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## Parabolic Partial Differential Equations (parabolic + partial_differential_equation)
## Selected Abstracts## The radial basis functions method for identifying an unknown parameter in a parabolic equation with overspecified data NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2007Mehdi DehghanAbstract Parabolic partial differential equations with overspecified data play a crucial role in applied mathematics and engineering, as they appear in various engineering models. In this work, the radial basis functions method is used for finding an unknown parameter p(t) in the inverse linear parabolic partial differential equation ut = uxx + p(t)u + ,, in [0,1] × (0,T], where u is unknown while the initial condition and boundary conditions are given. Also an additional condition ,01k(x)u(x,t)dx = E(t), 0 , t , T, for known functions E(t), k(x), is given as the integral overspecification over the spatial domain. The main approach is using the radial basis functions method. In this technique the exact solution is found without any mesh generation on the domain of the problem. We also discuss on the case that the overspecified condition is in the form ,0s(t)u(x,t)dx = E(t), 0 < t , T, 0 < s(t) < 1, where s and E are known functions. Some illustrative examples are presented to show efficiency of the proposed method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source] ## Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6-7 2009F. NobileAbstract We consider the problem of numerically approximating statistical moments of the solution of a time-dependent linear parabolic partial differential equation (PDE), whose coefficients and/or forcing terms are spatially correlated random fields. The stochastic coefficients of the PDE are approximated by truncated Karhunen,Ločve expansions driven by a finite number of uncorrelated random variables. After approximating the stochastic coefficients, the original stochastic PDE turns into a new deterministic parametric PDE of the same type, the dimension of the parameter set being equal to the number of random variables introduced. After proving that the solution of the parametric PDE problem is analytic with respect to the parameters, we consider global polynomial approximations based on tensor product, total degree or sparse polynomial spaces and constructed by either a Stochastic Galerkin or a Stochastic Collocation approach. We derive convergence rates for the different cases and present numerical results that show how these approaches are a valid alternative to the more traditional Monte Carlo Method for this class of problems. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Numerical studies of a nonlinear heat equation with square root reaction term NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2009Ron BuckmireAbstract Interest in calculating numerical solutions of a highly nonlinear parabolic partial differential equation with fractional power diffusion and dissipative terms motivated our investigation of a heat equation having a square root nonlinear reaction term. The original equation occurs in the study of plasma behavior in fusion physics. We begin by examining the numerical behavior of the ordinary differential equation obtained by dropping the diffusion term. The results from this simpler case are then used to construct nonstandard finite difference schemes for the partial differential equation. A variety of numerical results are obtained and analyzed, along with a comparison to the numerics of both standard and several nonstandard schemes. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 [source] ## Semilinear parabolic problem with nonstandard boundary conditions: Error estimates NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2003Marián SlodiAbstract We study a semilinear parabolic partial differential equation of second order in a bounded domain , , ,N, with nonstandard boundary conditions (BCs) on a part ,non of the boundary ,,. Here, neither the solution nor the flux are prescribed pointwise. Instead, the total flux through ,non is given, and the solution along ,non has to follow a prescribed shape function, apart from an additive (unknown) space-constant ,(t). We prove the well-posedness of the problem, provide a numerical method for the recovery of the unknown boundary data, and establish the error estimates. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 167,191, 2003 [source] ## Analysis of a block red-black preconditioner applied to the Hermite collocation discretization of a model parabolic equation NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2001Stephen H. BrillAbstract We are concerned with the numerical solution of a model parabolic partial differential equation (PDE) in two spatial dimensions, discretized by Hermite collocation. In order to efficiently solve the resulting systems of linear algebraic equations, we choose the Bi-CGSTAB method of van der Vorst (1992) with block Red-Black Gauss-Seidel (RBGS) preconditioner. In this article, we give analytic formulae for the eigenvalues that control the rate at which Bi-CGSTAB/RBGS converges. These formulae, which depend on the location of the collocation points, can be utilized to determine where the collocation points should be placed in order to make the Bi-CGSTAB/RBGS method converge as quickly as possible. Along these lines, we discuss issues of choice of time-step size in the context of rapid convergence. A complete stability analysis is also included. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:584,606, 2001 [source] ## An unconditionally stable and O(,2 + h4) order L, convergent difference scheme for linear parabolic equations with variable coefficients NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2001Zhi-Zhong SunAbstract M. K. Jain, R. K. Jain, and R. K. Mohanty presented a finite difference scheme of O(,2 + ,h2 + h4) for solving the one-dimensional quasilinear parabolic partial differential equation, uxx = f(x, t, u, ut, ux), with Dirichlet boundary conditions. The method, when applied to a linear constant coefficient case, was shown to be unconditionally stable by the Von Neumann method. In this article, we prove that the method, when applied to a linear variable coefficient case, is unconditionally stable and convergent with the convergence order O(,2 + h4) in the L, -norm. In addition, we obtain an asymptotic expansion of the difference solution, with which we obtain an O(,4 + ,2h4 + h6) order accuracy approximation after extrapolation. And last, we point out that the analysis method in this article is efficacious for complex equations. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:619,631, 2001 [source] ## Numerical analysis of the rectangular domain decomposition method INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 7 2009Younbae JunAbstract When solving parabolic partial differential equations using finite difference non-overlapping domain decomposition methods, one often uses the stripwise decomposition of spatial domain and it can be extended to the rectangular decomposition without further analysis. In this paper, we analyze the rectangular decomposition when the modified implicit prediction (MIP) algorithm is used. We show that the performance of the rectangular decomposition and the stripwise decomposition is different. We compare spectral radius, maximum error, efficiency, and total operations of the rectangular and the stripwise decompositions. We investigate the accuracy of the interface of the rectangular decomposition and the effects of the correction phase of the rectangular decomposition. Numerical experiments have been done in both two and three spatial dimensions and show that the rectangular decomposition is not better than the stripwise decomposition. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## An adaptive multiresolution method for parabolic PDEs with time-step control INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6 2009M. O. DominguesAbstract We present an efficient adaptive numerical scheme for parabolic partial differential equations based on a finite volume (FV) discretization with explicit time discretization using embedded Runge,Kutta (RK) schemes. A multiresolution strategy allows local grid refinement while controlling the approximation error in space. The costly fluxes are evaluated on the adaptive grid only. Compact RK methods of second and third order are then used to choose automatically the new time step while controlling the approximation error in time. Non-admissible choices of the time step are avoided by limiting its variation. The implementation of the multiresolution representation uses a dynamic tree data structure, which allows memory compression and CPU time reduction. This new numerical scheme is validated using different classical test problems in one, two and three space dimensions. The gain in memory and CPU time with respect to the FV scheme on a regular grid is reported, which demonstrates the efficiency of the new method. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Robust diagnosis and fault-tolerant control of distributed processes over communication networks INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, Issue 8 2009Sathyendra GhantasalaAbstract This paper develops a robust fault detection and isolation (FDI) and fault-tolerant control (FTC) structure for distributed processes modeled by nonlinear parabolic partial differential equations (PDEs) with control constraints, time-varying uncertain variables, and a finite number of sensors that transmit their data over a communication network. The network imposes limitations on the accuracy of the output measurements used for diagnosis and control purposes that need to be accounted for in the design methodology. To facilitate the controller synthesis and fault diagnosis tasks, a finite-dimensional system that captures the dominant dynamic modes of the PDE is initially derived and transformed into a form where each dominant mode is excited directly by only one actuator. A robustly stabilizing bounded output feedback controller is then designed for each dominant mode by combining a bounded Lyapunov-based robust state feedback controller with a state estimation scheme that relies on the available output measurements to provide estimates of the dominant modes. The controller synthesis procedure facilitates the derivation of: (1) an explicit characterization of the fault-free behavior of each mode in terms of a time-varying bound on the dissipation rate of the corresponding Lyapunov function, which accounts for the uncertainty and network-induced measurement errors and (2) an explicit characterization of the robust stability region where constraint satisfaction and robustness with respect to uncertainty and measurement errors are guaranteed. Using the fault-free Lyapunov dissipation bounds as thresholds for FDI, the detection and isolation of faults in a given actuator are accomplished by monitoring the evolution of the dominant modes within the stability region and declaring a fault when the threshold is breached. The effects of network-induced measurement errors are mitigated by confining the FDI region to an appropriate subset of the stability region and enlarging the FDI residual thresholds appropriately. It is shown that these safeguards can be tightened or relaxed by proper selection of the sensor spatial configuration. Finally, the implementation of the networked FDI,FTC architecture on the infinite-dimensional system is discussed and the proposed methodology is demonstrated using a diffusion,reaction process example. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Adaptive identification of two unstable PDEs with boundary sensing and actuation INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, Issue 2 2009Andrey SmyshlyaevAbstract In this paper we consider a problem of on-line parameter identification of parabolic partial differential equations (PDEs). In the previous study, on the actuation side, both distributed (SIAM J. Optim. Control 1997; 35:678,713; IEEE Trans. Autom. Control 2000; 45:203,216) and boundary (IEEE Trans. Autom. Control 2000; 45:203,216) actuations were considered in the open loop, whereas for the closed loop (unstable plants) only distributed one was addressed. On the sensing side, only distributed sensing was considered. The present study goes beyond the identification framework of (SIAM J. Optim. Control 1997; 35:678,713; IEEE Trans. Autom. Control 2000; 45:203,216) by considering boundary actuation for the unstable plants, resulting in the closed-loop identification, and also introducing boundary sensing. This makes the proposed technique applicable to a much broader range of practical problems. As a first step towards the identification of general reaction,advection,diffusion systems, we consider two benchmark plants: one with an uncertain parameter in the domain and the other with an uncertain parameter on the boundary. We design the adaptive identifier that consists of standard gradient/least-squares estimators and backstepping adaptive controllers. The parameter estimates are shown to converge to the true parameters when the closed-loop system is excited by an additional constant input at the boundary. The results are illustrated with simulations. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Optimal boundary control of cardiac alternans INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 2 2009Stevan DubljevicAbstract Alternation of normal electrical activity in the myocardium is believed to be linked to the onset of life-threatening ventricular arrhythmias and sudden cardiac death. In this paper, a spatially uniform unstable steady state of small amplitude of alternans described by parabolic partial differential equations (PDEs) is stabilized by boundary optimal control methods. A finite dimensional linear quadratic regulator (LQR) is utilized in both a full-state-feedback control structure and in a compensator design with the Luenberger observer, and it achieves global stabilization in a finite size tissue cable length. The ability to realize such control algorithm is analyzed based on the structure of the amplitude of alternans equation and the control methodology applied. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Predictive control of parabolic PDEs with state and control constraints INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 16 2006Stevan DubljevicAbstract This work focuses on predictive control of linear parabolic partial differential equations (PDEs) with state and control constraints. Initially, the PDE is written as an infinite-dimensional system in an appropriate Hilbert space. Next, modal decomposition techniques are used to derive a finite-dimensional system that captures the dominant dynamics of the infinite-dimensional system, and express the infinite-dimensional state constraints in terms of the finite-dimensional system state constraints. A number of model predictive control (MPC) formulations, designed on the basis of different finite-dimensional approximations, are then presented and compared. The closed-loop stability properties of the infinite-dimensional system under the low order MPC controller designs are analysed, and sufficient conditions that guarantee stabilization and state constraint satisfaction for the infinite-dimensional system under the reduced order MPC formulations are derived. Other formulations are also presented which differ in the way the evolution of the fast eigenmodes is accounted for in the performance objective and state constraints. The impact of these differences on the ability of the predictive controller to enforce closed-loop stability and state constraints satisfaction in the infinite-dimensional system is analysed. Finally, the MPC formulations are applied through simulations to the problem of stabilizing the spatially-uniform unstable steady-state of a linear parabolic PDE subject to state and control constraints. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## Alternating direction finite volume element methods for 2D parabolic partial differential equations NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2008Tongke WangAbstract On the basis of rectangular partition and bilinear interpolation, this article presents alternating direction finite volume element methods for two dimensional parabolic partial differential equations and gives three computational schemes, one is analogous to Douglas finite difference scheme with second order splitting error, the second has third order splitting error, and the third is an extended locally one dimensional scheme. Optimal L2 norm or H1 semi-norm error estimates are obtained for these schemes. Finally, two numerical examples illustrate the effectiveness of the schemes. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source] ## High order smoothing schemes for inhomogeneous parabolic problems with applications in option pricing NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2007A.Q.M. KhaliqAbstract A new family of numerical schemes for inhomogeneous parabolic partial differential equations is developed utilizing diagonal Padé schemes combined with positivity,preserving Padé schemes as damping devices. We also develop a split version of the algorithm using partial fraction decomposition to address difficulties with accuracy and computational efficiency in solving and to implement the algorithms in parallel. Numerical experiments are presented for several inhomogeneous parabolic problems, including pricing of financial options with nonsmooth payoffs.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source] |