Semilinear Parabolic Problems Research Articles

Numerical solution of a semi-linear inverse parabolic problem via HPM

In this work, we consider the problem of finding an unknown time-dependent function in a semi-linear parabolic equation with given initial, boundary and energy over-specified conditions. The homotopy perturbation method, a powerful technique, is applied to obtain the numerical solution of this problem. Using this method, a rapid convergent sequence can be constructed which tends to the exact solution of the problem. Some examples are presented to illustrate the strength of the method.

Numerical solution of a semi-linear inverse parabolic problem via HPM

In this work, we consider the problem of finding an unknown time-dependent function in a semi-linear parabolic equation with given initial, boundary and energy over-specified conditions. The homotopy perturbation method, a powerful technique, is applied to obtain the numerical solution of this problem. Using this method, a rapid convergent sequence can be constructed which tends to the exact solution of the problem. Some examples are presented to illustrate the strength of the method.

On the Stability of Solutions of Semilinear Elliptic Equations with Robin Boundary Conditions on Riemannian Manifolds

We investigate existence and nonexistence of stationary stable nonconstant solutions, i.e., patterns, of semilinear parabolic problems in bounded domains on Riemannian manifolds, satisfying Robin boundary conditions. These problems arise in several models in applications, in particular in mathematical biology. We point out the significance both of the nonlinearity and of geometric objects such as the Ricci curvature of the manifold, the second fundamental form of the boundary of the domain, and its mean curvature. Special attention is given to surfaces of revolution and to spherically symmetric manifolds, where we prove refined results.

Mixed Discontinuous Galerkin Time-Stepping Method for Semilinear Parabolic Optimal Control Problems

In this paper, we discuss the mixed discontinuous Galerkin (DG) finite element approximation to semilinear parabolic optimal control problems, where the discontinuous finite element method of the order r (r ? 0) is used for the time discretization and the Raviart---Thomas mixed finite element method of the order ? (? ? 0) is used for the space discretization. For ? ? 0, r = 0 or 1, we derive a priori error estimates for both the control variable and the state variables. Moveover, we derive a posteriori L2(0,T;L2(Ω)) error estimates for the scalar functions, assuming that only the underlying mesh is static.

A priori error estimates for nonstationary optimal control problems with gradient state constraints

AbstractWe are concerned with the discretization of semilinear parabolic optimization problems subject to constraints on the first derivative of the solution of the PDE. Constructing sequences of feasible controls for the dG(0)‐cG(1) discretization of the problem and relying on a quadratric growth condtion, we show convergence orders as temporal and spatial mesh tends to zero. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A semilinear parabolic problem with singular term at the boundary

In this paper, we deal with a class of semilinear parabolic problems related to a Hardy inequality with singular weight at the boundary.

On the continuation of solutions of non-autonomous semilinear parabolic problems

AbstractWe study non-autonomous parabolic equations with critical exponents in a scale of Banach spaces Eσ, σ ∈ [0,1 + μ). We consider a suitable E1+ε-solution and describe continuation properties of the solution. This concerns both a situation when the solution can be continued as an E1+ε-solution and a situation when the E1+ε-norm of the solution blows up, in which case a piecewise E1+ε-solution is constructed.

Wavelet based quasilinearization method for semi-linear parabolic initial boundary value problems

In this paper, numerical methods based on quasilinearization and Haar and Legendre wavelets to solve a class of semi linear parabolic initial boundary value problem (SPIBVP) have been presented. The Haar and Legendre wavelet methods have been successfully combined with quasilinearization to solve SPIBVP efficiently. The presented numerical scheme has been illustrated using appropriate examples including Fisher equation and the obtained results show that the proposed numerical scheme is robust and easy to apply.

A leapfrog semi-smooth Newton-multigrid method for semilinear parabolic optimal control problems

A new semi-smooth Newton multigrid algorithm is proposed for solving the discretized first order necessary optimality systems that characterizing the optimal solutions of a class of two dimensional semi-linear parabolic PDE optimal control problems with control constraints. A new computational scheme (leapfrog scheme) in time associated with the standard five-point stencil in space is established to achieve the second-order finite difference discretization. The convergence (or unconditional stability) of the proposed scheme is proved when assuming time-periodic solutions. Moreover, the derived well-structured discretized Jacobian matrices greatly facilitate the development of effective smoother in our multigrid algorithm. Numerical simulations are provided to illustrate the effectiveness of the proposed method, which validates the second-order accuracy in solution approximations and the optimal linear complexity of computing time.

Well-posedness and qualitative behaviour of a semi-linear parabolic Cauchy problem arising from a generic model for fractional-order autocatalysis.

In this paper, we examine a semi-linear parabolic Cauchy problem with non-Lipschitz nonlinearity which arises as a generic form in a significant number of applications. Specifically, we obtain a well-posedness result and examine the qualitative structure of the solution in detail. The standard classical approach to establishing well-posedness is precluded owing to the lack of Lipschitz continuity for the nonlinearity. Here, existence and uniqueness of solutions is established via the recently developed generic approach to this class of problem (Meyer & Needham 2015 The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations. London Mathematical Society Lecture Note Series, vol. 419) which examines the difference of the maximal and minimal solutions to the problem. From this uniqueness result, the approach of Meyer & Needham allows for development of a comparison result which is then used to exhibit global continuous dependence of solutions to the problem on a suitable initial dataset. The comparison and continuous dependence results obtained here are novel to this class of problem. This class of problem arises specifically in the study of a one-step autocatalytic reaction, which is schematically given by A→B at rate apbq (where a and b are the concentrations of A and B, respectively, with 0<p,q<1) and well-posedness for this problem has been lacking up to the present.

Identification of a memory kernel in a semilinear integrodifferential parabolic problem with applications in heat conduction with memory

In this contribution, the reconstruction of a solely time-dependent convolution kernel is studied in an inverse problem arising in the theory of heat conduction for materials with memory. The missing kernel is recovered from a measurement of the average of temperature. The existence, uniqueness and regularity of a weak solution is addressed. More specific, a new numerical algorithm based on Rothe’s method is designed. The convergence of iterates to the exact solution is shown.

An adaptive finite element algorithm for parabolic optimal control problems

In this paper, an adaptive fully discrete finite element algorithm of semilinear parabolic optimal control problems is investigated. On deriving a posteriori error estimates, we construct an adaptive finite element algorithm for parabolic optimal control problems. Some numerical examples are presented. It confirms that our adaptive algorithm is efficient and our theoretical results are correct.

Reconstruction of an unknown source parameter in a semilinear parabolic problem

In this paper, a semilinear parabolic problem with an unknown time-dependent source function p(t) is studied. This missing parameter is reconstructed from a given measurement of the total energy/mass in the domain. The existence and uniqueness of a solution in suitable function spaces is established under minimal regularity assumptions on the data. A numerical time-discrete scheme to approximate the unique weak solution and the unknown source parameter is designed and convergence of the approximations is proved. Finally, the theoretically obtained results are supported by a numerical experiment.

Asymptotic approximation for the solution to a semi-linear parabolic problem in a thick junction with the branched structure

We consider a semi-linear parabolic problem in a thick junction Ωε, which is the union of a domain Ω0 and a lot of joined thin trees situated ε-periodically along some manifold on the boundary of Ω0. The trees have finite number of branching levels. The following nonlinear Robin boundary condition ∂νvε+εαiμ(t,x2,vε)=εβgε is given on the boundaries of the branches from the i-th branching layer; {αi} and β are real parameters. The asymptotic analysis of this problem is made as ε→0, i.e., when the number of the thin trees infinitely increases and their thickness vanishes. In particular, the corresponding homogenized problem is found and the existence and uniqueness of its solution in an anisotropic Sobolev space of multi-sheeted functions is proved. We construct the asymptotic approximation for the solution vε and prove the corresponding asymptotic estimate in the space C([0,T];L2(Ωε))∩L2(0,T;H1(Ωε)), which shows the influence of the parameters {αi} and β on the asymptotic behavior of the solution.

Monotone iterative ADI method for semilinear parabolic problems

The paper deals with numerical solving semilinear parabolic problems based on a nonlinear alternating direction implicit (ADI) scheme. The convergence of the nonlinear ADI scheme to the continuous solution is proved. The existence and uniqueness of a solution of the nonlinear ADI scheme are established. A monotone iterative ADI method is constructed. An analysis of convergence of the monotone iterative ADI method to the solution of the nonlinear ADI scheme on the whole time interval is given. Numerical experiments are presented.

Determination of a solely time-dependent source in a semilinear parabolic problem by means of boundary measurements

A semilinear parabolic problem of second order with an unknown solely time-dependent source term is studied. The missing source is recovered from an additional integral measurement over the boundary. The global in time existence, uniqueness as well as the regularity of a solution are addressed. A new numerical scheme based on Rothe’s method is designed and convergence of iterates towards the exact solution is shown.

Blow-up in a semilinear parabolic problem with variable source under positive initial energy

In this paper, we consider a class of semilinear parabolic equations with variable source under homogeneous Dirichlet boundary conditions in a bounded domain with smooth boundary and For all we prove that the classical solutions to the above equation blow-up in finite time with arbitrary positive initial energy and suitable large initial values. This result extends a recent result by Wu et al. [Appl. Math. Lett. 2013;26:539–543] which asserts the blow-up of solutions for , provided that

SMONOTONE METHOD FOR COUPLED SYSTEM OF FINITE DIFFERENCE EQUATIONS OF TIME DEGENERATE PARABOLIC PROBLEMS

The aim of this paper is to develop monotone iterative scheme using the notion of upper-lower solutions for weakly coupled system of finite difference equations which corresponds to weakly coupled system of semilinear time degenerate parabolic Dirichlet initial boundary value problem (IBVP). Using upper and lower solutions as distinct initial iterations, two monotone convergent sequences from linear system are constructed. It is shown that these two sequences converge monotonically from above and below to maximal and minimal solutions respectively which lead to the existence-comparison and uniqueness results for the solution of the discrete nonlinear Dirichlet IBVP. Positivity lemma is the main ingredient used in the proof of these results. AMS Subject Classification: 65Q10

Reconstruction of a convolution kernel in a semilinear parabolic problem based on a global measurement

A semilinear parabolic problem of second order with an unknown time-convolution kernel is considered. The missing kernel is recovered from an additional integral measurement. The existence, uniqueness and regularity of a weak solution is addressed. We design a numerical algorithm based on Rothe’s method, derive a priori estimates and prove convergence of iterates towards the exact solution.

A Semi-linear Backward Parabolic Cauchy Problem with Unbounded Coefficients of Hamilton–Jacobi–Bellman Type and Applications to Optimal Control

We obtain weighted uniform estimates for the gradient of the solutions to a class of linear parabolic Cauchy problems with unbounded coefficients. Such estimates are then used to prove existence and uniqueness of the mild solution to a semi-linear backward parabolic Cauchy problem, where the differential equation is the Hamilton---Jacobi---Bellman equation of a suitable optimal control problem. Via backward stochastic differential equations, we show that the mild solution is indeed the value function of the controlled equation and that the feedback law is verified.