Quasilinear Partial Differential Equations Research Articles

Initial-Value Problem for a Higher-Order Quasilinear Partial Differential Equation

We examine an initial-value problem for a certain higher-order quasilinear partial differential equation. Expressing the partial differential operator as the superposition of first-order operators, we apply methods of solution of first-order equations. We prove the unique solvability of the initial-value problem considered.

Метод годографа для решения задачи о мелкой воде под твердой крышкой

One of the mathematical models describing the behavior of two horizontally infinite adjoining layers of an ideal incompressible liquid under a solid cover moving at different speeds is investigated. At a large difference in the layer velocities, the Kelvin-Helmholtz instability occurs, which leads to a distortion of the interface. At the initial point in time, the interface is not necessarily flat. From a mathematical point of view, the behavior of the liquid layers is described by a system of four quasilinear equations, either hyperbolic or elliptic, in partial derivatives of the first order. Some type shallow water equations are used to construct the model. In the simple version of the model considered in this paper, in the spatially one-dimensional case, the unknowns are the boundary between the liquid layers h(x,t) and the difference in their velocities γ(x,t). The main attention is paid to the case of elliptic equations when |h|<1 and γ>1. An evolutionary Cauchy problem with arbitrary sufficiently smooth initial data is set for the system of equations. The explicit dependence of the Riemann invariants on the initial variables of the problem is indicated. To solve the Cauchy problem formulated in terms of Riemann invariants, a variant of the hodograph method based on a certain conservation law is used. This method allows us to convert a system of two quasilinear partial differential equations of the first order to a single linear partial differential equation of the second order with variable coefficients. For a linear equation, the Riemann-Green function is specified, which is used to construct a two-parameter implicit solution to the original problem. The explicit solution of the problem is constructed on the level lines (isochrons) of the implicit solution by solving a certain Cauchy problem for a system of ordinary differential equations. As a result, the original Cauchy problem in partial derivatives of the first order is transformed to the Cauchy problem for a system of ordinary differential equations, which is solved by numerical methods. Due to the bulkiness of the expression for the Riemann-Green function, some asymptotic approximation of the problem is considered, and the results of calculations, and their analysis are presented.

Quasi linear parabolic pde posed on a network with non linear Neumann boundary condition at vertices

The purpose of this article is to study quasi linear parabolic partial differential equations of second order, posed on a bounded network, satisfying a nonlinear and non dynamical Neumann boundary condition at the vertices. We prove the existence and the uniqueness of a classical solution.

Systems of reflected stochastic PDEs in a convex domain: Analytical approach

In this paper, we establish an existence and uniqueness result for the system of quasilinear stochastic partial differential equations (SPDEs for short) with reflection in a convex domain in RK by analytical approach. The method is based on the approximation of the penalized systems of SPDEs.

Solving high-dimensional forward-backward doubly SDEs and their related SPDEs through deep learning

Forward-backward doubly stochastic differential equations (FBDSDEs) are related to a type of quasi-linear parabolic stochastic partial differential equations (SPDEs). We propose a deep learning-based numerical algorithm for solving such equations. Using deep neural networks as approximations of the controls, we can deal with high-dimensional cases. Numerical experiments are carried out to demonstrate the accuracy and efficiency of the proposed numerical algorithm.

Global existence for a semi-discrete scheme of some quasilinear hyperbolic balance laws

In this article, we show the global existence of solution for some semi-discrete finite difference schemes of some model system of quasilinear hyperbolic balance laws with the Cattaneo law. In [10] and [11] the first author proposed the energy method for the structure-preserving fully discrete numerical schemes for semilinear partial differential equations. The result here gives its application to quasilinear partial differential equations.

Vanishing relaxation time dynamics of the Jordan Moore-Gibson-Thompson equation arising in nonlinear acoustics

The (third order in time) JMGT equation [Jordan (J Acoust Soc Am 124(4):2491–2491, 2008) and Cattaneo (C Sulla conduzione del calore Atti Sem Mat Fis Univ Modena 3:83–101, 1948)] is a nonlinear (quasi-linear) partial differential equation (PDE) model introduced to describe a nonlinear propagation of sound in an acoustic medium. The important feature is that the model avoids the infinite speed of propagation paradox associated with a classical second-order in time equation referred to as Westervelt equation. Replacing Fourier’s law by Maxwell–Cattaneo’s law gives rise to the third-order in time derivative scaled by a small parameter \(\tau >0\), the latter represents the thermal relaxation time parameter and is intrinsic to the medium where the dynamics occur. In this paper, we provide an asymptotic analysis of the third-order model when \(\tau \rightarrow 0 \). It is shown that the corresponding solutions converge in a strong topology of the phase space to a limit which is the solution of Westervelt equation. In addition, rate of convergence is provided for solutions displaying higher-order regularity. This addresses an open question raised in [20], where a related JMGT equation has been studied and weak star convergence of the solutions when \(\tau \rightarrow 0\) has been established. Thus, our main contribution is showing strong convergence on infinite time horizon, along with related rates of convergence valid on a finite time horizon. The key to unlocking the difficulty owns to a tight control and propagation of the “smallness” of the initial data in carrying the estimates at three different topological levels. The rate of convergence allows one then to estimate the relaxation time needed for the signal to reach the target. The interest in studying this type of problems is motivated by a large array of applications arising in engineering and medical sciences.

Freidlin--Wentzell Type Large Deviation Principle for Multiscale Locally Monotone SPDEs

This work is concerned with a Freidlin--Wentzell type large deviation principle for a family of multiscale quasilinear and semilinear stochastic partial differential equations. Employing the weak c...

A Class of Second-Order Geometric Quasilinear Hyperbolic PDEs and Their Application in Imaging

Motivated by important applications in image processing, we study a class of second-order geometric quasilinear hyperbolic partial differential equations (PDEs). This is inspired by the recent deve...

Stochastic Camassa-Holm equation with convection type noise

We consider a stochastic Camassa-Holm equation driven by a one-dimensional Wiener process with a first order differential operator as diffusion coefficient. We prove the existence and uniqueness of local strong solutions of this equation. In order to do so, we transform it into a random quasi-linear partial differential equation and apply Kato's operator theory methods. Some of the results have potential to find applications to other nonlinear stochastic partial differential equations.

On the equivalence of pathwise mild and weak solutions for quasilinear SPDEs

The main goal of this work is to relate weak and pathwise mild solutions for parabolic quasilinear stochastic partial differential equations (SPDEs). Extending in a suitable way techniques from the theory of nonautonomous semilinear SPDEs to the quasilinear case, we prove the equivalence of these two solution concepts.

Nonclassical symmetry analysis, conservation laws of one-dimensional macroscopic production model and evolution of nonlinear waves

In this article we consider a system of first order quasilinear hyperbolic partial differential equations (PDEs) governing the one-dimensional macroscopic production model which describes high-volume product flows. We first show that the governing system admits nonlinear self-adjointness property. Further, we construct various conservation laws of the given system by using the direct multipliers method and also each of its classical symmetries yields a conservation law of the original system due to Ibragimov's formulation. Then we perform the nonclassical symmetry analysis and compute several hidden symmetries. Using these nonclassical symmetries, we obtain some new exact solutions and discuss their physical behavior graphically. Finally, the evolution of characteristic shock and weak discontinuity are studied by using one of the obtained solutions.

Reconstruction of the parameter in parabolic partial differential equations using Haar wavelet method

PurposeIn the past few years, Haar wavelet-based numerical methods have been applied successfully to solve linear and nonlinear partial differential equations. This study aims to propose a wavelet collocation method based on Haar wavelets to identify a parameter in parabolic partial differential equations (PDEs). As Haar wavelet is defined in a very simple way, implementation of the Haar wavelet method becomes easier than the other numerical methods such as finite element method and spectral method. The computational time taken by this method is very less because Haar matrices and Haar integral matrices are stored once and used for each iteration. In the case of Haar wavelet method, Dirichlet boundary conditions are incorporated automatically. Apart from this property, Haar wavelets are compactly supported orthonormal functions. These properties lead to a huge reduction in the computational cost of the method.Design/methodology/approachThe aim of this paper is to reconstruct the source control parameter arises in quasilinear parabolic partial differential equation using Haar wavelet-based numerical method. Haar wavelets possess various properties, for example, compact support, orthonormality and closed form expression. The main difficulty with the Haar wavelet is its discontinuity. Therefore, this paper cannot directly use the Haar wavelet to solve partial differential equations. To handle this difficulty, this paper represents the highest-order derivative in terms of Haar wavelet series and using successive integration this study obtains the required term appearing in the problem. Taylor series expansion is used to obtain the second-order partial derivatives at collocation points.FindingsAn efficient and accurate numerical method based on Haar wavelet has been proposed for parameter identification in quasilinear parabolic partial differential equations. Numerical results are obtained from the proposed method and compared with the existing results obtained from various finite difference methods including Saulyev method. It is shown that the proposed method is superior than the conventional finite difference methods including Saulyev method in terms of accuracy and CPU time. Convergence analysis is presented to show the accuracy of the proposed method. An efficient algorithm is proposed to find the wavelet coefficients at target time.Originality/valueThe outcome of the paper would have a valuable role in the scientific community for several reasons. In the current scenario, the parabolic inverse problem has emerged as very important problem because of its application in many diverse fields such as tomography, chemical diffusion, thermoelectricity and control theory. In this paper, higher-order derivative is represented in terms of Haar wavelet series. In other words, we represent the solution in multiscale framework. This would enable us to understand the solution at various resolution levels. In the case of Haar wavelet, this paper can achieve a very good accuracy at very less resolution levels, which ultimately leads to huge reduction in the computational cost.

The virtual element method for a minimal surface problem

In this paper we consider the Virtual Element discretization of a minimal surface problem, a quasi-linear elliptic partial differential equation modeling the problem of minimizing the area of a surface subject to a prescribed boundary condition. We derive an optimal error estimate and present several numerical tests assessing the validity of the theoretical results.

A family of quasi-variable meshes high-resolution compact operator scheme for Burger's-Huxley, and Burger's-Fisher equation

We describe a quasi-variable meshes implicit compact finite-difference discretization having an accuracy of order four in the spatial direction and second-order in the temporal direction for obtaining numerical solution values of generalized Burger’s-Huxley and Burger’s-Fisher equations. The new difference scheme is derived for a general one-dimension quasi-linear parabolic partial differential equation on a quasi-variable meshes network to the extent that the magnitude of local truncation error of the high-order compact scheme remains unchanged in case of uniform meshes network. Practically, quasi-variable meshes high-order compact schemes yield more precise solution compared with uniform meshes high-order schemes of the same magnitude. A detailed exposition of the new scheme has been introduced and discussed the Fourier analysis based stability theory. The computational results with generalized Burger’s-Huxley equation and Burger’s-Fisher equation are obtained using quasi-variable meshes high-order compact scheme and compared with a numerical solution using uniform meshes high-order schemes to demonstrate capability and accuracy.

Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method

<p style='text-indent:20px;'>The mathematical model of a semiconductor device is described by a coupled system of three quasilinear partial differential equations. The mixed finite element method is presented for the approximation of the electrostatic potential equation, and the characteristics finite element method is used for the concentration equations. First, we estimate the mixed finite element and the characteristics finite element method solution in the sense of the <inline-formula><tex-math id="M1">$ L^q $</tex-math></inline-formula> norm. To linearize the full discrete scheme of the problem, we present an efficient two-grid method based on the idea of Newton iteration. The two-grid algorithm is to solve the nonlinear coupled equations on the coarse grid and then solve the linear equations on the fine grid. Moreover, we obtain the <inline-formula><tex-math id="M2">$ L^{q} $</tex-math></inline-formula> error estimates for this algorithm. It is shown that a mesh size satisfies <inline-formula><tex-math id="M3">$ H = O(h^{1/2}) $</tex-math></inline-formula> and the two-grid method still achieves asymptotically optimal approximations. Finally, the numerical experiment is given to illustrate the theoretical results.</p>

High-resolution compact numerical method for the system of 2D quasi-linear elliptic boundary value problems and the solution of normal derivatives on an irrational domain with engineering applications

In this paper, we present a novel approach to attain fourth-order approximate solution of 2D quasi-linear elliptic partial differential equation on an irrational domain. In this approach, we use nine grid points with dissimilar mesh in a single compact cell. We also discuss appropriate fourth-order numerical methods for the solution of the normal derivatives on a dissimilar mesh. The method has been protracted for solving system of quasi-linear elliptic equations. The convergence analysis is discussed to authenticate the proposed numerical approximation. On engineering applications, we solve various test problems, such as linear convection–diffusion equation, Burgers’equation, Poisson equation in singular form, NS equations, bi- and tri-harmonic equations and quasi-linear elliptic equations to show the efficiency and accuracy of the proposed methods. A comprehensive comparative computational experiment shows the accuracy, reliability and credibility of the proposed computational approach.

Convergence of the SQP method for quasilinear parabolic optimal control problems

Based on the theoretical framework recently proposed by Bonifacius and Neitzel (Math Control Relat Fields 8(1):1–34, 2018. https://doi.org/10.3934/mcrf.2018001) we discuss the sequential quadratic programming (SQP) method for the numerical solution of an optimal control problem governed by a quasilinear parabolic partial differential equation. Following well-known techniques, convergence of the method in appropriate function spaces is proven under some common technical restrictions. Particular attention is payed to how the second order sufficient conditions for the optimal control problem and the resulting L^2-local quadratic growth condition influence the notion of “locality” in the SQP method. Further, a new regularity result for the adjoint state, which is required during the convergence analysis, is proven. Numerical examples illustrate the theoretical results.

Future Expectations Modeling, Random Coefficient Forward–Backward Stochastic Differential Equations, and Stochastic Viscosity Solutions

In this paper we study a class of infinite horizon fully coupled forward–backward stochastic differential equations (FBSDEs) with random coefficients that are stimulated by various continuous time future expectations models. Under standard Lipschitz and monotonicity conditions and by means of the contraction mapping principle, we establish existence and uniqueness of an adapted solution, and we obtain results regarding the dependence of this solution on the data of the problem. Making further the connection with finite horizon quasilinear backward stochastic partial differential equations via a generalization of the well known four-step-scheme, we are led to the notion of stochastic viscosity solutions. As an application of this framework, we also provide a stochastic maximum principle for the optimal control problem of such FBSDEs, which in the linear-quadratic Markovian case boils down to the solvability of an infinite horizon fully coupled system of forward-backward Ricatti differential equations.

Model-Free Backward and Forward Nonlinear PDEs for Implied Volatility

The authors derive backward and forward nonlinear partial differential equations that govern the implied volatility of a contingent claim whenever the latter is well defined. This would include at least any contingent claim written on a positive stock price whose payoff at a possibly random time is convex. The authors also discuss suitable initial and boundary conditions for those partial differential equations. Finally, we demonstrate how to solve them numerically by using an iterative finite-difference approach. TOPICS:Volatility measures, options Key Findings • In this article, we derive backward and forward quasilinear parabolic partial differential equations (PDEs) that govern the implied volatility of a contingent claim whenever the latter is well defined. Alternatively, we have derived a forward nonlinear hyperbolic PDE of the first order, which also governs evolution of the implied volatility in (K, T, Z) space. We discuss suitable initial and boundary conditions for those PDEs. • We develop an iterative numerical method to solve the PDEs by using a finite-difference approach. The method is of second order of approximation in both space and time, is unconditionally stable, and preserves positivity of the solution. • Using this method, we compute the PDE implied volatility and find that our intuition behind the main idea of the article is correct. In other words, performance of the finite-difference solver exceeds that of the traditional approach by factor of 40. However, this result is subject to some details, which are highlighted in the article.