Related Partial Differential Equations Research Articles

Weak solutions to coupled quadratic forward backward stochastic differential equations and Sobolev solutions to their related partial differential equations

We establish existence and uniqueness of a weak (in law sense) solution to coupled forward–backward stochastic differential equations (FBSDEs), with possibly discontinuous diffusion coefficient. We assume that the coefficient of the backward component is of quadratic growth. We first prove the existence of a Sobolev solution for the related partial differential equation (PDE) in the Sobolev space , by using compactness arguments. Next, we use Itô–Krylov's formula to get the existence of weak solution to our considered FBSDE. For the uniqueness part, we first establish the weak uniqueness of the FBSDEs then deduce the uniqueness of its related PDE by using the Feynmann–Kac formula.

Projected incidence trends of need for long-term care in German men and women from 2011 to 2021.

The German Federal Statistical Office routinely collects and reports aggregated numbers of people in need of long-term care (NLTC) stratified by age and sex. Age- and sex-specific prevalence of NLTC from 2011 to 2021 is reported as well. One estimation of the incidence rate of NLTC based on the age- and sex-specific prevalence exists that did not explore possible trends in incidence [based on MRR (mortality rate ratio)], which is important for an adequate projection of the future number of people with NLTC. We aim to explore possible trends in age-specific incidence of NLTC in German men and women from 2011 to 2021 based on different scenarios about excess mortality (in terms of MRR). The incidence of NLTC was calculated based on an illness-death model and a related partial differential equation based on data from the Federal Statistical Office. Estimation of annual percent change (APC) of the incidence rate was conducted in eight scenarios. There are consistent indications for trends in incidence for men and women aged 50-79 years with APC in incidence rate of more than +9% per year (up to nearly 19%). For ages 80+ the APC is between +0.4% and +12.5%. In all scenarios, women had higher age-specific APCs than men. We performed the first analysis of APC in the age- and sex-specific incidence rate of NLTC in Germany and revealed an increasing trend in the incidences. With these findings, a future prevalence of NLTC can be estimated which may exceed current prognoses.

On the investigation of a one-dimensional blood flow model in elastic arteries under symmetry analysis

In this paper, we consider a one-dimensional model of blood flow along the compliant arteries. With the help of the invariant function, we construct and classify the optimal system of subalgebras. Next, we reduced the given system of partial differential equations (PDEs) to the system of ordinary differential equations (ODEs) for each subalgebra and subsequently solved them exactly. Further, we investigate the evolutionary behavior of the average blood flow velocity and the cross-sectional area of the arteries; under the influence of the physical parameter [Formula: see text] graphically. Furthermore, we construct nonlocally related PDEs for the given system of PDEs, consisting of inverse potential systems (IPS) and potential systems. Finally, we classify the nonlocal symmetries arising from the potential system and IPS.

Classification of nonlocal symmetries and exact solutions for 3 × 3 Chaplygin gas equation with conservation laws

In this paper, we study the one-dimensional isentropic compressible Euler system for the Chaplygin gas through Lie symmetry analysis. The one-dimensional optimal subalgebras are classified using the adjoint transformation and the invariant functions. We derived several new exact solutions from the optimal subalgebras and investigated the physical behavior of some solutions graphically. Next, a tree of nonlocally related partial differential equations (PDEs) is presented and we classify the nonlocal symmetry of the given system. Futher, some nontrivial exact solutions for the given model are constructed using nonlocal symmetries. Furthermore, using the traveling wave transformation, which is invariant under the symmetry group, we obtain solutions of the nature of peakon-type and kink-type solitons. Then, conservation laws are constructed through the direct multipliers method. Finally, the evolutionary behavior of a C1-wave is investigated using one of the developed solutions.

Hydrogen diffusion into caprock: A semi-analytical solution and a hydrogen loss criterion

Depleted gas reservoirs can provide a gigaton capacity and the necessary infrastructure for large scale hydrogen storage. Therefore, underground storage of hydrogen in these geological formations is considered a valuable option. However, the high molecular diffusion of hydrogen from these storage sites is a major concern. In this paper, the idea of hydrogen diffusion through the caprock during geological storage was investigated using a series of one- and three-dimensional numerical models. In these models, the caprock and overlying formations were considered as a semi-infinite system, while the interaction between the reservoir and the caprock was defined by the pressure and gas composition at the boundary. Hydrogen diffusion was based on chemical potential differences and the influence of pre-diffused hydrocarbon gas in the caprock was considered. Furthermore, based on the analogies between the related partial differential equations of diffusion and spontaneous imbibition processes, a simple yet accurate semi-analytical solution was proposed. The presented semi-analytical solution can be used to predict the cumulative hydrogen loss over time. This solution states that the hydrogen loss increases proportionally with the square root of time. Using the semi-analytical solution, a criterion for loss by molecular hydrogen diffusion was presented that indicates the fraction of hydrogen lost by diffusion at a given time. The results obtained based on the diffusion fluxes showed that the hydrogen loss can be underestimated if the existence of hydrocarbon gas in the caprock is ignored. The analysis also indicated that hydrogen loss is directly proportional to the interface between rock and reservoir exposed to hydrogen, the caprock porosity, the gas saturation of the caprock, the square root of the diffusion coefficient, and the square root of time. While the presence of hydrocarbon gas facilitates diffusion, the thermodynamic effects at high pressures lead to a comparatively lower molar density in the caprock than in the matrix. Thus, a lower final loss of the injected hydrogen can be expected at higher pressures.

Stochastic zero-sum differential games and backward stochastic differential equations

Abstract In this paper, we study the stochastic zero-sum differential game in finite horizon in a general case. We first prove that the BSDE associated with a specific generator (the Hamiltonian function for the game) has a unique solution. Then we characterize the value function as that solution to prove the existence of a saddle point for the game. Finally, in the Markovian framework, we show that the value function is the unique viscosity solution for the related partial differential equation.

Conservation laws and some new exact solutions for traffic flow model via symmetry analysis

In this paper, we investigate the traffic flow model with congested phase through local and nonlocal symmetry analysis. Firstly, we derive the Lie group of transformations and show that it admits four one-dimensional optimal algebras. Next, by similarity reductions, we construct several exact solutions for each subalgebras as well as analyze the relation of group parameters. Furthermore, we construct a tree representing nonlocally related partial differential equations (PDEs) consisting inverse potential systems (IPS) and potential systems. Then, we prove that the traffic flow model yields one nonlocal symmetry and hence we derive an exact solution for the given system. Moreover, we generate the conservation laws for the governing systems through the nonlinear self-adjoint property. Finally, we study the nonlinear behaviors like weak discontinuity (C1-wave), characteristic shock for the physical model, and the impact of the anticipation factor on their evolutionary behavior graphically.

Continuum Covariance Propagation for Understanding Variance Loss in Advective Systems

Motivated by the spurious variance loss encountered during covariance propagation in atmospheric and other large-scale data assimilation systems, we consider the problem for state dynamics governed by the continuity and related hyperbolic partial differential equations. This loss of variance is often attributed to reduced-rank representations of the covariance matrix, as in ensemble methods for example, or else to the use of dissipative numerical methods. Through a combination of analytical work and numerical experiments, we demonstrate that significant variance loss, as well as gain, typically occurs during covariance propagation, even at full rank. The cause of this unusual behavior is a discontinuous change in the continuum covariance dynamics as correlation lengths become small, for instance in the vicinity of sharp gradients in the velocity field. This discontinuity in the covariance dynamics arises from hyperbolicity: the diagonal of the kernel of the covariance operator is a characteristic surface for advective dynamics. Our numerical experiments demonstrate that standard numerical methods for evolving the state are not adequate for propagating the covariance, because they do not capture the discontinuity in the continuum covariance dynamics as correlations lengths tend to zero. Our analytical and numerical results demonstrate in the context of mass conservation that this leads to significant, spurious variance loss in regions of mass convergence and gain in regions of mass divergence. The results suggest that developing local covariance propagation methods designed specifically to capture covariance evolution near the diagonal may prove a useful alternative to current methods of covariance propagation.

Numerical analysis of two-dimensional Navier–Stokes equations with additive stochastic forcing

Abstract We propose and study a temporal and a spatio-temporal discretisation of the two-dimensional stochastic Navier–Stokes equations in bounded domains supplemented with no-slip boundary conditions. Considering additive noise, we base its construction on the related nonlinear random partial differential equation, which is solved by a transform of the solution of the stochastic Navier–Stokes equations. We show a strong rate (up to) $1$ in probability for a corresponding discretisation in space and time (and space-time).

Topology optimization for surface flows

Surface flows can represent the motions of the viscous and incompressible fluid at the solid/fluid interfaces. This paper presents a topology optimization approach for surface flows and extends the design space of topology optimization of fluidic structures onto the curved surfaces in the forms of 2-manifolds corresponding to the geometrical configurations of the solid/fluid interfaces. The presented approach is implemented by filling a porous medium onto the 2-manifolds. An artificial Darcy friction is correspondingly added to the area force term of the surface Navier-Stokes equations used to describe the surface flows and the physical area forces are penalized to eliminate their existence in the fluidic regions and to avoid the invalidity of the porous medium based topology optimization model. Topology optimization for the steady and unsteady surface flows has been executed by iteratively evolving the impermeability of the porous medium, where the impermeability is interpolated by the material density derived from a design variable. The related partial differential equations are solved by using the surface finite element method. Numerical examples have been provided to demonstrate this topology optimization approach for the surface flows, including the boundary velocity driven flows, the area force driven flows and the convection-diffusion flows.

Forward-backward doubly stochastic systems and classical solutions of path-dependent stochastic PDEs

In this paper, a class of non-Markovian forward-backward doubly stochastic systems is studied. By using the technique of functional Itô (or path-dependent) calculus, the relationship between the systems and related path-dependent quasi-linear stochastic partial differential equations (SPDEs in short) is established, and the well-known nonlinear stochastic Feynman-Kac formula of Pardoux and Peng [Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Relat. Fields 98 (1994), pp. 209–227.] is developed to the non-Markovian situation. Moreover, we obtain the differentiability of the solution to the forward-backward doubly stochastic systems and some properties of solutions to the path-dependent SPDEs.

The nonlocal potential transformation method for solitary wave packets of a shock-breaking dynamics system

In this work, we employ the potential similarity transformation method to derive some solitary wave packet solutions for the Calogero-Bogoyavlenskii-Schiff (CBS) equation. Exploiting nonsingular local multipliers, a set of local conservation laws is presented for the equation. The nonlocally related partial differential equation (PDE) systems were observed. Considering the nonlocal-related PDE systems, we present 46 analytic solutions of the CBS equation that involves several arbitrary functions, enabling the spread of the equation’s solution to understand Riemann waves dynamics extensively. These are helpful in severalapplications such as electromagnetic waves in an optical tsunami in fibers, magneto-sonic and ion in plasma, the acoustic waves in compressible fluids, surface, and interval waves in oceans, tidal and tsunami in rivers.

Generalized Donaldson functionals and related nonlinear partial differential equations

Generalized Donaldson functionals and related nonlinear partial differential equations

Multidimensional transonic shock waves and free boundary problems

We are concerned with free boundary problems arising from the analysis of multidimensional transonic shock waves for the Euler equations in compressible fluid dynamics. In this expository paper, we survey some recent developments in the analysis of multidimensional transonic shock waves and corresponding free boundary problems for the compressible Euler equations and related nonlinear partial differential equations (PDEs) of mixed type. The nonlinear PDEs under our analysis include the steady Euler equations for potential flow, the steady full Euler equations, the unsteady Euler equations for potential flow, and related nonlinear PDEs of mixed elliptic–hyperbolic type. The transonic shock problems include the problem of steady transonic flow past solid wedges, the von Neumann problem for shock reflection–diffraction, and the Prandtl–Meyer problem for unsteady supersonic flow onto solid wedges. We first show how these longstanding multidimensional transonic shock problems can be formulated as free boundary problems for the compressible Euler equations and related nonlinear PDEs of mixed type. Then we present an effective nonlinear method and related ideas and techniques to solve these free boundary problems. The method, ideas, and techniques should be useful to analyze other longstanding and newly emerging free boundary problems for nonlinear PDEs.

Mean-field backward stochastic differential equations with reflection and related nonlocal PDEs in a convex domain

In this paper, we present the unique solvability of a type of reflected mean-field backward stochastic differential equation (RMFBSDE) in a convex domain. Moreover, employing the techniques of stochastic flow and penalization, we present a probabilistic interpretation in the Sobolev sense for a kind of reflected nonlocal partial differential equation (PDE) in a convex domain. The results of this paper provide the foundation of studying control problems of constrained mean-field systems.

Symmetry considerations for differential equation formulations from classical and fractional Lagrangians

The utility of Noether’s classical theorem on differential equations extended to a generalized nonclassical theorem is the focus of this paper. After addressing a couple of standard related Partial Differential Equation (P.D.E.) formulations from classical Lagrangians, it culminates into a non-classical formulation of the diffusion equation in one spatial dimension from a fractional Lagrangian. Comparisons and contrasts between techniques for the classical and fractional formulations, as done here, facilitate the basic computational methods required for building analytical results. A noteworthy interface between Distribution theory, Trace theory and Lie symmetry theory is a key point of interest in this study.

Well-posedness for some non-linear SDEs and related PDE on the Wasserstein space

In this paper, we investigate the well-posedness of the martingale problem associated to non-linear stochastic differential equations (SDEs) in the sense of McKean-Vlasov under mild assumptions on the coefficients as well as classical solutions for a class of associated linear partial differential equations (PDEs) defined on [0,T]×Rd×P2(Rd), for any T>0, P2(Rd) being the Wasserstein space (i.e. the space of probability measures on Rd with a finite second-order moment). The martingale problem is addressed by a fixed point argument on a suitable complete metric space, under some mild regularity assumptions on the coefficients that covers a large class of interaction. Also, new well-posedness results in the strong sense are obtained from the previous analysis. Under additional assumptions, we then prove the existence of the associated density and investigate its smoothness property. In particular, we establish some Gaussian type bounds for its derivatives. We eventually address the existence and uniqueness for the related linear Cauchy problem with irregular terminal condition and source term.

Analysis of BBM solitary wave interactions using the conserved quantities

In this paper, a simple, robust, fast and effective method based on the conserved quantities is developed to approximate and analyse the shape, structure and interaction characters of the solitary waves described by the Benjamin–Bona–Mahony (BBM) equation. Due to the invariant character of the conserved quantities, there is no need to solve the related complex nonlinear partial differential BBM equation to simulate the interactions between the solitary waves at the most merging instance. Good accuracy of the proposed method has been found when compared with the numerical method for the solitary wave interactions with different initial incoming wave shapes. The conserved quantity method developed in this work can serve as an ideal tool to benchmark numerical solvers, to perform the stability analysis, and to analyse the interacting phenomena between solitary waves.

Comparison of modified ADM and classical finite difference method for some third-order and fifth-order KdV equations

Abstract The KdV equation, which appears as an asymptotic model in physical systems ranging from water waves to plasma physics, has been studied. In this paper, we are concerned with dispersive nonlinear KdV equations by using two reliable methods: Shehu Adomian decomposition method (STADM) and the classical finite difference method for solving three numerical experiments. STADM is constructed by combining Shehu’s transform and Adomian decomposition method, and the nonlinear terms can be easily handled using Adomian’s polynomials. The Shehu transform is used to accelerate the convergence of the solution series in most cases and to overcome the deficiency that is mainly caused by unsatisfied conditions in other analytical techniques. We compare the approximate and numerical results with the exact solution for the two numerical experiments. The third numerical experiment does not have an exact solution and we compare profiles from the two methods vs the space domain at some values of time. This study provides us with information about which of the two methods are effective based on the numerical experiment chosen. Knowledge acquired will enable us to construct methods for other related partial differential equations such as stochastic Korteweg-de Vries (KdV), KdV-Burgers, and fractional KdV equations.

Regularity and Strict Positivity of Densities for the Nonlinear Stochastic Heat Equation

In this paper, we establish a necessary and sufficient condition for the existence and regularity of the density of the solution to a semilinear stochastic (fractional) heat equation with measure-valued initial conditions. Under a mild cone condition for the diffusion coefficient, we establish the smooth joint density at multiple points. The tool we use is Malliavin calculus. The main ingredient is to prove that the solutions to a related stochastic partial differential equation have negative moments of all orders. Because we cannot prove u ( t , x ) ∈ D ∞ u(t,x)\in \mathbb {D}^\infty for measure-valued initial data, we need a localized version of Malliavin calculus. Furthermore, we prove that the (joint) density is strictly positive in the interior of the support of the law, where we allow both measure-valued initial data and unbounded diffusion coefficient. The criteria introduced by Bally and Pardoux are no longer applicable for the parabolic Anderson model. We have extended their criteria to a localized version. Our general framework includes the parabolic Anderson model as a special case.