Nonhomogeneous Parabolic Equations Research Articles

Applying Proper Orthogonal Decomposition to Parabolic Equations: A Reduced Order Numerical Approach

n this paper we present a low-order numerical scheme developed using the Proper Orthogonal (POD) method to address non-homogeneous parabolic equations in both one and two dimensions. The proposed schemes leverage the POD technique to reduce the computational complexity associated with solving these equations while maintaining accuracy. By employing POD, the high-dimensional problem is approximated by a reduced set of models, allowing for a more efficient representation of the system dynamics. The application of this method to non-homogenous parabolic equations offers a promising approach for enhancing the computational efficiency of simulations in diverse fields, such as fluid dynamics, heat conduction, and reaction-diffusion processes. The presented numerical scheme demonstrates its efficacy in achieving accurate results with significantly reduced computational costs, making it a valuable tool for applications demanding efficient solutions to non-homogeneous parabolic equations in one and two dimensions.

Local Hölder regularity for the general non-homogeneous parabolic equations

In this paper we obtain the local Hölder regularity estimates of the gradient of weak solutions for the following general non-homogeneous parabolic equations with the logarithmic term in divergence formut−div((ADu⋅Du)p−22ln⁡(e+(ADu⋅Du)12)ADu)=div(|f|p−2ln⁡(e+|f|)) in ΩT, where ΩT:=Ω×(0,T) with T>0 and Ω is an open bounded domain in Rn, under some proper conditions on A and f. We would like to point out that our result in the present work is a generalized version of the known results for the classical parabolic p-Laplacian equation without the logarithmic term.

Null Controllability for a Degenerate Population Equation with Memory

In this paper we consider the null controllability for a population model depending on time, on space and on age. Moreover, the diffusion coefficient degenerate at the boundary of the space domain. The novelty of this paper is that for the first time we consider the presence of a memory term, which makes the computations more difficult. However, under a suitable condition on the kernel we deduce a null controllability result for the original problem via new Carleman estimates for the adjoint problem associated to a suitable nonhomogeneous parabolic equation.

An Intrinsic Harnack inequality for some non-homogeneous parabolic equations in non-divergence form

In this paper, we establish a scale invariant Harnack inequality for some inhomogeneous parabolic equations in a suitable intrinsic geometry dictated by the nonlinearity. The class of equations that we consider correspond to the parabolic counterpart of the equations studied by Julin in [10] where a generalized Harnack inequality was obtained which quantifies the strong maximum principle. Our version of parabolic Harnack (see Theorem 1.2) when restricted to the elliptic case is however quite different from that in [10]. The key new feature of this work is an appropriate modification of the stack of cubes covering argument which is tailored for the nonlinearity that we consider.

Estimation of Heat Flux in Two-Dimensional Nonhomogeneous Parabolic Equation Based on a Sufficient Descent Levenberg–Marquard Algorithm

The main work of this paper focuses on identifying the heat flux in inverse problem of two-dimensional nonhomogeneous parabolic equation, which has wide application in the industrial field such as steel-making and continuous casting. Firstly, the existence of the weak solution of the inverse problem is discussed. With the help of forward solution and dual equation, this paper proves the Lipchitz continuity of the cost function and derives the Lipchitz constant. Furthermore, in order to accelerate the convergence rate and reduce the running time, this paper presents a sufficient descent Levenberg–Marquard algorithm with adaptive parameter (SD-LMAP) to solve this inverse problem. At last, compared with other methods, the results of simulation experiment show that this algorithm can obviously reduce the running time and iterative number.

Analytical evaluation of transformation matrix between functional spaces and its application in obtaining exact solutions of thermo-mechanical stresses in disks

Thermo-mechanical stresses developed in a disk, inner boundary of which is subject to the nonstationary thermo-mechanical loadings due to variation in temperature and pressure inside the disk is considered in the presented paper. Problems of theoretical analysis are divided and described as follows: The first problem implies analytical solution of the heat conduction equation with nonhomogeneous boundary conditions of Robin type, and the second problem deals with obtaining analytical solution of dynamic equation of displacement for nonisothermal elastic body (Navier equation) with traction boundary conditions in the plane strain case of uncoupled thermo-elasticity. Finite Hankel transform is used to solve the problems. Two sets of orthogonal basis functions necessary for integral transform are obtained. The first one corresponds to the boundary conditions written for the heat conduction equation, and the second fits the boundary conditions for Navier equation. Integrals defining elements of transformation matrix between the two sets of basis functions are evaluated analytically. In general, convenient method of obtaining analytical solution of linear hyperbolic equation with nonhomogeneous term, which itself is solution of nonhomogeneous parabolic equation, is elaborated within the presented work. Using finite difference method, discrete counterparts of the described problems are constructed and, via the built-in ordinary differential equation (ODE) solvers in MATLAB, numerical solutions are obtained, which exhibit excellent accordance with analytical ones.

Applications of General Residual Power Series Method to Differential Equations with Variable Coefficients

This paper is devoted to studying the analytical series solutions for the differential equations with variable coefficients. By a general residual power series method, we construct the approximate analytical series solutions for differential equations with variable coefficients, including nonhomogeneous parabolic equations, fractional heat equations in 2D, and fractional wave equations in 3D. These applications show that residual power series method is a simple, effective, and powerful method for seeking analytical series solutions of differential equations (especially for fractional differential equations) with variable coefficients.

Effect of desorption nonequilibrium on structure of shock and rarefaction waves in coal bed

The authors propose a physico-mathematical model to describe nonequilibrium filtration and diffusion of gas in coal in the framework of heterogeneous medium mechanics considering physical desorption. The model consists of a set of nonhomogeneous parabolic equations with. In terms of problems on propagation of compression and rarefaction waves, effect of nonequilibrium desorption on parameters of these waves is revealed. The problem on initiation and steady-state of a shock wave in coal is solved.

Local Controllability to Trajectories of the Magnetohydrodynamic Equations

In this paper we prove the local controllability to trajectories of the three dimensional magnetohydrodynamic equations by means of two internal controls, one in the velocity equations and the other in the magnetic field equations and both localized in an arbitrary small subset with not empty interior of the domain. This paper improves the previous results (Barbu et al. in Comm Pure Appl Math 56:732-783, 2003; Barbu et al. in Adv Differ Equ 10:481-504, 2005 ;H avet al. in Adv Differ Equ 11:893-929, 2006 ;H avin SIAM J Control Optim 46:1802-1830, 2007) where the second control is not localized and it allows to deduce the local controllability to trajectories with boundary controls. The proof relies on the Carleman inequality for the Stokes system of Imanuvilov et al. (Carleman estimates for second order nonhomogeneous parabolic equations, preprint) to deal with the velocity equations and on a new Carleman inequality for a Dynamo-type equation to deal with the magnetic field equations. Mathematics Subject Classification. Primary 76W05, 93B05; Secondary 76D55, 35Q35.

Homogenization of the parabolic cauchy problem in the Sobolev class H1(Rd)

Homogenization in the small period limit for the solution ue of the Cauchy problem for a parabolic equation in Rd is studied. The coefficients are assumed to be periodic in Rd with respect to the lattice ɛG. As ɛ → 0, the solution u ɛ converges in L2(Rd) to the solution u0 of the effective problem with constant coefficients. The solution u ɛis approximated in the norm of the Sobolev space H 1(Rd) with error O( ɛ); this approximation is uniform with respect to the L2-norm of the initial data and contains a corrector term of order ɛ. The dependence of the constant in the error estimate on time t is given. Also, an approximation in H 1(Rd) for the solution of the Cauchy problem for a nonhomogeneous parabolic equation is obtained.

Boundary-value problem for a nonhomogeneous parabolic equation with Orlicz right side

Conditions for the existence with probability one of the solution of the boundary-value problem for a nonhomogeneous parabolic equation of mathematical physics with random right side and zero initial conditions are found.

Application of He's homotopy perturbation method to nonhomogeneous parabolic problems

In this paper, the solutions of the one dimensional non-homogeneous parabolic equations with a variable coefficient. are obtained by means of homotopy perturbation method (HPM). HPM results show good approximation with the exact solution.

Exact solutions to one dimensional non-homogeneous parabolic problems by the homogeneous Adomian decomposition method

In this paper, we propose a reliable modification of Adomian decomposition method, namely the homogeneous Adomian decomposition method (HADM), that solves one dimensional non-homogeneous parabolic equations with a variable coefficient. The effectiveness of this method is verified through illustrative examples.

Asymptotic Behavior of Parabolic Equations Arising from One-Dimensional Null-Recurrent Diffusions

This work is concerned with the asymptotic behavior of homogeneous and nonhomogeneous parabolic equations arising from one-dimensional null-recurrent diffusion processes. First, we review the concepts of regularity, recurrence, and positive recurrence of Markov processes and recall the connections of these concepts with properties of solutions of the corresponding differential equations. Next, we examine the rate of convergence of the solutions of both homogeneous and nonhomogeneous parabolic equations when the initial function and the forcing function are integrable with respect to the invariant measure. Weaker and verifiable conditions compared with the existing work in the literature are obtained. Then the corresponding problems when the initial and forcing functions are not integrable with respect to the invariant measure are dealt with. Convergence under suitable scaling for the solutions of the parabolic equations is proved, and the explicit limit is obtained.

Side wall effects in centrifugal separation of mixtures

The flow of a centrifugally separating mixture of particles (droplets) and fluid is considered, with emphasis on the azimuthal no-slip requirement on the side wall of the cylindrical container. In the framework of the ‘‘mixture’’ averaged equations—when the Ekman number E, particle Taylor number β, and relative density difference ε are small—a single partial differential nonhomogeneous parabolic equation is obtained for the angular velocity ω(r,t) (relative to the rotating vessel). Analytical and numerical solutions are presented for the dilute limit and for the more general case, respectively. It is shown that, during the separation process, ω develops a quasisteady boundary layer of thickness δ(λ/(λ−1))1/2 on r=1, only if λ>1, where δ=H1/2E1/4 is the typical Stewartson layer scale, λ=E1/2/εβH, and H is the dimensionless height of the container. If λ≤1, however, the side wall affects ω in an unsteady, diffusing viscous domain. Correspondingly significant variations in the axial flux are induced by the Ekman layers. Comparisons to numerical results (produced by a code discussed elsewhere) of the full two-fluid equations display a favorable agreement.

Bounded solutions of linear periodic abstract parabolic equations

SynopsisWe study the asymptotic behaviour of the parabolic evolution operatorG(t,s) associated with a family of generators of analytic semigroups {A(t);t∊ ℝ} in general Banach space, under the periodicity assumptionA(t) =A(t+T). We find new maximal regularity results for the bounded solutions of abstract nonhomogeneous parabolic equations in unbounded time intervals.