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This article considers the $H_{\infty }$ sampled-data fuzzy observer (SDFO) design problem for nonlinear parabolic partial differential equation (PDE) systems under spatially local averaged measurements (SLAMs). Initially, the nonlinear PDE system is accurately represented by the Takagi–Sugeno (T–S) fuzzy PDE model. Then, based on the T–S fuzzy PDE model, an SDFO under SLAMs is constructed for the state estimation. To attenuate the effect of the exogenous disturbance and the design disturbance, an $H_{\infty }$ SDFO design under SLAMs is developed in terms of linear matrix inequalities by utilizing Lyapunov functional and inequality techniques, which can guarantee the exponential stability and satisfy an $H_{\infty }$ performance for the estimation error fuzzy PDE system. Finally, simulation results on the state estimation of the FitzHugh–Nagumo equation are given to support the presented $H_{\infty }$ SDFO design method.

The paper deals with non-classical initial-boundary value problems for parabolic equations with a fractional Laplacian. We study the existence and uniqueness of a mild solution to our problem. The ...

In this paper we establish a stability result for the nonnegative local weak solutions to $$\begin{aligned} u_t= \text {div} \big (|Dw|^{p-2}Dw\big ) , \quad p>1 \end{aligned}$$ where $$w= \frac{u^\gamma -1}{\gamma }$$ and $$\gamma = \frac{m+p-2}{p-1}$$ , as $$|\gamma |\rightarrow 0$$ .

In this paper, the weighted average-based differential quadrature method is presented for solving one-dimensional homogeneous first-order non-linear parabolic partial differential equation. First, the given solution domain is discretized by using uniform discretization grid point. Next, by using Taylor series expansion we obtain central finite difference discretization of the partial differential equation involving with temporal variable associated with weighted average of partial derivative concerning spatial variable. From this, we obtain the system of nonlinear ordinary differential equations and it is linearized by using the quasilinearization method. Then by using the polynomial-based differential quadrature method for approximating derivative involving with spatial variable at specified grid point, we obtain the system of linear equation. Then they obtained linear system equation is solved by using the LU matrix decomposition method. To validate the applicability of the proposed method, two model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm and maximum absolute error norm and the local behavior of the solution is captured exactly. Numerical versus exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root mean square error norm and maximum absolute error norm presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving the non-linear parabolic equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution.

This article extends the semidiscrete maximal $L^p$-regularity results in Li (2019, Analyticity, maximal regularity and maximum-norm stability of semi-discrete finite element solutions of parabolic equations in nonconvex polyhedra. Math. Comp., 88, 1--44) to multistep fully discrete finite element methods for parabolic equations with more general diffusion coefficients in $W^{1,d+\beta }$, where $d$ is the dimension of space and $\beta>0$. The maximal angles of $R$-boundedness are characterized for the analytic semigroup $e^{zA_h}$ and the resolvent operator $z(z-A_h)^{-1}$, respectively, associated to an elliptic finite element operator $A_h$. Maximal $L^p$-regularity, an optimal $\ell ^p(L^q)$ error estimate and an $\ell ^p(W^{1,q})$ estimate are established for fully discrete finite element methods with multistep backward differentiation formulae.

In this paper, we examine the solvability of a new class of nonclassical direct problems for multidimensional loaded parabolic equations with Cauchy data. We obtain sufficient conditions for the solvability of the problem; the proof is based on the method of weak approximation. By an example, we demonstrate the application of the theorem proved to the study of inverse problems for multidimensional parabolic equations with Cauchy data.

In this paper, we consider a loaded parabolic equation of a special form in an unbounded domain with Cauchy data. The equation is one-dimensional and its right-hand side depends on the unknown function u(t, x) and traces of this function and its derivatives by the spatial variable at a finite number of different points of space. Such equations appear after the reduction of some identification problems for coefficients of one-dimensional parabolic equations with Cauchy data to auxiliary direct problems. We obtain sufficient conditions of the global solvability and sufficient conditions of the solvability of the problem considered in a small time interval. We search for solutions in the class of sufficiently smooth bounded functions. We examine the uniqueness of the classical solution found and prove the corresponding sufficient conditions. We also obtain an a priori estimate of a solution that guarantees the continuous dependence of the solution on the right-hand side of the equation and the initial conditions.

In this study, fractional stochastic pseudo-parabolic equations driven by fractional Brownian motion are investigated. This work aims at establishing existence, uniqueness, regularity results for m...

International audience; In this work we determine the second-order coefficient in a parabolic equation from the knowledge of a single final data. Under assumptions on the concentration of eigenvalues of the associated elliptic operator, and the initial state, we show the uniqueness of solution, and we derive a Lipschitz stability estimate for the inversion when the final time is large enough. The Lipschitz stability constant grows exponentially with respect to the final time, which makes the inversion ill-posed. The proof of the stability estimate is based on a spectral decomposition of the solution to the parabolic equation in terms of the eigenfunctions of the associated elliptic operator, and an ad hoc method to solve a nonlinear stationary transport equation that is itself of interest.

In this paper, we consider the weak Galerkin finite element approximations of second order linear parabolic problems in two dimensional convex polygonal domains under the low regularities of the solutions. Optimal order error estimates in L 2 ( L 2 ) and L 2 ( H 1 ) norms are shown to hold for both the spatially discrete continuous time and the discrete time weak Galerkin finite element schemes, which allow using the discontinuous piecewise polynomials on finite element partitions with the arbitrary shape of polygons with certain shape regularity. The fully discrete scheme is based on first order in time Euler method. We have derived O ( h r + 1 ) in L 2 ( L 2 ) norm and O ( h r ) in L 2 ( H 1 ) norm when the exact solution u ∈ L 2 ( 0 , T ; H r + 1 ( Ω ) ) ∩ H 1 ( 0 , T ; H r − 1 ( Ω ) ) , for some r ≥ 1 . Numerical experiments are reported for several test cases to justify our theoretical convergence results.

In this paper, we show how to construct an asymptotic representation of the fundamental solution to the Cauchy problem for degenerate linear parabolic equations. DOI 10.1134/S1061920821020047

In this paper we study both analytic and numerical solutions of option pricing equations using systems of orthogonal polynomials. Using a Galerkin-based method, we solve the parabolic partial diferential equation for the Black-Scholes model using Hermite polynomials and for the Heston model using Hermite and Laguerre polynomials. We compare obtained solutions to existing semi-closed pricing formulas. Special attention is paid to the solution of Heston model at the boundary with vanishing volatility.

In this paper, we present results of the development of certain parallel numerical methods for solving three-dimensional evolutionary and stationary problems of diffusion and heat transfer. We present a detailed description of a special, explicit iteration scheme for parabolic equations and discuss a multigrid technology used for solving elliptic equations and implicit schemes for parabolic equations.

We study a new mathematical model of locally nonequilibrium processes of heat, mass, and momentum transfer taking into account the relaxation phenomena based on hyperbolic and parabolic equations. We also propose a method aimed at getting a priori Schauder-type estimates. The unique solvability of the problem with inner-boundary nonlocal conditions is established.

In this work, a fast numerical method is considered to solve the space-fractional semilinear parabolic equations on closed surfaces. It is a challenge in that how to define the space fractional operator and corresponding semilinear parabolic equation with the energy functional defined on surface. To overcome it, using the local tangential space, we construct the spectral approximation for the space fractional operator on surfaces and apply the matrix transfer technique to avoid the difficulty of fractional nonlocality. The main advantage of the matrix transfer method is the completed diagonal representation of fractional operators from eigenvalue decomposition. Moreover, the time-discrete error estimates are presented as well as the energy stability. Various numerical examples are carried out to verify the theoretical results.

By studying parabolic equations in mixed-norm spaces, we prove the existence and uniqueness of strong solutions to stochastic differential equations driven by Brownian motion with coefficients in spaces with mixed-norm, which extends Krylov and Rockner’s result in Krylov (Probab. Theory Rel. Fields. 131(2)154–196, 2005) and Zhang’s result in Zhang (Electron. J. Probab. 16,1096–1116, 2011).

In this paper we propose an elementary proof of the Harnack inequality for linear parabolic equations in non-divergence form.

We demonstrate a measure theoretical approach to the local regularity of weak supersolutions to elliptic and parabolic equations in divergence form. In the first part, we show that weak supersolutions become lower semicontinuous after redefinition on a set of measure zero. The proof relies on a general principle, i.e. the De Giorgi type lemma, which offers a unified approach for a wide class of elliptic and parabolic equations, including an anisotropic elliptic equation, the parabolic p-Laplace equation, and the porous medium equation. In the second part, we shall show that for parabolic problems the lower semicontinuous representative at an instant can be recovered pointwise from the “ess lim inf” of past times. We also show that it can be recovered by the limit of certain integral averages of past times. The proof hinges on the expansion of positivity for weak supersolutions. Our results are structural properties of partial differential equations, independent of any kind of comparison principle.

In this paper, we are concerned with the existence, uniqueness and long-time behavior of the solutions for a parabolic equation with nonlocal diffusion even if the reaction term is not Lipschitz-continuous at $ 0 $ and grows superlinearly or exponentially at $ +\infty $. First, we give a special sub-supersolution pair for some parabolic equations with nonlocal diffusion and establish the method of sub-supersolution. Second, using the sub-supersolution method, we prove the existence, uniqueness and long-time behavior of positive solutions. Finally, some one-dimensional numerical experiments are presented.

A porous medium equation is a nonlinear parabolic partial differential equation that presents many physical occurrences. The solutions of the porous medium equation are important to facilitate the investigation on nonlinear processes involving fluid flow, heat transfer, diffusion of gas-particles or population dynamics. As part of the development of a family of efficient iterative methods to solve the porous medium equation, the Half-Sweep technique has been adopted. Prior works in the existing literature on the application of Half-Sweep to successfully approximate the solutions of several types of mathematical problems are the underlying motivation of this research. This work aims to solve the one-dimensional porous medium equation efficiently by incorporating the Half-Sweep technique in the formulation of an unconditionally-stable implicit finite difference scheme. The noticeable unique property of Half-Sweep is its ability to secure a low computational complexity in computing numerical solutions. This work involves the application of the Half-Sweep finite difference scheme on the general porous medium equation, until the formulation of a nonlinear approximation function. The Newton method is used to linearize the formulated Half-Sweep finite difference approximation, so that the linear system in the form of a matrix can be constructed. Next, the Successive Over Relaxation method with a single parameter was applied to efficiently solve the generated linear system per time step. Next, to evaluate the efficiency of the developed method, deemed as the Half-Sweep Newton Successive Over Relaxation (HSNSOR) method, the criteria such as the number of iterations, the program execution time and the magnitude of absolute errors were investigated. According to the numerical results, the numerical solutions obtained by the HSNSOR are as accurate as those of the Half-Sweep Newton Gauss-Seidel (HSNGS), which is under the same family of Half-Sweep iterations, and the benchmark, Newton-Gauss-Seidel (NGS) method. The improvement in the numerical results produced by the HSNSOR is significant, and requires a lesser number of iterations and a shorter program execution time, as compared to the HSNGS and NGS methods.

This article employs the observer-based output feedback control technique to deal with dynamic compensator design for a linear $N$ -D parabolic partial differential equation (PDE) with multiple local piecewise control inputs and multiple noncollocated local piecewise observation outputs. These control inputs and observation outputs are provided by only few actuators and noncollocated sensors active over partial areas (or entire) of the spatial domain. An observer-based dynamic feedback compensator is constructed for exponential stabilization of the linear PDE. Poincare–Wirtinger inequality and its variant in $N$ -D spatial domain are presented for the closed-loop stability analysis. By the Lyapunov direct method and Poincare–Wirtinger inequality and its variant, sufficient conditions on the existence of such feedback compensator of the linear PDE are developed, and presented in term of standard linear matrix inequalities. Well-posedness is also analyzed for both open-loop PDE and resulting closed-loop coupled PDEs within the $C_0$ -semigroup framework. Finally, numerical simulation results are presented to support the proposed design method.

This paper deals with the application of Stackelberg-Nash strategies to the control to quasi-linear parabolic equations in dimensions 1D, 2D, or 3D. We consider two followers, intended to solve a Nash multi-objective equilibrium; and one leader satisfying the controllability to the trajectories.

In this paper, an adaptive robust boundary controller is designed for an unstable heat equation with external disturbance flowing into the control end. The disturbance and uncertainty effect is can...

The paper gives sharp spectral gap conditions for existence of inertial manifolds for abstract semilinear parabolic equations with non-self-adjoint leading part. Main attention is paid to the case where this leading part have Jordan cells which appear after applying the so-called Kwak transform to various important equations such as 2D Navier–Stokes equations, reaction-diffusion-advection systems, etc. The different forms of Kwak transforms and relations between them are also discussed.

Interface partial differential equations (PDEs) are very important in science and engineering. A new mass preserving solution-flux scheme is proposed in this paper for solving parabolic multi-layer interface problems. In the scheme, the domain is divided into staggered meshes for layers. At grid points in each subdomain, the solution-flux scheme is proposed to approximate the equation. However, due to the interface jump conditions, it is difficult to define the approximate fluxes at the irregular points next to interfaces for satisfying mass conservation for the scheme across the interfaces. The important feature of our work is that at the irregular grid points, the novel corrected approximate fluxes from two sides of the interface are proposed by combining with the interface jump conditions at interfaces, which ensure the developed solution-flux scheme mass conservative while keeping the same accuracy. We prove theoretically that our scheme satisfies mass conservation in the discrete form over the whole domain for the parabolic interface equations with multi-layers. Numerical experiments show mass conservation and convergence orders of our scheme and numerical results in multi-layer media show its excellent performance.

This article is devoted to the stabilization via adaptive feedback for a class of uncertain coupled reaction–diffusion dynamics in the actuation path of an ordinary differential equation (ODE) system. The system under investigation, a class of coupled parabolic partial differential equation (PDE)–ODE systems, is more representative since the dynamics in actuation path (i.e., the PDE subsystem) are coupled rather than uncoupled parabolic equations and hence includes those in the related literature as special cases. Moreover, serious parametric uncertainties are present in both PDE and ODE subsystems, which bring obstacles to the traditional methods on this issue. To solve the control problem, an infinite-dimensional backstepping transformation with time-varying matrix-valued kernel functions is adopted to change the original system into a new one. Then, an adaptive state-feedback controller is designed for the new system, which guarantees that all the closed-loop system states are bounded while regulating the original system states to zero. Finally, a simulation example is provided to illustrate the effectiveness of the theoretical results.

In this paper, we prove the existence of Radon measure-valued solutions for nonlinear degenerate parabolic equations with nonnegative bounded Radon measure data. Moreover, we show the uniqueness of the measure-valued solutions when the Radon measure as a forcing term is diffuse with respect to the parabolic capacity and the Radon measure as a initial value is diffuse with respect to the Newtonian capacity. We also deduce that the concentrated part of the solution with respect to the Newtonian capacity depends on time.

We establish a Schn$\ddot{\text{u}}$rer's convergence result and then apply it to obtain the existence of solutions on the second boundary value problem for a family of special Lagrangian equations

We study nonlinear stability of pulled fronts in scalar parabolic equations on the real line of arbitrary order, under conceptual assumptions on existence and spectral stability of fronts. In this general setting, we establish sharp algebraic decay rates and temporal asymptotics of perturbations to the front. Some of these results are known for the specific example of the Fisher-KPP equation, and our results can thus be viewed as establishing universality of some aspects of this simple model. We also give a precise description of how the spatial localization of perturbations to the front affects the temporal decay rate, across the full range of localizations for which asymptotic stability holds. Technically, our approach is based on a detailed study of the resolvent operator for the linearized problem, through which we obtain sharp linear time decay estimates that allow for a direct nonlinear analysis., 37 pages, 3 figures

We consider systems of \begin{document}$ n $\end{document} parabolic equations coupled in zero or first order terms with \begin{document}$ m $\end{document} scalar controls acting through a control matrix \begin{document}$ B $\end{document} . We are interested in stabilization with a control in feedback form. Our approach relies on the approximate controllability of the linearized system, which in turn is related to unique continuation property for the adjoint system. For the unique continuation we establish algebraic Kalman type conditions.

A notion of parabolic C-subsolutions is introduced for parabolic equations, extending the theory of C-subsolutions recently developed by B. Guan and more specifically G. Sz\'ekelyhidi for elliptic equations. The resulting parabolic theory provides a convenient unified approach for the study of many geometric flows.

This paper considers two problems: the initial boundary value problem of nonlinear Caputo time-fractional pseudo-parabolic equations with fractional Laplacian, and the Cauchy problem (initial value problem) of Caputo time-fractional pseudo-parabolic equations. For the first problem with the source term satisfying the globally Lipschitz condition, we establish the local well-posedness theory including existence, uniqueness and regularity of the local solution, and the further local existence theory related to the finite time blow-up are also obtained for the problem with logarithmic nonlinearity. For the second problem with the source term satisfying the globally Lipschitz condition, we prove the global existence theorem.

We analyze the convergence of a multilevel Markov chain Monte Carlo (MLMCMC) algorithm for the Bayesian estimation of solution functionals for linear, parabolic partial differential equations subje...

This paper deals with the existence and non-xistence of the solutions to the Cauchy problem of a class of quasilinear parabolic equations with a gradient term. We establish Fujita-type blowup theorems and determine the critical Fujita exponent in terms of spatial dimension, the asymptotic behavior of the coefficients of the gradient term at infinity, the exponents of spatial positions in the coefficients of the time-derivative term, and the source term. In particular, we classify the critical case as the blowup case.

We consider a spatial economic growth model with trade costs whose spatial-temporal dynamics of the capital stock are governed by a parabolic partial differential equation and analyze the effects of the trade costs on the asymptotic behaviors of the capital in the framework of spatial AK model, where the technology is time-varying. Taking advantage of the good properties of the Green function, we formulate an explicit solution to the auxiliary partial differential equation of the original problem and derive the sufficient conditions on the consumption function that guarantee the existence and the uniqueness of the solution and obtain the convergence properties of the capital in the long run for the spatial AK model with trade costs.

Parabolic equation methods are a robust and efficient tool for modeling long-range acoustic propagation in range-dependent waveguides. A lesser known, but equally effective, application of parabolic equations is to the scattering problem. In this paper, we study the applicability and accuracy of the multisector parabolic equation approach to scattering from impenetrable objects – first developed approximately two decades ago for convex objects and recently extended to more complex scatterers – in waveguides in two and three dimensions. We benchmark the method in two dimensions in a range-dependent waveguide consisting of two different fluid media and find very good agreement with finite element methods for both forward scattering and backscattering, including the case where the scatterer is located at the interface between the media. Finally, we provide an example of scattering in a large three-dimensional waveguide, which would be extremely computationally intensive using alternate approaches, but is practical using the parabolic equation method.

In this paper, we consider the initial-boundary value problem to the non-isothermal incompressible liquid crystal system with both variable density and temperature. Global well-posedness of strong solutions is established for initial data being small perturbation around the equilibrium state. As the tools in the proof, we establish the maximal regularities of the linear Stokes equations and parabolic equations with variable coefficients and a rigid lemma for harmonic maps on bounded domains. This paper also generalizes the result in [ 5 ] to the inhomogeneous case.

We consider a reaction-diffusion model of biological invasion in which the evolution of the population is governed by several parameters among them the intrinsic growth rate \begin{document}$ \mu(x) $\end{document} . The knowledge of this growth rate is essential to predict the evolution of the population, but it is a priori unknown for exotic invasive species. We prove uniqueness and unconditional Lipschitz stability for the corresponding inverse problem, taking advantage of the positivity of the solution inside the spatial domain and studying its behaviour near the boundary with maximum principles. Our results complement previous works by Cristofol and Roques [ 11 , 13 ].

This paper employs observer-based feedback control technique to discuss the design problem of output feedback fuzzy controllers for a class of nonlinear coupled systems of a parabolic partial differential equation (PDE) and an ordinary differential equation (ODE), where both ODE output and pointwise PDE observation output (i.e., only PDE state information at some specified positions of the spatial domain) are available. Initially, a Takagi-Sugeno (T-S) fuzzy PDE-ODE model is constructed to exactly represent the nonlinear coupled system via the local sector nonlinearity approach. Then, based on the T-S fuzzy PDE-ODE model, a Lunberger-type fuzzy observer is constructed by the ODE output and the pointwise PDE observation output to estimate the nonlinear coupled system state. A fuzzy controller is proposed by the estimated state. Sufficient conditions for existence of the proposed observer-based fuzzy controller are developed such that the exponential stability of the augmented closed-loop system is ensured, and presented in terms of bilinear matrix inequalities (BMIs). A local algorithm via the convex optimization technique is proposed to solve these BMIs. Finally, numerical simulation results are provided to support the proposed design method.

An error estimate is presented for a fully discrete, linearized and stabilized finite element method for solving the coupled system of nonlinear hyperbolic and parabolic equations describing incompressible flow with variable density in a two-dimensional convex polygon. In particular, the error of the numerical solution is split into the temporal and spatial components, separately. The temporal error is estimated by applying discrete maximal Lp-regularity of time-dependent Stokes equations, and the spatial error is estimated by using energy techniques based on the uniform regularity of the solutions given by semi-discretization in time.