampling) can therefore be taken into account by allowing the conductance function to vary depending on the location in the spatial and temporal coordinate space. Secondly, by defining the conductance as a second-rank tensor, the non-homogeneous diffusion equation gains a truly anisotropic character which is essential to emulate and handle certain aspects of complex data systems. The method presented is suitable for image enhancement and segmentation of single- or multi-valued images. We demonstrate the efficiency of the proposed framework by applying it to anatomical and velocity-encoded cine volumetric (4-D) MR images of the left ventricle. Spatial and temporal a priori knowledge about the shape and dynamics of the heart is incorporated into the diffusion process. We compare our results to those obtained with other diffusion schemes and exhibit the improvement in regions of the image with low contrast and low signal-to-noise ratio.

Limiting spheroid size as a function of growth factor source location

Solutions C( r) of the time-independent nonhomogeneous diffusion equation for three different piecewise-uniform source terms are used to examine the limiting size of multicell spheroids using a simple model which reproduces concentration-dependent mitotic behavior. A condition is derived under which nontrivial solutions do not exist (in all three cases), and a condition for the existence of a unique nontrivial solution is established for the case of growth-modifying factor (GMF) production throughout the spheroid. Qualitative behavior of the limiting size is established as a function of various physiological parameters. Of fundamental importance is the assumed GMF concentration threshold θ, near which (i.e., as C → θ −) mitosis, and hence spheroid growth, is generally strongly inhibited.

On certain properties of the electric vector potential in eddy-current problems. Part 2

The boundary-value problem for eddy currents induced in a magnetic solid by an external magnetic field is considered. It is shown that the Lorentz or Coulomb condition in general cannot be applied for systems comprising magnetic solids of finite extents. These conditions are applicable for systems comprising nonmagnetic solids. Therefore the nonhomogeneous diffusion equation can be derived only for the electric vector potential in systems comprising non-magnetic solids.