Quasilinear Partial Differential Equations Research Articles

Numerical analysis of a nonsmooth quasilinear elliptic control problem: I. Explicit second-order optimality conditions

In this paper, we derive explicit second-order necessary and sufficient optimality conditions of a local minimizer to an optimal control problem for a quasilinear second-order partial differential equation with a piecewise smooth but not differentiable nonlinearity in the leading term. The key argument rests on the analysis of level sets of the state. Specifically, we show that if a function vanishes on the boundary and its the gradient is different from zero on a level set, then this set decomposes into finitely many closed simple curves. Moreover, the level sets depend continuously on the functions defining these sets. We also prove the continuity of the integrals on the level sets. In particular, Green’s first identity is shown to be applicable on an open set determined by two functions with nonvanishing gradients. In the second part to this paper, the explicit sufficient second-order conditions will be used to derive error estimates for a finite-element discretization of the control problem.

On hierarchical competition through reduction of individual growth

We consider a population organised hierarchically with respect to size in such a way that the growth rate of each individual depends only on the presence of larger individuals. As a concrete example one might think of a forest, in which the incidence of light on a tree (and hence how fast it grows) is affected by shading by taller trees. The classic formulation of a model for such a size-structured population employs a first order quasi-linear partial differential equation equipped with a non-local boundary condition. However, the model can also be formulated as a delay equation, more specifically a scalar renewal equation, for the population birth rate. After discussing the well-posedness of the delay formulation, we analyse how many stationary birth rates the equation can have in terms of the functional parameters of the model. In particular we show that, under reasonable and rather general assumptions, only one stationary birth rate can exist besides the trivial one (associated to the state in which there are no individuals and the population birth rate is zero). We give conditions for this non-trivial stationary birth rate to exist and analyse its stability using the principle of linearised stability for delay equations. Finally, we relate the results to the alternative, partial differential equation formulation of the model.

Generalized RKM methods for solving fifth-order quasi-linear fractional partial differential equation

Abstract Fractional differential equations (FDEs) are used for modeling the natural phenomena and interpretation of many life problems in the fields of applied science and engineering. The mathematical models which include different types of differential equations are used in some fields of applied sciences like biology, diffusion, electronic circuits, damping laws, fluid mechanics, and many others. The derivation of modern analytical or numerical methods for solving FDEs is a significant problem. However, in this article, we introduce a novel approach to generalize Runge Kutta Mechee (RKM) method for solving a class of fifth-order fractional partial differential equations (FPDEs) by combining numerical RKM techniques with the method of lines. We have applied the developed approach to solve some problems involving fifth-order FPDEs, and then, the numerical and analytical solutions for these problems have been compared. The comparisons in the implementations have proved the efficiency and accuracy of the developed RKM method.

Finite element solution of crack-tip fields for an elastic porous solid with density-dependent material moduli and preferential stiffness

In this paper, the finite element solutions of crack-tip fields for an elastic porous solid with density-dependent material moduli are presented. Unlike the classical linearized case in which material parameters are globally constant under a small strain regime, the stiffness of the model presented in this paper can depend upon the density with a modeling parameter. The proposed constitutive relationship appears linear in the Cauchy stress and linearized strain independently. From a subclass of the implicit constitutive relation, the governing equation is bestowed via the balance of linear momentum, resulting in a quasi-linear partial differential equation (PDE) system. Using the classical damped Newton’s method, the sequence of linear problems is then obtained, and the linear PDEs are discretized through a bilinear continuous Galerkin-type finite element method. We perform a series of numerical simulations for material bodies with a single edge-crack subject to a variety of loading types (i.e. the pure mode-I, II, and mixed-mode). Numerical solutions demonstrate that the modeling parameter in our proposed model can control preferential mechanical stiffness with its sign and magnitude along with the change of volumetric strain. This study can provide a mathematical and computational foundation to further model the quasi-static and dynamic evolution of cracks, utilizing the density-dependent moduli model and its modeling framework.

Optimal control of a nonsmooth PDE arising in the modeling of shear–thickening fluids

This paper focuses on the analysis of an optimal control problem governed by a nonsmooth quasilinear partial differential equation that models a stationary incompressible shear-thickening fluid. We start by studying the directional differentiability of the non-smooth term within the state equation as a prior step to demonstrate the directional differentiability of the solution operator. Thereafter, we establish a primal first order necessary optimality condition (Bouligand (B) stationarity), which is derived from the directional differentiability of the solution operator. By using a local regularization of the nonsmooth term and carrying out an asymptotic analysis thereafter, we rigourously derive a weak stationarity system for local minima. By combining the B- and weak stationarity conditions, and using the regularity of the Lagrange multiplier, we are able to obtain a strong stationarity system that includes a characterization of the Lagrange multiplier on the active and inactive sets.

A finite element implementation of the incompressible Schrödinger flow method

As first proposed by Madelung in 1926, the analogy between quantum mechanics and hydrodynamics has been known for a long time; however, its potentialities and the possibility of using the characteristic equations of quantum mechanics to simulate the behavior of inviscid fluids have not been thoroughly investigated in the past. In this methodology, the incompressible Euler equations are thus substituted by the Schrödinger equation, turning a quasi-linear Partial Differential Equation into a linear one, an algorithm known in the literature as Incompressible Schrödinger Flow. Previous works on the subject used the Fast Fourier Transform method to solve this problem, obtaining promising results, especially in predicting vortex dynamics; this paper aims to implement this novel approach into a Finite Element framework to find a more general formulation better suited for future application on complex geometries and on test cases closer to real-world applications. Simple case studies are presented in this work to analyze the potentialities of this method: the results obtained confirm that this method could potentially have some advantages over traditional Computational Fluid Dynamics method, especially for what concerns computational savings related to the required time discretization, whilst also introducing new aspects of the algorithm, mainly related to boundary conditions, not addressed in previous works.

CONDIONALLY CORRECT INITIAL VALUES PROBLEM FOR THE SYSTEM OF QUASILINEAR PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER

In the theory of systems of quasilinear partial differential equations of the first order the main questions are the solvability of initial values problem and justification of the approximate methods.In this paper a class of systems of quasilinear partial differential equations of the first order is considered, for which the concept of a generalized solution is introduced. The approach of introducing a generalized solution of conditionally correct problem related to the classical Cauchy problem is also proposed. In the case of smooth initial functions, the generalized solution at least locally coincides with the classical solution, and in the general case, such a solution loses its smoothness and turns into a singular function.There is a class of problems that have a unique solution. Estimates of the convergence speed in approximate methods of solving such problems have been obtained. Approximate methods in this approach can be arbitrary in nature, for example, finite-difference methods, smoothing method, or viscosity method. A description of constructions for building an infinitenumber of solutions for the same class of such problems is also given.

A class of third order quasilinear partial differential equations describing spherical or pseudospherical surfaces

Third order equations, which describe spherical surfaces (ss) or pseudospherical surfaces (pss), of the formνzt−λzxxt=A(z,zx,zxx)zxxx+B(z,zx,zxx), with ν, λ ∈ R, ν2+λ2≠0 and A2+B2≠0, are considered. These equations are equivalent to the structure equations of a metric with Gaussian curvature K=1 or K=−1, respectively. Alternatively they can be seen as the compatibility condition of an associated su(2)-valued or sl(2,R)-valued linear problem, also referred to as a zero curvature representation. Under certain assumptions we obtain an explicit classification for equations of the considered form that describe ss or pss, in terms of some arbitrary differentiable functions. Several examples of such equations, which describe also a number of already known equations, are provided by suitably choosing the arbitrary functions. In particular, the problem of determining sequences of conservation laws, either in the ss or pss case, is discussed and illustrated by some examples.

The [formula omitted] transformation for solving quasi-linear partial differential equations

The Z1-Z4 transformations proposed in our previous papers have acquired remarkable success in the field of linear partial differential equations, general solutions of a large number of typical equations were obtained for the first time, and exact solutions of many definite solution problems of general significance were also successfully obtained. In this paper we will present the Z5 transformation and use it to get the general solutions of a variety of first-order quasi-linear partial differential equations and, for the first time, to obtain analytical solutions of a class of second-order quasi-linear partial differential equations containing arbitrary functions. Using the general solutions, we obtain exact solutions of two definite solution problems. By comparison we find that the general solutions of some first-order quasi-linear partial differential equations that can be obtained using the characteristic equation method are incomplete.

A space-time pseudospectral method for solving multi-dimensional quasi-linear parabolic partial differential (Burgers') equations

The viscous Burgers' equation is a kind of quasi-linear important partial differential equation appeared in many fields. The nonlinear term contained in this equation results in difficulty in finding its high-precision numerical solution. In this paper, a space-time pseudospectral method for solving multi-dimensional Burgers' equations is proposed based on the Lagrange polynomials. The method is employed in time and space both at Chebyshev- Gauss- Lobbato(CGL) points. The spectral coefficients are found in such a way that the residual becomes minimum. The given problem is reduced to a system of nonlinear algebraic equations, which is solved by Newton-Raphson's method. Error bounds on discrete L2-norm and Sobolev norm (Hp) are presented. The computational experiments are carried out to corroborate the theoretical results and to compare the present method with existing methods in the literature. Moreover, some model examples of multi-dimensional Burgers' equations are tested and a detailed comparison of the proposed method with various other methods is also given. When using the proposed method, some interesting phenomenon appears with the coefficient of kinematic viscous.

Riemann solutions of two-layered blood flow model in arteries

This study investigates the solutions of the Riemann problem for a two-layered blood flow model which is modeled by a system of quasi-linear hyperbolic partial differential equations (PDEs) obtained by vertically averaging the Euler equations over each layer. We explore the elementary waves, namely shock wave, rarefaction wave and contact discontinuity wave on the basis of method of characteristics. Further, we establish the existence and uniqueness of the corresponding local Riemann solution. Across the contact discontinuity wave, the areas of two nonlinear algebraic equations are determined by using the Newton–Raphson method of two variables in all possible wave combinations. A precise analytical method is used to display a detailed vision of the solution for this model inside a specified space domain and some certain time frame.

A continuum model for morphology formation from interacting ternary mixtures: Simulation study of the formation and growth of patterns

Our interest lies in exploring the ability of a coupled nonlocal system of two quasilinear parabolic partial differential equations to produce phase separation patterns. The obtained patterns are referred here as morphologies. Our target system is derived in the literature as the rigorous hydrodynamic limit of a suitably scaled interacting particle system of Blume–Capel-type driven by Kawasaki dynamics. The system describes in a rather implicit way the interaction within a ternary mixture that is the macroscopic counterpart of a mix of two populations of interacting solutes in the presence of a solvent. Our discussion is based on the qualitative behavior of numerical simulations of finite volume approximations of smooth solutions to our system and their quantitative postprocessing in terms of two indicators (correlation and structure factor calculations). Our results show many similar qualitative features (e.g. general shape and approximate coarsening rates) which have been observed in previous works on the stochastic Blume–Capel model with three interacting species. The properties of the obtained morphologies (shape, connectivity, and so on) can play a key role in, e.g., the design of the active layer for efficient organic solar cells.

High resolution numerical algorithm of order (2, 4, 4) for 2D quasi-linear hyperbolic equations based on half-step discretization on an unequal grid

This study suggests a novel three-level implicit half-step scheme of order two and four in time and space direction respectively, utilizing an unequal grid for the approximation of 2D quasi-linear hyperbolic partial differential equations on an unequal grid. The stability analysis of the model Telegraphic equation for unequal grid has been discussed and it has been demonstrated that the presented method is unconditionally stable for the Telegraphic equation. The alternating direction implicit (ADI) techniques are discussed for linear difference equations on a dissimilar domain. The developed method investigated several well-known physical problems on an unequal mesh to demonstrate its accuracy and efficiency.

A perturbed fractional p-Kirchhoff problem with critical nonlinearity

We consider a quasilinear partial differential equation governed by the p-Kirchhoff fractional operator. By using variational methods, we prove several results concerning the existence of solutions and their stability properties with respect to some parameters.

DYNAMIC MODEL OF THERMAL PROCESSES IN A LIQUID FLOWING THROUGH A CHANNEL

The article presents a dynamic thermal process model for an incompressible fluid moving through a channel of arbitrary cross-sectional shape. The study focuses on expressing the non-steady law of conservation of internal energy of the fluid by considering the dependence of parameters on both spatial coordinates and time. The one-dimensional spatial formulation of the problem is simplified by disregarding any changes in flow parameters in the direction perpendicular to the channel axis. The resulting equation represents a non-stationary, non-homogeneous, quasi-linear hyperbolic partial differential equation, which can be solved with the method of characteristics in the case of constant coefficients. The solution consists of two distinct solutions that are glued along the line x/t=v, with isolines shown in the figure provided. The research is relevant for the design of heat exchangers, rocket and aircraft engines, and other heat engineering devices.

Qualitative Properties of Solutions of Equations and Inequalities with KPZ-Type Nonlinearities

For quasilinear partial differential and integrodifferential equations and inequalities containing nonlinearities of the Kardar—Parisi—Zhang type, various (old and recent) results on qualitative properties of solutions (such as the stabilization of solutions, blow-up phenomena, long-time decay of solutions, and others) are presented. Descriptive examples demonstrating the Bitsadze approach (the technique of monotone maps) applied in this research area are provided.

The structure group for quasi-linear equations via universal enveloping algebras

We replace trees by multi-indices as an index set of the abstract model space to tackle quasi-linear singular stochastic partial differential equations. We show that this approach is consistent with the postulates of regularity structures when it comes to the structure group, which arises from a Hopf algebra and a comodule. Our approach, where the dual of the abstract model space naturally embeds into a formal power series algebra, allows to interpret the structure group as a Lie group arising from a Lie algebra consisting of derivations on this power series algebra. These derivations in turn are the infinitesimal generators of two actions on the space of pairs (non-linearities, functions of space-time mod constants). We also argue that there exist pre-Lie algebra and Hopf algebra morphisms between our structure and the tree-based one in the cases of branched rough paths (Grossman-Larson, Connes-Kreimer) and of the stochastic heat equation.

Penalty approximation method for a double obstacle quasilinear parabolic variational inequality problem

<p style='text-indent:20px;'>In this paper, we first establish a surjectivity result for the sum of a maximal monotone mapping and a generalized pseudomontone mapping by using the generalized Yosida approximation technique. Then, we study a double obstacle quasilinear parabolic variational inequality problem <inline-formula><tex-math id="M1">\begin{document}$ {{\rm{(VIP)}}} $\end{document}</tex-math></inline-formula> by using the surjectivity result and penalty approximation method. In order to deal with the double obstacle constraints, we construct a quasilinear parabolic partial differential penalty equation, then we obtain the solvability of the quasilinear parabolic partial differential penalty equation. Moreover, we show that the set of the solutions to the penalty equation is bounded and every weak cluster point of this set is a solution of the problem <inline-formula><tex-math id="M2">\begin{document}$ {{\rm{(VIP)}}} $\end{document}</tex-math></inline-formula>. At last, as an application, we obtain numerical solutions of two double obstacle parabolic variational inequality problems by using the power penalty approximation method. Through the figures in the examples, we can intuitively see the different numerical solutions of the problems at different times.</p>

Sobolev space weak solutions to one kind of quasilinear parabolic partial differential equations related to forward-backward stochastic differential equations

This paper is concerned with the Sobolev type weak solutions of one class of second order quasilinear parabolic partial differential equations (PDEs, for short). First of all, similar to Feng, Wang and Zhao [9] and Wu and Yu [29], we use a family of coupled forward-backward stochastic differential equations (FBSDEs, for short) which satisfy the monotonous assumption to represent the classical solutions of the quasilinear PDEs. Then, based on the classical solutions of a family of PDEs approximating the weak solutions of the quasilinear PDEs, we prove the existence of the weak solutions. Moreover, the principle of norm equivalence is employed to link FBSDEs and PDEs to obtain the uniqueness of the weak solutions. In summary, we provide a probabilistic interpretation for the weak solutions of quasilinear PDEs, which enriches the theory of nonlinear PDEs.

Начально-краевая задача о течении в кровеносном сосуде

The solution of the initial boundary value problem for a system of two quasi-linear hyperbolic equations, which describes a rotationally symmetric vortex-free flow of a viscous incompressible fluid in an infinite cylindrical domain (pipe, blood vessel) is constructed. It is assumed that the side surface of the domain is a free (soft, compliant) boundary on which the kinematic condition is set. The dynamic condition at the boundary is modeled by some constitutive relation – the dependence of the pressure P on the cross-sectional area S of the cylindrical domain. To study the flow an asymptotic model based on the shallow water theory is used. When constructing the solution of the initial-boundary (as well as initial and boundary) problem for a system of two quasilinear partial differential equations of the first order, the hodograph method based on the conservation law is used. A variant of the polynomial constitutive relation is chosen for the study: P~S2β (β>0). In the case of the initial data problem which specified at the initial time the Riemann-Green function, allowing to construct an implicit analytical solution, and an algorithm (numerical) for constructing an explicit solution of the original problem are given. The main attention is focused on the special case P~S2, for which all the formulas necessary to construct a solution are written in explicit form. For P~S2, several variants of conditions (initial and boundary value) are considered for the initial boundary value problem, which allow us to trace the evolution of the solution in detail. Numerical calculations demonstrating the motion of shock waves and wave fronts are given. The results obtained, in practice, can be used to describe flows in blood vessels, as well as reliable tests for testing compu tational algorithms intended to solve such problems, in particular, systems of quasi-linear hyperbolic equations.