Nonlinear Parabolic Partial Differential Equations Research Articles

A selfdual variational principle with minimal hypothesis and applications to stationary, dynamic and stochastic equations

We revisit the self-dual variational approach to the resolution of non-linear stationary, parabolic and stochastic partial differential equations involving a monotone non-linearity, by relaxing the hypothesis under which it is applicable, in two directions: The prohibitive coercivity condition which was problematic for applications to evolution equations, be them deterministic or stochastic, and the restrictive “regularity” property that required the domain of the non-linear operator under study to be large. The new principle is applied to give more direct proofs for the existence of weak solutions for incompressible Navier-Stokes equations in their stationary, dynamic or stochastic versions.

Dynamic Fuzzy Boundary Output Feedback Control for Nonlinear Delayed Parabolic Partial Differential Equation Systems Under Noncollocated Boundary Measurement

For nonlinear space-varying parabolic partial differential equation systems (PPDESs) with random time-varying delay, this paper introduces a dynamic fuzzy boundary output feedback (DFBOF) control under non-collocated boundary measurement (NCBM). Initially, the nonlinear delayed PPDESs are represented by Takagi-Sugeno (T–S) fuzzy models and random time-varying delay is considered by taking the influence of uncertain factors, which belongs to two intervals in a probabilistic way. Since the system state is not fully available and NCBM makes the boundary control design very difficult, a fuzzy observer under NCBM is presented to surmount the design difficulty. Subsequently, an observer-based fuzzy boundary controller is proposed and spatial linear matrix inequality (SLMI) based sufficient conditions to ensure mean square exponential stability are obtained for closed-loop delayed PPDESs by utilizing the Lyapunov direct method and Wirtinger inequality. Then, to solve the SLMIs, the feasibility conditions of DFBOF controller design for nonlinear delayed PPDES are expressed in LMIs. Lastly, two examples are offered to demonstrate the validity of the presented dynamic fuzzy boundary control approach.

System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected Obstacles and Oblique Derivative Boundary Conditions on Non-Smooth Time-Dependent Domains

System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected Obstacles and Oblique Derivative Boundary Conditions on Non-Smooth Time-Dependent Domains

Diffusion - Convection Equation of Galactic Cosmic Rays (GCR) in the Atmosphere and Its Analytical, Numerical Solutions by Using Finite Elements Method Using Parker Transport Equation

This study depicts that to find the Analytical and numerical solutions of Diffusion – Convections Equations of Galactic Cosmic Rays (GCRs) by Finite Element method (FEM) and also to find the Energy Equation of GCRs by using a part of Parker’s transport equation. This considers moreover centres on the exactness and acknowledgment of the FEM strategy by utilizing dissemination blunder, scattering mistake and add up to blunder investigation. The comes about are depicted both graphically and in a unthinkable frame, which essentially guarantees the method’s legitimacy and the algorithm’s proficiency to maintain the exactness, effortlessness, and nonlinear Convection-Diffusion Equation conditions. The proposed method may be connected for tackling any nonlinear convection diffusion equation. We concentrate on assaying the confluence and stability of the nonlinear parabolic partial differential equation. This study focuses on the delicacy and acceptance of FEM method by exercising dispersion error dissipation error, and total error analysis.

Spline collocation methods for solving some types of nonlinear parabolic partial differential equations

Spline collocation methods for solving some types of nonlinear parabolic partial differential equations

A fully nonlinear Feynman–Kac formula with derivatives of arbitrary orders

We present an algorithm for the numerical solution of nonlinear parabolic partial differential equations. This algorithm extends the classical Feynman–Kac formula to fully nonlinear partial differential equations, by using random trees that carry information on nonlinearities on their branches. It applies to functional, non-polynomial nonlinearities that are not treated by standard branching arguments, and deals with derivative terms of arbitrary orders. A Monte Carlo numerical implementation is provided.

Nonlocal fully nonlinear parabolic differential equations arising in time-inconsistent problems

We prove the local well-posedness results, i.e. existence, uniqueness, and stability, of the solutions to a class of nonlocal fully nonlinear parabolic partial differential equations (PDEs), where there is an external time parameter t on top of the temporal and spatial variables (s,y) and thus the problem could be considered as a flow of equations. The nonlocality comes from the dependence on the unknown function and its first- and second-order derivatives evaluated at not only the local point (t,s,y) but also at the diagonal line of the time domain (s,s,y). Such equations arise from time-inconsistent problems in game theory or behavioral economics, where the observations and preferences are (reference-)time-dependent. We first study the linearized version of the nonlocal PDEs with an innovative construction of appropriate norms and Banach spaces and contraction mappings over which. With fixed-point arguments, we obtain the solvability of nonlocal linear PDEs and establish a Schauder-type prior estimate for the solutions. Then, by the linearization method, we establish the well-posedness under the fully nonlinear case. Moreover, we reveal that the solution of a nonlocal fully nonlinear parabolic PDE is an adapted solution to a flow of second-order forward-backward stochastic differential equations.

Editorial: Tensor numerical methods and their application in scientific computing and data science

Editorial: Tensor numerical methods and their application in scientific computing and data science

Half-sweep Modified SOR Approximation of A Two-dimensional Nonlinear Parabolic Partial Differential Equation

The sole subject of this numerical analysis was the half-sweep modified successive over-relaxation approach (HSMSOR), which takes the form of an iterative formula. This study computed a class of two-dimensional nonlinear parabolic partial differential equations subject to Dirichlet boundary conditions numerically using the implicit-type finite difference scheme. The computational cost optimization was considered by converting the traditional implicit finite difference approximation into the half-sweep finite difference approximation. The implementation required inner-outer iteration cycles, the second-order Newton method, and a linearization technique. The created HSMSOR is utilized to obtain an approximation of the linearized equations system through the inner iteration cycle. In contrast, the problem's numerical solutions are obtained using the outer iteration cycle. The study examined the local truncation error and the stability, convergence, and method analysis. Results from three initial-boundary value issues showed that the proposed method had competitive computational costs compared to the existing method.

A numerical method for parabolic complementarity problem

<abstract><p>In this paper, we study the numerical solution of a parabolic complementarity problem which is a widely used model in many fields, such as option pricing, risk measures, etc. Using a power penalty method we represent the complementarity problem as a nonlinear parabolic partial differential equation (PDE). Then, we use the trapezoidal rule as the time discretization, for which we have to solve a nonlinear equation at each time step. We solve such a nonlinear equation by the fixed-point iteration and in this methodology solving a tridiagonal linear system is the major computation. We present an efficient backward substitution algorithm to handle this linear system. Numerical results are given to illustrate the advantage of the proposed algorithm (compared to the built-in command backslash in Matlab) in terms of CPU time.</p></abstract>

A stable numerical scheme for pricing American put options under irrational behavior with rationality parameter

This study investigates the irrational behavior of American put options holders that results in exercising options at non-optimal times. Investors usually react to market information and consequently market movements. These emotional reactions lead to exercising options strategy at a time that might not be optimal. In this situation, we consider irrational behavior in the option pricing problem. For this, we used the proposed intensity-based models with stochastic intensity parameters. Under these models, the option pricing problem leads to a nonlinear parabolic partial differential equation (PDE) with an additional term to the PDE of the American option under rational strategy (classical American option with optimal exercise strategy) due to the intensity functions of models. In this paper, we are interested in finding a stable solution for the resulting PDE using a finite element method. For this, we show the stability of the proposed finite element method by proving some theoretical results. Our numerical experiments demonstrate the accuracy and efficiency of the proposed method to obtain fast solutions for the pricing problem of American put options under irrational behavior.

Orthogonal cubic splines for the numerical solution of nonlinear parabolic partial differential equations

In this paper, a new orthogonal basis for the space of cubic splines has been introduced. A linear combination of cubic orthogonal splines is considered to approximate the functions in which the coefficients are calculated with numerically stable formulae. Applications to the numerical solutions of some parabolic partial differential equations are given, in which the approximations are obtained using the first and second integral of orthogonal splines which leads to an efficient solution procedure. The convergence analysis in the approximate scheme is investigated. A comparison of the obtained numerical solutions with some other papers indicates that the presented method is reliable and yields result with good accuracy. The main parts of our study are as follows:•We propose a robust approach based on the orthogonal cubic splines procedure in conjunction with the operational matrix.•The convergence in the approximate scheme is analyzed.•Numerical examples show that the proposed method is very accurate.

Optimal subalgebras and conservation laws with exact solutions for biological population model

In the present study, we focus on the (2+1) dimensional normal biological population model (NBPM), which describes the population migration of species. We employ Lie symmetry analysis to the given nonlinear degenerate parabolic partial differential equation (PDE), which shows substantial advancement and upgraded results over other analytical techniques in determining some classes of exact solutions. Using the symmetry group of transformations, we construct the one-dimensional and two-dimensional optimal subalgebras for the NBPM. Further, we present the reduced ordinary differential equation(ODE) for each one-dimensional optimal subalgebras and construct some exact solutions for the physical model. Furthermore, we illustrate the physical behaviour of the model graphically through the obtained exact solutions. Lastly, applying the multipliers method, we develop some new conservation laws yielding some potential systems which are nonlocally related to the given PDE.

Partial Differential Equations

The workshop covered topics in nonlinear elliptic and parabolic Partial Differential Equations as well as topics in Geometric Measure Theory, touching topics such as geometric variational problems and minimal surfaces, geometric flows, free boundaries and the structure of nodal sets of eigenfunctions as well as real and complex Monge–Ampère equations.

Hybrid‐driven‐based fuzzy secure filtering for nonlinear parabolic partial differential equation systems with cyber attacks

SummaryThe problem of hybrid‐driven fuzzy filtering for nonlinear semi‐linear parabolic partial differential equation systems with dual cyber attacks is investigated. First, a Takagi‐Sugeno (T‐S) fuzzy model is employed to reconstruct the original nonlinear systems. Second, a hybrid‐driven mechanism is applied for filter design to balance the systems' performance and the limited network resource consumption under dual cyber attacks composed of deception and denial of service attacks. Based on the Lyapunov direct method, sufficient conditions to guarantee the stability of the augmented system are obtained, and the parameters of the designed filter are earned with explicit form. Finally, a simulation example with robust analysis and comparative analysis is provided to illustrate the effectiveness of the proposed method.

A learning scheme by sparse grids and Picard approximations for semilinear parabolic PDEs

Abstract Relying on the classical connection between backward stochastic differential equations and nonlinear parabolic partial differential equations (PDEs), we propose a new probabilistic learning scheme for solving high-dimensional semilinear parabolic PDEs. This scheme is inspired by the approach coming from machine learning and developed using deep neural networks in Han et al. (2018, Solving high-dimensional partial differential equations using deep learning. Proc. Natl. Acad. Sci., 115, 8505–8510. Our algorithm is based on a Picard iteration scheme in which a sequence of linear-quadratic optimization problem is solved by means of stochastic gradient descent algorithm. In the framework of a linear specification of the approximation space, we manage to prove a convergence result for our scheme, under some smallness condition. In practice, in order to be able to treat high-dimensional examples, we employ sparse-grid approximation spaces. In the case of periodic coefficients and using pre-wavelet basis functions, we obtain an upper bound on the global complexity of our method. It shows, in particular, that the curse of dimensionality is tamed in the sense that in order to achieve a root mean squared error of order $\varepsilon $, for a prescribed precision $\varepsilon $, the complexity of the Picard algorithm grows polynomially in $\varepsilon ^{-1}$ up to some logarithmic factor $|\!\log (\varepsilon )|$, whose exponent grows linearly with respect to the PDE dimension. Various numerical results are presented to validate the performance of our method, and to compare them with some recent machine learning schemes proposed in E et al. (2017, Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Commun. Math. Stat., 5, 349–380) and Huré et al. (2020, Deep backward schemes for high-dimensional nonlinear PDEs. Math. Comput., 89, 1547–1579).

Travelling Wave Analysis of a Diffusive COVID-19 Model

In this paper, a mathematical model based on a system of nonlinear parabolic partial differential equations is developed to investigate the effect of human mobility on the dynamics of coronavirus 2019 (COVID-19) disease. Positivity and boundedness of the model solutions are shown. The existence of the disease-free, the endemic equilibria, and the travelling wave solutions of the model are shown. From the numerical analysis, it is shown that human mobility plays a crucial role in the disease transmission. Therefore, interventions that affect diffusion (human mobility), such as lock-down, travel restrictions, and cessation of movement, may play a significant role in controlling and preventing the spread of COVID-19.

Relative Stability in the Sup-Norm and Input-to-State Stability in the Spatial Sup-Norm for Parabolic PDEs

In this article, we introduce the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">relative <inline-formula><tex-math notation="LaTeX">$\mathcal {K}$</tex-math></inline-formula>-equi-stability</i> (RKES) to characterize the uniformly continuous dependence of (weak) solutions on external disturbances for nonlinear parabolic partial differential equations (PDEs). Based on the RKES, we prove the input-to-state stability (ISS) in the spatial sup-norm for a class of nonlinear parabolic PDEs with either Dirichlet or Robin boundary disturbances. An example concerned with a superlinear parabolic PDE with Robin boundary condition is provided to illustrate the obtained ISS results. Besides, as an application of the notion of RKES, we conduct stability analysis for a class of parabolic PDEs in cascade coupled over the domain or on the boundary of the domain, in the spatial and time sup-norm, and in the spatial sup-norm, respectively. The technique of De Giorgi iteration is extensively used in the proof of the results presented in this article.

Adaptive Event-Triggered Control of Networked Fuzzy PDE Systems Under Hybrid Cyber Attacks

This study focuses on the security control problem of networked fuzzy partial differential equation (PDE) systems under hybrid cyber attacks. First, the nonlinear parabolic PDE system is exactly represented by a Takagi–Sugeno fuzzy PDE model. Second, with taking the insecure transmission network into consideration, a more practical stochastic hybrid cyber attacks model for PDE-based networked control systems is introduced. This model contains both denial-of-service attacks and deception attacks in a unified framework. Then, based on spatially pointwise nonuniform sampled-data measurements, an adaptive event-triggered control scheme with an adjustable threshold is newly designed to counter the hybrid cyber attacks’ impact while saving communication resources. Moreover, by constructing a novel Lyapunov–Krasovskii functional and employing some inequality techniques, stability conditions with less conservatism are derived. Finally, the developed method is applied to the Fisher equation to demonstrate its viability and superiority.

John–Nirenberg inequalities for parabolic BMO

AbstractWe discuss a parabolic version of the space of functions of bounded mean oscillation related to a doubly nonlinear parabolic partial differential equation. Parabolic John–Nirenberg inequalities, which give exponential decay estimates for the oscillation of a function, are shown in the natural geometry of the partial differential equation. Chaining arguments are applied to change the time lag in the parabolic John–Nirenberg inequality. We also show that the quasihyperbolic boundary condition is a necessary and sufficient condition for a global parabolic John–Nirenberg inequality. Moreover, we consider John–Nirenberg inequalities with medians instead of integral averages and show that this approach gives the same class of functions as the original definition.