Degenerate Partial Differential Equations Research Articles

Self-Similar Solutions of a Multidimensional Degenerate Partial Differential Equation of the Third Order

When studying the boundary value problems’ solvability for some partial differential equations encountered in applied mathematics, we frequently need to create systems of partial differential equations and explicitly construct linearly independent solutions explicitly for these systems. Hypergeometric functions frequently serve as solutions that satisfy these systems. In this study, we develop self-similar solutions for a third-order multidimensional degenerate partial differential equation. These solutions are represented using a generalized confluent Kampé de Fériet hypergeometric function of the third order.

Decoupling numerical method based on deep neural network for nonlinear degenerate interface problems

Many practical problems, including modeling composite materials, nuclear waste disposal, oil reservoir simulations, and flows in porous medium, commonly involve interface problems. However, the solution to interface problems with discontinuous coefficients of PDEs using fully decoupled numerical methods is challenging. The main objective is to solve the interface problems with fully decoupled numerical methods. This paper proposes an efficient decoupled numerical method for solving degenerate interface problems with double singularities. First, we divide the whole domain into singular and regular subdomains. Then, we use the Deep Neural Network (DNN) to find the solution on the singular subdomain and approximate the solution on the regular subdomain using the finite difference method. The scheme combines the solutions of singular and regular subdomains, which is an exciting idea. The key to the new approach is to split nonlinear degenerate partial differential equations with an interface into two independent boundary value problems based on deep learning. In this way, the expansion of the solution on the singular domain does not contain undetermined parameters, and two independent boundary value problems can be solved with any well-known traditional numerical methods. The main advantage of the proposed scheme is that we not only get the order of convergence of the degenerate interface problems on the whole domain, but we also can calculate VERY BIG jump ratio (such as 1012:1 or 1:1012) for the interface problems including degenerate and non-degenerate cases. Finally, with examples, we demonstrate the efficiency and accuracy of methods for 1 and 2D problems. It is also interesting that the proposed method is valid for the interface problems with degenerate and non-degenerate cases, we show it with some examples.

Qualitative analysis of solutions for a degenerate partial differential equations model of epidemic spread dynamics

Compartmental models are widely used in mathematical epidemiology to describe the dynamics of infectious diseases or in mathematical models of population genetics. In this study, we study a time-dependent Susceptible-Infectious-Susceptible (SIS) Partial Differential Equation (PDE) model that is based on a diffusion-drift approximation of a probability density from a well-known discrete-time Markov chain model. This SIS-PDE model is conservative due to the degeneracy of the diffusion term at the origin. The main results of this article are the qualitative behavior of weak solutions, the dependence of the local asymptotic property of these solutions on initial data, and the existence of Dirac delta function type solutions. Moreover, we study the long-term behavior of solutions and confirm our analysis with numerical computations.

Convergence rate for degenerate partial and stochastic differential equations via weak Poincaré inequalities

We employ weak hypocoercivity methods to study the long-term behavior of operator semigroups generated by degenerate Kolmogorov operators with variable second-order coefficients, which solve the associated abstract Cauchy problem. We prove essential m-dissipativity of the operator, which extends previous results and is key to the rigorous analysis required. We give estimates for the L2-convergence rate by using weak Poincaré inequalities. As an application, we obtain estimates for the (sub-)exponential convergence rate of solutions to the corresponding degenerate Fokker-Planck equations and of weak solutions to the corresponding degenerate stochastic differential equation with multiplicative noise.

PROPERTIES OF THE ABEL-POISSON TRANSFORMATION OF FORMAL HERMITE SERIES

In the paper we investigate the properties of the Abel-Poisson transformation of the Hermite formal series (differentiability property, boundary properties). Such series are identified with linear continuous functionals defined on the space $S_{1/2}^{1/2}$, which belongs to spaces of type $S$. The space $S_{1/2}^{1/2}$ coincides with the class of analytic vectors of the harmonic oscillator -- the operator $d^2/dx^2+x^2$, which is integral and self-adjoint in the Hilbert space $L_2(\mathbb{R})$. An explicit form of the function, which is the core of the Abel--Poisson transformation, was found, and the properties of this function were investigated. The application of such transformation is given when studying the well-posedness of the Cauchy problem for a degenerate partial differential equation.

Solution of a Semi-Boundary-Value Problem for a First-Order Degenerate Partial Differential Equation

A first-order partial differential equation with constant irreversible coefficients in a Banach space is considered. In the particular case of a finite-dimensional space, the initial-boundary-value problem with irreversible matrix coefficients has no solution; hence, we pose Showalter-type conditions. Due to the regularity of the operator pencil, the equation splits into differential equations in subspaces and given conditions lead to initial conditions in subspaces. A solution to the problem is constructed and an example is provided.

Homogenization of periodic elliptic degenerate PDEs with non-linear Neumann boundary condition

In this paper, a semi-linear elliptic partial differential equation (PDE) with non linear Neumann boundary condition and rapidly oscillating coefficients is homogenized. The novelty of our result lies in the fact that we allow the second order part of the differential operator to be degenerate in some portion of R d . Our fully probabilistic method is based on the connection between PDEs and backward stochastic differential equations (BSDEs) with random terminal time and the weak convergence of a class of diffusion processes.

A fitted finite volume method for stochastic optimal control problems in finance

In this article, we provide a numerical method based on fitted finite volume method to approximate the Hamilton-Jacobi-Bellman (HJB) equation coming from stochastic optimal control problems in one and two dimensional domain. The computational challenge is due to the nature of the HJB equation, which may be a second-order degenerate partial differential equation coupled with optimization. For such problems, standard scheme such as finite difference losses its monotonicity and therefore the convergence toward the viscosity solution may not be guarantee. In the work, we discretize the HJB equation using the fitted finite volume method, which has for main feature to tackle the degeneracy of the equation. The time discretisation is performed using the Implicit Euler method, which is unconditionally stable. We show that matrices resulting from spatial discretization and temporal discretization are M-matrices. The optimization problem is solved at every time step using iterative method. Numerical results are presented to show the robustness of the fitted finite volume numerical method comparing to the standard finite difference method.

On Existence, Uniqueness and Two-Scale Convergence of a Model for Coupled Flows in Heterogeneous Media

This paper is concerned with the global existence, uniqueness and homogenization of degenerate partial differential equations with integral conditions arising from coupled transport processes and chemical reactions in three-dimensional highly heterogeneous porous media. Existence of global weak solutions of the microscale problem is proved by means of semidiscretization in time deriving a priori estimates for discrete approximations needed for proofs of existence and convergence theorems. It is further shown that the solution of the microscale problem is two-scale convergent to that of the upscaled problem as the scale parameter goes to zero. In particular, we focus our efforts on the contribution of the so-called first order correctors in periodic homogenization. Finally, under additional assumptions, we consider the problem of the uniqueness of the solution to the homogenized problem.

Solution of the mixed formulation for generalized Forchheimer flows of isentropic gases

This paper focuses on the generalized Forchheimer flows of isentropic gas in a porous medium, described by a system of two nonlinear degenerate partial differential equations of first order. We prove the existence and uniqueness of the Dirichlet problem for the stationary problem. The technique of semi-discretization in time is used to prove the existence for the time-dependent problem.

Self-Similar Solutions of Some Model Degenerate Partial Differential Equations of the Second, Third, and Fourth Order

When studying boundary value problems for some partial differential equations arising in applied mathematics, we often have to study the solution of a system of partial differential equations satisfied by hypergeometric functions and find explicit linearly independent solutions for the system. In this study, we construct self-similar solutions of some model degenerate partial differential equations of the second, third, and fourth order. These self-similar solutions are expressed in terms of hypergeometric functions.

Numerical analysis of degenerate Kolmogorov equations of constrained stochastic Hamiltonian systems

In this work, we propose a method to compute numerical approximations of the invariant measures and Rice’s formula (frequency of threshold crossings) for a certain type of stochastic Hamiltonian system constrained by an obstacle and subjected to white or colored noise. As an alternative to probabilistic Monte-Carlo simulations, our approach relies on solving a class of degenerate partial differential equations with non-local Dirichlet boundary conditions, as derived in Mertz et al. (2018). A functional analysis framework is presented; regularization and approximation by the finite element method is applied; numerical experiments on these are performed and show good agreement with probabilistic simulations.

High-order exponential spline method for pricing European options

ABSTRACTThe key purpose of the present work is to constitute an analysis of a numerical method for a degenerate partial differential equation, called the Black–Scholes equation, governing European option pricing. The method is based on exponential spline spatial discretization and an explicit finite-difference time-stepping technique. We establish the convergence and an error bound for the solutions of the fully discretized system. The numerical and graphical results elucidate that the suggested approach is very straightforward and accurate.

Numerical methods for PDE models related to pricing and expected lifetime of an extraction project under uncertainty

Numerical techniques for solving some mathematical models related to a mining extraction project under uncertainty are proposed. The mine valuation is formulated as a complementarity problem associated to a degenerate second order partial differential equation (PDE), which incorporates the option to abandon the project. The probability of completion and the expected lifetime of the project are the respective solutions of problems governed by similar degenerated PDE operators. In all models, the underlying stochastic factors are the commodity price and the remaining resource. After justifying the required boundary conditions on the computational bounded domain, the proposed numerical techniques mainly consist of a Crank-Nicolson characteristics method for the time discretization to cope with the convection dominating setting and Lagrange finite elements for the discretization in the commodity and resource variables, with the additional use of an augmented Lagrangian active set method for the complementarity problem. Some numerical examples are discussed to illustrate the performance of the methods and models.

DG framework for pricing European options under one-factor stochastic volatility models

The modern theory of option pricing is based on models introduced almost 50 years ago. These models, however, are not able to capture real market behaviour sufficiently well. One line of extensions consists of introducing an additional variable into the model, the so-called stochastic volatility. Since such models lead to the (semi) closed-form solution only rarely, some form of a numerical approximation can be essential. In this paper we study a general one-factor stochastic volatility model for the pricing of European options. A standard mathematical approach to this problem leads to a degenerate partial differential equation completed by boundary and terminal conditions. We formulate this problem in a variational sense and prove the existence and the uniqueness of a weak solution. Further, a robust numerical procedure based on the discontinuous Galerkin approach is proposed to improve the numerical valuation process. The performance of the procedure is accompanied with theoretical results and documented using reference experiments with the emphasis on investigation of the behaviour of option values with respect to the different mesh sizes as well as polynomial orders of approximation.

Poincaré inequalities for Sobolev spaces with matrix-valued weights and applications to degenerate partial differential equations

For bounded domains Ω, we prove that the Lp-norm of a regular function with compact support is controlled by weighted Lp-norms of its gradient, where the weight belongs to a class of symmetric non-negative definite matrix-valued functions. The class of weights is defined by regularity assumptions and structural conditions on the degeneracy set, where the determinant vanishes. In particular, the weight A is assumed to have rank at least 1 when restricted to the normal bundle of the degeneracy set S. This generalization of the classical Poincaré inequality is then applied to develop a robust theory of first-order Lp-based Sobolev spaces with matrix-valued weight A. The Poincaré inequality and these Sobolev spaces are then applied to produce various results on existence, uniqueness and qualitative properties of weak solutions to boundary-value problems for degenerate elliptic, degenerate parabolic and degenerate hyperbolic partial differential equations (PDEs) of second order written in divergence form, where A is calibrated to the matrix of coefficients of the second-order spatial derivatives. The notion of weak solution is variational: the spatial states belong to the matrix-weighted Sobolev spaces with p = 2. For the degenerate elliptic PDEs, the Dirichlet problem is treated by the use of the Poincaré inequality and Lax–Milgram theorem, while the treatment of the Cauchy–Dirichlet problem for the degenerate evolution equations relies only on the Poincaré inequality and the parabolic and hyperbolic counterparts of the Lax–Milgram theorem.

Improvement of numerical approximation of coupled multiphase multicomponent flow with reactive geochemical transport in porous media

In this paper, we consider a parallel finite volume algorithm for modeling complex processes in porous media that include multiphase flow and geochemical interactions. Coupled flow and reactive transport phenomena often occur in a wide range of subsurface systems such as hydrocarbon reservoir production, groundwater management, carbon dioxide sequestration, nuclear waste repository or geothermal energy production. This work aims to develop and implement a parallel code coupling approach for non-isothermal multiphase multicomponent flow and reactive transport simulation in the framework of the parallel open-source platform DuMuX. Modeling such problems leads to a highly nonlinear coupled system of degenerate partial differential equations to algebraic or ordinary differential equations requiring special numerical treatment. We propose a sequential fully implicit scheme solving firstly a multiphase compositional flow problem and then a Direct Substitution Approach (DSA) is used to solve the reactive transport problem. Both subsystems are discretized by a fully implicit cell-centred finite volume scheme and then an efficient sequential coupling has been implemented in DuMuX. We focus on the stability and robustness of the coupling process and the numerical benefits of the DSA approach. Parallelization is carried out using the DUNE parallel library package based on MPI providing high parallel efficiency and allowing simulations with several tens of millions of degrees of freedom to be carried out, ideal for large-scale field applications involving multicomponent chemistry. As we deal with complex codes, we have tested and demonstrated the correctness of the implemented software by benchmarking, including the MoMaS reactive transport benchmark, and comparison to existing simulations in the literature. The accuracy and effectiveness of the approach is demonstrated through 2D and 3D numerical simulations. Parallel scalability is investigated for 3D simulations with different grid resolutions. Numerical results for long-term fate of injected CO2 for geological storage are presented. The numerical results have demonstrated that this approach yields physically realistic flow fields in highly heterogeneous media and showed that this approach performs significantly better than the Sequential Iterative Approach (SIA).

Extension of a Regularization Based Time-adaptive Numerical Method for a Degenerate Diffusion-Reaction Biofilm Growth Model to Systems Involving Quorum Sensing

We extend a regularization based numerical method for a highly degenerate partial differential equation that describes biofilm growth to systems of PDEs describing biofilms with several particulate substances. The example for which we develop the method is a quorum sensing biofilm which consists of down- and up-regulated biomass fractions. We carry out computational studies to assess the effect of the regularization parameter, a grid refinement study and report briefly on parallel performance of our code under OpenMP on desktop workstations.

On micropolar elastic cusped prismatic shells

A huge literature is devoted to the study of cusped prismatic shells on the basis of the classical theory of elasticity. It was stimulated by the works of I. Vekua. I. Vekua considered very important to carry out investigations of boundary value and initial boundary value problems for such bodies, since they are connected with degenerate partial differential equations and, therefore, are not classical, in general. The present paper is devoted to cusped prismatic shells on the basis of the theory of micropolar elasticity. Namely, on the basis of the N=0 approximation of hierarchical models for micropolar elastic cusped prismatic shells constructed by the I. Vekua dimension reduction method.

A compact exponential method for the efficient numerical simulation of the dewetting process of viscous thin films

In this work, we propose a discrete mathematical system to model the evolution of the thickness of two-dimensional viscous thin films subject to a dewetting process. The continuous model under consideration is a degenerate partial differential equation that generalizes the classical thin film equation, and considers the inclusion of a singular potential. The analytical model is discretized using an exponential method that is capable of preserving the positive character of the approximations. In addition, the explicit nature of our approach results in an economic computer implementation which produces fast simulations. We provide some illustrative examples on the dynamics of the growth of thin films in the presence/absence of a dewetting process. The qualitative results exhibit the appearance of typical patterns obtained in experimental settings. The technique was validated against Bhattacharya’s method and a standard explicit discretization of the mathematical model.