Deterministic Partial Differential Equations Research Articles

One-dimensional Kardar-Parisi-Zhang and Kuramoto-Sivashinsky universality class: Limit distributions.

Tracy-Widom and Baik-Rains distributions appear as universal limit distributions for height fluctuations in the one-dimensional Kardar-Parisi-Zhang (KPZ) stochastic partial differential equation (PDE). We obtain the same universal distributions in the spatiotemporally chaotic, nonequilibrium, but statistically steady state of the one-dimensional Kuramoto-Sivashinsky (KS) deterministic PDE, by carrying out extensive pseudospectral direct numerical simulations to obtain the spatiotemporal evolution of the KS height profile h(x,t) for different initial conditions. We establish, therefore, that the statistical properties of the one-dimensional (1D) KS PDE in this state are in the 1D KPZ universality class.

T-IFISS: a toolbox for adaptive FEM computation

T-IFISS is a finite element software package for studying finite element solution algorithms for deterministic and parametric elliptic partial differential equations. The emphasis is on self-adaptive algorithms with rigorous error control using a variety of a posteriori error estimation techniques. The open-source MATLAB framework provides a computational laboratory for experimentation and exploration, enabling users to quickly develop new discretizations and test alternative algorithms. The package is also valuable as a teaching tool for students who want to learn about state-of-the-art finite element methodology.

Pathwise mild solutions for quasilinear stochastic partial differential equations

Stochastic partial differential equations (SPDEs) have become a key modeling tool in applications. Yet, there are many classes of SPDEs, where the existence and regularity theory for solutions is not completely developed. Here we contribute to this aspect and prove the existence of mild solutions for a broad class of quasilinear Cauchy problems, including - among others - cross-diffusion systems as a key application. Our solutions are local-in-time and are derived via a fixed point argument in suitable function spaces. The key idea is to combine in a suitable way the classical theory of deterministic quasilinear parabolic partial differential equations (PDEs) with recent theory of evolution semigroups. We also show, how to apply our theory to the Shigesada-Kawasaki-Teramoto (SKT) model. Furthermore, we provide examples of blow-up and ill-posed operators, which can occur after finite-time showing that solutions can only be local-in-time for general quasilinear SPDEs, while they might be global-in-time for special subclasses of problems.

Traveling wave solutions of some important Wick-type fractional stochastic nonlinear partial differential equations

In this article, exact traveling wave solutions of a Wick-type stochastic nonlinear Schr\"{o}dinger equation and of a Wick-type stochastic fractional Regularized Long Wave-Burgers (RLW-Burgers) equation have been obtained by using an improved computational method. Specifically, the Hermite transform is employed for transforming Wick-type stochastic nonlinear partial differential equations into deterministic nonlinear partial differential equations with integral and fraction order. Furthermore, the required set of stochastic solutions in the white noise space is obtained by using the inverse Hermite transform. Based on the derived solutions, the dynamics of the considered equations are performed with some particular values of the physical parameters. The results reveal that the proposed improved computational technique can be applied to solve various kinds of Wick-type stochastic fractional partial differential equations.

On Bernstein processes generated by hierarchies of linear parabolic systems in [formula omitted

In this article we investigate the properties of Bernstein processes generated by infinite hierarchies of forward–backward systems of decoupled linear deterministic parabolic partial differential equations defined in Rd, where d is arbitrary. An important feature of those systems is that the elliptic part of the parabolic operators may be realized as an unbounded Schrödinger operator with compact resolvent in standard L2-space. The Bernstein processes we are interested in are in general non-Markovian, may be stationary or non-stationary and are generated by weighted averages of measures naturally associated with the pure point spectrum of the operator. We also introduce time-dependent trace-class operators which possess several attributes of density operators in Quantum Statistical Mechanics, and prove that the statistical averages of certain bounded self-adjoint observables usually evaluated by means of such operators coincide with the expectation values of suitable functions of the underlying processes. In the particular case where the given parabolic equations involve the Hamiltonian of an isotropic system of quantum harmonic oscillators, we show that one of the associated processes is identical in law with the periodic Ornstein–Uhlenbeck process.

On Mean Field Games with Common Noise based on Stable-Like Processes

In this paper, we study Mean Field games with common noise based on nonlinear stable-like processes. The MFG limit is specified by a single quasi-linear deterministic infinite-dimensional partial differential second order backward equation. The main result is that any its solution provides an 1/N-Nash equilibrium for the initial game of N agents. Our approach is based on interpreting the common noise as a kind of binary interaction of agents and our previous results on regularity and sensitivity with respect to the initial conditions of the solution to the nonlinear stochastic differential equations of McKean-Vlasov type generated by stable-like processes.

On mean field games with common noise and McKean-Vlasov SPDEs

We formulate the MFG limit for N interacting agents with a common noise as a single quasi-linear deterministic infinite-dimensional partial differential second order backward equation. We prove that any (regular enough) solution provides an -Nash-equilibrium profile for the initial N-player game. We use the method of stochastic characteristics to provide the link with the basic models of MFG with a major player. We develop two auxiliary theories of independent interest: sensitivity and regularity analysis for McKean-Vlasov SPDEs and the -convergence rate for the propagation of chaos property of interacting diffusions.

A novel approach for stochastic solutions of wick-type stochastic time-fractional Benjamin–Bona–Mahony equation for modeling long surface gravity waves of small amplitude

Objectives: In the paper, two new reliable analytical methods have been devised for getting new exact analytical solutions of wick-type stochastic time-fractional Benjamin-Bona-Mahony (BBM) equation. Moreover, the Hermite transform andinverse Hermite transform have been utilized for converting fractional stochastic differential equation to deterministic fractional partial differential equation and vice versa respectively. Here for reducing fractional partial differential equations (FPDE) to the ordinary differential equation (ODE), fractional complex transform has been utilized.Methods: The authors have used a newly proposed method and Kudryshov method for getting the solutions for wick-type stochastic time-fractional Benjamin-Bona-Mahony (BBM) equation.Results: By using two reliable methods, here, the authors find the new exact solutions for the governing equations.Conclusion: Two new approaches to find solutions of the aforementioned equation have been established. Also, the new exact solutions have been obtained for stochastic differential equation by using two methods.

An efficient SPDE approach for El Niño

We consider the numerical approximation of stochastic partial differential equations (SPDEs) based models for a quasi-periodic climate pattern in the tropical Pacific Ocean known as El Niño phenomenon. We show that for these models the mean and the covariance are given by a deterministic partial differential equation and by an operator differential equation, respectively. In this context, we provide a numerical framework to approximate these parameters directly. We compare this method to stochastic differential equations and SPDEs based models from the literature solved by Taylor methods and stochastic Galerkin methods, respectively. Numerical results for different scenarios taking as a reference measured data of the years 2014 and 2015 (last Niño event) validate the efficiency of our approach.

Solving of Some Random Partial Differential Equations by Using Differential Transformation Method and Laplace- Padé Method

In this study, the solutions of random partial differential equations are examined. The parameters and the initial conditions of the random component partial differential equations are investigated with Beta distribution. A few examples are given to illustrate the efficiency of the solutions obtained with the random Differential Transformation Method (rDTM). Functions for the expected values and the variances of the approximate analytical solutions of the random equations are obtained. Random Differential Transformation Method is applied to examine the solutions of these partial differential equations and MAPLE software is used for the finding the solutions and drawing the figures. Also the Laplace- Pade Method is used to improve the convergence of the solutions. The results for the random component partial differential equations with Beta distribution are analysed to investigate effects of this distribution on the results. Random characteristics of the equations are compared with the results of the deterministic partial differential equations. The efficiency of the method for the random component partial differential equations is investigated by comparing the formulas for the expected values and variances with results from the simulations of the random equations.

Computing Optimal Properties of Drugs Using Mathematical Models of Single Channel Dynamics

Abstract Single channel dynamics can be modeled using stochastic differential equations, and the dynamics of the state of the channel (e.g. open, closed, inactivated) can be represented using Markov models. Such models can also be used to represent the effect of mutations as well as the effect of drugs used to alleviate deleterious effects of mutations. Based on the Markov model and the stochastic models of the single channel, it is possible to derive deterministic partial differential equations (PDEs) giving the probability density functions (PDFs) of the states of the Markov model. In this study, we have analyzed PDEs modeling wild type (WT) channels, mutant channels (MT) and mutant channels for which a drug has been applied (MTD). Our aim is to show that it is possible to optimize the parameters of a given drug such that the solution of theMTD model is very close to that of the WT: the mutation’s effect is, theoretically, reduced significantly.We will present the mathematical framework underpinning this methodology and apply it to several examples. In particular, we will show that it is possible to use the method to, theoretically, improve the properties of some well-known existing drugs.

Solving Stochastic LQR Problems by Polynomial Chaos

We consider the infinite dimensional stochastic linear quadratic optimal control problem for the infinite horizon case. We provide a numerical framework for solving this problem using a polynomial chaos expansion approach. By applying the method of chaos expansions to the state equation, we obtain a system of deterministic partial differential equations in terms of the coefficients of the state and the control variables. We set up a control problem for each equation, which results in a set of infinite horizon deterministic linear quadratic regulator problems. We prove the optimality of the solution expressed in terms of the expansion of these coefficients compared to the direct approach. We perform numerical experiments which validate our approach and compare the finite and infinite horizon case.

Parameter estimation in SDEs via the Fokker–Planck equation: Likelihood function and adjoint based gradient computation

In this paper we consider the problem of identifying parameters in stochastic differential equations. For this purpose, we transform the originally stochastic and nonlinear state equation to a deterministic linear partial differential equation for the transition probability density. We provide an appropriate likelihood cost function for parameter fitting, and derive an adjoint based approach for the computation of its gradient.

A Fully Parallelizable Space-Time Multilevel Monte Carlo Method for Stochastic Differential Equations with Additive Noise

In this work a combination of parallelizable space-time multigrid methods for deterministic parabolic partial differential equations with multilevel Monte Carlo methods for stochastic differential equations with additive noise is developed. Instead of applying the backward Euler--Maruyama scheme sequentially for every time step, the basic idea for the considered space-time method is to solve a large linear system at once, for which a parallelizable multigrid algorithm is constructed that inherits the space-time hierarchy of the multilevel Monte Carlo method. Overall, this results in a fully parallelizable algorithm with respect to space, time, and probability. As model problems for the numerical testing of the proposed method serve in finite dimensions the Ornstein--Uhlenbeck process and in infinite dimensions the stochastic heat equation in 1, 2, and 3 space dimensions.

On the time evolution of Bernstein processes associated with a class of parabolic equations

In this article dedicated to the memory of Igor D. Chueshov, I first summarize in a few words the joint results that we obtained over a period of six years regarding the long-time behavior of solutions to a class of semilinear stochastic parabolic partial differential equations. Then, as the beautiful interplay between partial differential equations and probability theory always was close to Igor's heart, I present some new results concerning the time evolution of certain Markovian Bernstein processes naturally associated with a class of deterministic linear parabolic partial differential equations. Particular instances of such processes are certain conditioned Ornstein-Uhlenbeck processes, generalizations of Bernstein bridges and Bernstein loops, whose laws may evolve in space in a non trivial way. Specifically, I examine in detail the time development of the probability of finding such processes within two-dimensional geometric shapes exhibiting spherical symmetry. I also define a Faedo-Galerkin scheme whose ultimate goal is to allow approximate computations with controlled error terms of the various probability distributions involved.

Quasi-Linear (Stochastic) Partial Differential Equations with Time-Fractional Derivatives

In this paper we develop a method to solve (stochastic) evolution equations on Gelfand triples with time-fractional derivative based on monotonicity techniques. Applications include deterministic and stochastic quasi-linear partial differential equations with time-fractional derivatives, including time-fractional (stochastic) porous media equations (including the case where the Laplace operator is also fractional) and $p$-Laplace equations as special cases.

Exact solutions for the Wick-type stochastic Kersten-Krasil’shchik coupled KdV-mKdV equations

In this article, exact solutions of Wick-type stochastic Kersten-Krasil’shchik coupled KdV-mKdV equations have been obtained by using the Jacobian elliptic function expansion method. We have used the Hermite transform for transforming the Wick-type stochastic Kersten-Krasil’shchik coupled KdV-mKdV equation into a deterministic partial differential equation. Also, we have applied the inverse Hermite transform for obtaining a set of stochastic solutions in the white noise space.

Multi-level Monte Carlo weak Galerkin method with nested meshes for stochastic Brinkman problem

This paper is devoted to the numerical analysis of a multi-level Monte Carlo weak Galerkin (MLMCWG) approximation with nested meshes for solving stochastic Brinkman equations with two dimensional spatial domain. With weak gradient operator and a stabilizer at hand, the weak Galerkin (WG) technique is a high-order accurate and stable method which can easily handle deterministic partial differential equations with complex geometries, flows with jump fluid viscosity coefficients or high-contrast permeability fields given by each sample. The multi-level Monte Carlo (MLMC) technique with nested meshes balances the sampling error and the spatial approximation error, where the computational cost can be sharply reduced to log-linear complexity with respect to the degree of freedom in spatial direction. The nested meshes requirement is introduced here in order to simplify the analysis, which can be generalized to MLMC with non-nested meshes. Error estimates are derived in terms of the spatial meshsize and the number of samples. The numerical tests are provided to illustrate the behavior of the MLMCWG method and verify our theoretical results regarding optimal convergence of the approximate solutions.

A Two-Level Sparse Grid Collocation Method for Semilinear Stochastic Elliptic Equation

Abstract In this work, we investigate a novel two-level discretization method for the elliptic equations with random input data. Motivated by the two-grid method for deterministic nonlinear partial differential equations introduced by Xu [36], our two-level discretization method uses a two-grid finite element method in the physical space and a two-scale stochastic collocation method with sparse grid in the random domain. Specifically, we solve a semilinear equations on a coarse mesh 𝒯 H ⁢ ( D ) {\mathcal{T}_{H}(D)} with small scale of sparse collocation points η ⁢ ( L , N ) {\eta(L,N)} and solve a linearized equations on a fine mesh 𝒯 h ⁢ ( D ) {\mathcal{T}_{h}(D)} using large scale of sparse collocation points η ⁢ ( ℓ , N ) {\eta(\ell,N)} (where η ⁢ ( L , N ) , η ⁢ ( ℓ , N ) {\eta(L,N),\eta(\ell,N)} are the numbers of sparse grid with respect to different levels L , ℓ {L,\ell} in N dimensions). Moreover, an error correction on the coarse mesh with large scale of collocation points is used in the method. Theoretical results show that when h ≈ H 3 , η ⁢ ( ℓ , N ) ≈ ( η ⁢ ( L , N ) ) 3 {{h\approx H^{3},\eta(\ell,N)\approx(\eta(L,N))^{3}}} , the novel two-level discretization method achieves the same convergence accuracy in norm ∥ ⋅ ∥ ℒ ρ 2 ⁢ ( Γ ) ⊗ ℒ 2 ⁢ ( D ) {\|\cdot\|_{\mathcal{L}_{\rho}^{2}(\Gamma)\otimes\mathcal{L}^{2}(D)}} ( ℒ ρ 2 ⁢ ( Γ ) {\mathcal{L}_{\rho}^{2}(\Gamma)} is the weighted ℒ 2 {\mathcal{L}^{2}} space with ρ a probability density function) as that for the original semilinear problem directly by sparse grid stochastic collocation method with 𝒯 h ⁢ ( D ) {\mathcal{T}_{h}(D)} and large scale collocation points η ⁢ ( ℓ , N ) {\eta(\ell,N)} in random spaces.

Efficient solution algorithms for multiaxial probabilistic elasto‐plastic constitutive simulations of soils

SummaryA Fokker‐Planck‐Kolmogorov (FPK) equation approach has recently been developed to probabilistically solve any elastic‐plastic constitutive equation with uncertain material parameters by transforming the nonlinear, stochastic constitutive rate equation into a linear, deterministic partial differential equation (PDE) and thereby simplifying the numerical solution process. For an uniaxial problem, conventional numerical techniques, such as the finite difference or finite element methods, may be used to solve the resulting univariate FPK PDE. However, for a multiaxial problem, an efficient algorithm is necessary for tractability of the numerical solution of the multivariate FPK PDE. In this paper, computationally efficient algorithms, based on a Fourier spectral approach, are presented for solving FPK PDEs in (stress) space and (pseudo) time, having space‐independent but time‐dependent coefficients and both space‐ and time‐dependent coefficients, that commonly arise in probabilistic elasto‐plasticity. The algorithms are illustrated by probabilistically simulating 2 common laboratory constitutive experiments in geotechnical engineering, namely, the unconfined compression test and the unconsolidated undrained triaxial compression test.