Semilinear PDEs Research Articles

A semilinear diffusion PDE with variable order time-fractional Caputo derivative subject to homogeneous Dirichlet boundary conditions

A semilinear diffusion PDE with variable order time-fractional Caputo derivative subject to homogeneous Dirichlet boundary conditions

Rigorous Computation of Solutions of Semilinear PDEs on Unbounded Domains via Spectral Methods

Rigorous Computation of Solutions of Semilinear PDEs on Unbounded Domains via Spectral Methods

RBF–based IMEX finite difference schemes for pricing option under liquidity switching

In this work, we introduce two accurate and efficient finite difference methods based on radial basis functions (RBF–FD) for pricing European and American options under liquidity shocks. The problem is formulated as the semi-linear complementarity and PDE systems for American and European options, respectively. In the context of temporal semi-discretization, we provide two backward difference formulas of order one and two (BDF1 & BDF2). Furthermore, we discuss the stability and convergence properties of the proposed methods. For the American option, to solve the semi-linear system of complementarity problems (LCPs), we combine the RBF-FD approaches with an operator splitting (OS) method. To illustrate the efficiency and accuracy of the suggested methods, we provide numerical examples for both European and American call options and verify them with the existing work in the literature. In numerical discussion, we show the Greeks (Delta & Gamma) plots for the American options.

Validated integration of semilinear parabolic PDEs

Validated integration of semilinear parabolic PDEs

The Gevrey class implicit mapping theorem with application to UQ of semilinear elliptic PDEs

This paper is concerned with a regularity analysis of parametric operator equations with a perspective on uncertainty quantification. We study the regularity of mappings between Banach spaces near branches of isolated solutions that are implicitly defined by a residual equation. Under [Formula: see text]-Gevrey assumptions on the residual equation, we establish [Formula: see text]-Gevrey bounds on the Fréchet derivatives of the locally defined data-to-solution mapping. This abstract framework is illustrated in a proof of regularity bounds for a semilinear elliptic partial differential equation with parametric and random field input.

Finite element error analysis of affine optimal control problems

This paper is concerned with error estimates for the numerical approximation for affine optimal control problems subject to semilinear elliptic PDEs. To investigate the error estimates, we focus on local minimizers that satisfy certain local growth conditions. The local growth conditions we consider in this paper appeared recently in the context of solution stability and contain the joint growth of the first and second variation of the objective functional. These growth conditions are especially meaningful for affine control constrained optimal control problems because the first variation can satisfy a local growth, which is not the case for unconstrained problems. The main results of this paper are the achievement of error estimates for the numerical approximations generated by a finite element scheme with piecewise constant controls or a variational discretization scheme. Even though the growth conditions considered are weaker than those appearing in the recent literature on finite element error estimates for affine problems, this paper substantially improves the existing error estimates for both the optimal controls and the states when a Hölder-type growth is assumed.

Non-Collocated Output Feedback Control for the Semi-Linear Parabolic PDEs Under Input and Output Quantization

Non-Collocated Output Feedback Control for the Semi-Linear Parabolic PDEs Under Input and Output Quantization

Error estimates for the discretization of bilinear control problems governed by semilinear elliptic PDEs

Error estimates for the discretization of bilinear control problems governed by semilinear elliptic PDEs

Classical solution of path-dependent mean-field semilinear PDEs

Classical solution of path-dependent mean-field semilinear PDEs

A domain-dependent stability analysis of reaction–diffusion systems with linear cross-diffusion on circular domains

In this study, we present theoretical considerations of, and analyse, the effects of circular geometry on the stability analysis of semi-linear parabolic PDEs of reaction–diffusion type with linear cross-diffusion for a two-component system on circular domains. The highlights of our theoretical and computational findings are: (i) By employing linear stability analysis for a two-component reaction–diffusion system with linear cross-diffusion on circular disc domains, we derive necessary and sufficient conditions for the system to exhibit cross-diffusion driven-instability, dependent on the length scale of the geometry. These analytical studies involve cross-diffusion and circular geometry to unravel analytical conditions for the full computational classification of the parameter spaces that allow the system to exhibit Turing, Hopf and transcritical patterns. (ii) We compute parameter spaces on which patterns are formed only due to linear cross-diffusion as well as due to a critical domain length. These spaces do not exist in the absence of cross-diffusion nor when the conditions on the domain length are violated. (iii) To support our theoretical findings, finite element simulations illustrating the formation of spot patterns on circular domains are presented. Model parameter values are selected from parameter spaces that are induced by cross-diffusion, thereby supporting linear cross-diffusion coupled with reaction–diffusion theory as a candidate mechanism for pattern formation. (iv) A by-product of this study, is that an activator-depleted reaction–diffusion system with linear cross-diffusion on circular domains, appears to favour the formation of spot patterns for most of the parameter values chosen. Such patterns are reminiscent of those observed on stingrays, which form on approximately circular domains during growth development.

On solving elliptic obstacle problems by constant abs-linearization

We consider optimal control problems governed by an elliptic variational inequality of the first kind, namely the obstacle problem. The variational inequality is treated by penalization, which leads to optimization problems governed by a nonsmooth semi-linear elliptic PDE. The CALi algorithm is then applied for the efficient solution of these nonsmooth optimization problems. The special feature of the optimization algorithm CALi is the treatment of the nonsmooth Lipschitz-continuous operators abs, max and min, which allows to explicitly exploit the nonsmooth structure. Stationary points are located by appropriate decomposition of the optimization problem into so-called smooth constant abs-linearized problems. Each of these constant abs-linearized problems can be solved by classical means. The comprehensive algorithmic concept is presented, and its performance is discussed through examples.

Deep high-order splitting method for semilinear degenerate PDEs and application to high-dimensional nonlinear pricing models

Deep high-order splitting method for semilinear degenerate PDEs and application to high-dimensional nonlinear pricing models

The invariant manifold approach applied to global long-time dynamics of FitzHugh-Nagumo systems

We consider the FitzHugh-Nagumo system equipped with boundary condition of Dirichlet type on some two/three-dimensional domains. This system describes the signal transmission across the axonal membrane in neurophysiology. It is a semilinear parabolic PDE for the voltage variable coupled with a first-order ODE of space-time type for the recovery variable. We prove that there exists a finite-dimensional global manifold in the case of the fast recovery variable. Since the manifold is uniformly attracting, it gives geometric insight into the global long-time dynamics of the solutions. The proof is based on an abstract invariant manifold theorem for dynamical systems on a Banach space.

L∞-Convergence Analysis of a Finite Element Linear Schwarz Alternating Method for a Class of Semi-Linear Elliptic PDEs

In this paper, we prove uniform convergence of the standard finite element method for a Schwarz alternating procedure for a class of semi-linear elliptic partial differential equations, in the context of linear iterations and non-matching grids. More precisely, making use of the subsolution-based concept, we prove that finite element Schwarz iterations converge, in the maximum norm, to the true solution of the PDE. We also give numerical results to validate the theory. This work introduces a new approach and generalizes the one in [14] as it encompasses a larger class of problems.

Regularity of time-periodic solutions to autonomous semilinear hyperbolic PDEs

This paper concerns autonomous boundary value problems for 1D semilinear hyperbolic PDEs. For time-periodic classical solutions, which satisfy a certain non-resonance condition, we show the following: If the PDEs are continuous with respect to the space variable x and C∞-smooth with respect to the unknown function u, then the solution is C∞-smooth with respect to the time variable t, and if the PDEs are C∞-smooth with respect to x and u, then the solution is C∞-smooth with respect to t and x. The same is true for appropriate weak solutions. Moreover, we show examples of time-periodic functions, which do not satisfy the non-resonance condition, such that they are weak, but not classical solutions, and such that they are classical solutions, but not C∞-smooth, neither with respect to t nor with respect to x, even if the PDEs are C∞-smooth with respect to x and u. For the proofs we use Fredholm solvability properties of linear time-periodic hyperbolic PDEs and a result of E. N. Dancer about regularity of solutions to abstract equivariant equations.

Cost-optimal adaptive iterative linearized FEM for semilinear elliptic PDEs

We consider scalar semilinear elliptic PDEs where the nonlinearity is strongly monotone, but only locally Lipschitz continuous. We formulate an adaptive iterative linearized finite element method (AILFEM) which steers the local mesh refinement as well as the iterative linearization of the arising nonlinear discrete equations. To this end, we employ a damped Zarantonello iteration so that, in each step of the algorithm, only a linear Poisson-type equation has to be solved. We prove that the proposed AILFEM strategy guarantees convergence with optimal rates, where rates are understood with respect to the overall computational complexity (i.e., the computational time). Moreover, we formulate and test an adaptive algorithm where also the damping parameter of the Zarantonello iteration is adaptively adjusted. Numerical experiments underline the theoretical findings.

Non-existence of patterns for a class of weighted degenerate operators

A classical result by Casten-Holland and Matano asserts that constants are the only positive and stable solutions to semilinear elliptic PDEs subject to homogeneous Neumann boundary condition in bounded convex domains. In other terms, this result asserts that stable patterns do not exist in convex domains.In this paper we consider a weighted version of the Laplace operator, where the weight may be singular or degenerate at the origin, and prove the nonexistence of patterns, extending the results by Casten-Holland and Matano to general weak solutions (not necessarily stable) and under a suitable assumption on the nonlinearity and the domain.Our results exhibit some intriguing behavior of the problem according to the weight and the geometry of the domain. Indeed, our main results follow from a geometric assumption on the second fundamental form of the boundary in terms of a parameter which describes the degeneracy of the operator. As a consequence, we provide some examples and show that nonexistence of patterns may occur also for non convex domains whenever the weight is degenerate.

Symmetry of solutions to semilinear PDEs on Riemannian domains

This paper deals with symmetry phenomena for solutions to the Dirichlet problem involving semilinear PDEs on Riemannian domains. We shall present a rather general framework where the symmetry problem can be formulated and provide some evidence that this framework is completely natural by pointing out some results for stable solutions. The case of manifolds with density, and corresponding weighted Laplacians, is inserted in the picture from the very beginning.

Hierarchical exact controllability of hyperbolic equations and Dunford-Pettis' theorem

This paper deals with the Stackelberg-Nash strategies for boundary control problems for linear and semilinear wave equations. Assuming that we can act on the system through a hierarchy of controls, to each leader, we associate a Nash equilibrium (two followers) corresponding to a bi-objective optimal control problem. Then we look for a leader that solves an exact boundary controllability problem. We mainly consider the case of a semilinear hyperbolic PDE where all the controls (leader and followers) act on small parts of the boundary. In view of the lack of regularity, the arguments usually invoked to prove controllability do not work here. Accordingly, we carry out a fixed-point approach based on weak compactness in $L^1$. To this purpose, we apply Dunford-Pettis' Theorem. We also consider the case where the followers act in small subsets of the domain.

Qualitative Properties of Solutions to a Class of Sixth-Order Equations

In this paper, we present a detailed study of a class of sixth-order semilinear PDEs: existence, regularity and uniqueness. The uniqueness results are a consequence of a maximum principle called the P-function method.