Problems For Partial Differential Equations Research Articles

On a nonlinear boundary-value problem for a third-order partial differential equations

On a nonlinear boundary-value problem for a third-order partial differential equations

A finite element method for Dirichlet boundary control of elliptic partial differential equations

This paper introduces a new variational formulation for Dirichlet boundary control problem of elliptic partial differential equations, based on observations that the state and adjoint state are related through the control on the boundary of the domain, and that such a relation may be imposed in the variational formulation of the adjoint state. Well-posedness (unique solvability and stability) of the variational problem is established in the $H^{1}(\Omega)\times H_{0}^{1}(\Omega)$ space for the respective state and adjoint state. A finite element method based on this formulation is analyzed. It is shown that the conforming $k-$th order finite element approximations to the state and the adjoint state, in the respective $L^{2}$ and $H^{1}$ norms converge at the rate of order $k-1/2$ on quasi-uniform mesh for conforming element of order $k$. Numerical examples are presented to validate the theory.

Periodic problem for non-instantaneous impulsive partial differential equations

<abstract><p>We obtain a new maximum principle of the periodic solutions when the corresponding impulsive equation is linear. If the nonlinear is quasi-monotonicity, we study the existence of the minimal and maximal periodic mild solutions for impulsive partial differential equations by using the perturbation method, the monotone iterative technique and the method of upper and lower solution. We give an example in last part to illustrate the main theorem.</p></abstract>

Свойства одного матрично-дифференциального оператора высокого порядка

The article considers a linear matrix-differential operator of the n-th order of the form A^n. For it and for the operator (A ̃^(-1) )^n, an analytical expression is derived, for which an operator analog of the Newton binomial is obtained. A lemma on the solution of a linear equation is given. It is used in the study of the abstract Cauchy problem for an algebro-differential equation in a Banach space with the cube of the operator A at the highest derivative. The operator A has the property of having 0 as a normal eigenvalue. Conditions for the existence and uniqueness of the solution are determined; the solution is found, for which the method of cascade splitting of the equation and conditions into the corresponding equations and conditions in subspaces of lower dimensions is used. As an application, the results obtained for n=3 are used in solving a mixed problem for a fourth-order partial differential equation. These equations include the generalized shallow water wave equation and the generalized Liouville equation.

A Stable Difference Scheme for a Third-Order Partial Differential Equation

The nonlocal boundary-value problem for a third-order partial differential equation$$ \left\{\begin{array}{l}\frac{d^3u(t)}{dt^3}+A\frac{du(t)}{dt}=f(t),\kern1em 0<t<1,\\ {}u(0)=\gamma u\left(\lambda \right)+\varphi, \kern1em {u}^{\prime }(0)=\alpha {u}^{\prime}\left(\lambda \right)+\psi, \left|\gamma \right|<1,\\ {}{u}^{{\prime\prime} }(0)=\beta {u}^{{\prime\prime}}\left(\lambda \right)+\xi, \kern1em \left|1+\beta \alpha \right|>\left|\alpha +\beta \right|,0<\lambda \le 1\end{array}\right. $$in a Hilbert space H with a self-adjoint positive definite operator A is considered. A stable three-step difference scheme for the approximate solution of the problem is presented. The main theorem on stability of this difference scheme is established. As applications, the stability estimates for the solution of difference schemes of the approximate solution of three nonlocal boundary-value problems for third-order partial differential equations are obtained. Numerical results for one- and two-dimensional third-order partial differential equations are provided.

Meshfree numerical integration for some challenging multi-term fractional order PDEs

&lt;abstract&gt;&lt;p&gt;Fractional partial differential equations (PDEs) have key role in many physical, chemical, biological and economic problems. Different numerical techniques have been adopted to deal the multi-term FPDEs. In this article, the meshfree numerical scheme, Radial basis function (RBF) is discussed for some time-space fractional PDEs. The meshfree RBF method base on the Gaussian function and is used to test the numerical results of the time-space fractional PDE problems. Riesz fractional derivative and Grünwald-Letnikov fractional derivative techniques are used to deal the space fractional derivative terms while the time-fractional derivatives are iterated by Caputo derivative method. The accuracy of the suggested scheme is analyzed by using $ L_\infty $-norm. Stability and convergence analysis are also discussed.&lt;/p&gt;&lt;/abstract&gt;

HYBRID FDM-STABILIZED LANCZOS-TYPE IN SOLVING PDE PROBLEMS

This study investigates the combination of finite difference method (FDM) and the stabilized Lanczos method to solve various partial differential equation (PDE) problems. This combination is wrapped in the algorithms called hybrid FDMRMEIEMLA and hybrid FDM-RLMinRes. FDM is the discretization method which converts the PDEs into algebraic formula, whereas both RMEIEMLA and RLMinRes are known as the stabilized Lanczos methods in solving largescale problems of SLEs. Their hybrids enable us to find the solutions of PDE problems accurately. There are at least three types of PDEs solved in this study, namely Helmholtz, wave, and heat equations. The convergence rate of our methods computed using the residual norms || b - Axk ||. Numerical results showed that our proposed methods performed well in solving the various PDEs with small residual norms.

Conservation Laws -- a Source for Distortionless Propagation and Time Delays

Since the very first paper of J. Bernoulli in 1728, a connection exists between initial boundary value problems for hyperbolic Partial Differential Equations (PDE) in the plane (with a single space coordinate accounting for wave propagation) and some associated Functional Equations (FE). From the point of view of dynamics and control (to be specific, of dynamics for control) both type of equations generate dynamical and controlled dynamical systems. The functional equations may be difference equations (in continuous time), delay-differential (mostly of neutral type) or even integral/integro-differential. It is possible to discuss dynamics and control either for PDE or FE since both may be viewed as self contained mathematical objects. A more recent topic is control of systems displaying conservation laws. Conservation laws are described by nonlinear hyperbolic PDE belonging to the class ``lossless'' (conservative); their dynamics and control theory is well served by the associated energy integral. It is however not without interest to discuss association of some FE. Lossless implies usually distortionless propagation hence one would expect here also lumped time delays. The paper contains some illustrating applications from various fields: nuclear reactors with circulating fuel, canal flows control, overhead crane, drilling devices, without forgetting the standard classical example of the nonhomogeneous transmission lines for distortionless and lossless propagation. Specific features of the control models are discussed in connection with the control approach wherever it applies.

Recent progress in the conductivity reconstruction in Calderón’s problem

In this work, we study a nonlinear inverse problem for an elliptic partial differential equation known as the Calderón problem or the inverse conductivity problem. We give a short survey on the reconstruction question of conductivity from measurements on the boundary by covering the main currently known results regarding the isotropic problem with full data in two and higher dimensions. We present Nachman’s reconstruction procedure and summarize the theoretical progress of the technique to more recent results in the field. An open problem of significant interest is proposed to check whether it is possible to extend the method for Lipschitz conductivities.

Automatic pre- and postconditions for partial differential equations

Based on an automata-theoretic and algebraic framework, we study equational reasoning for Initial Value Problems (ivps) of polynomial Partial Differential Equations (pdes). We first characterize the solutions of a pde system Σ in terms of the final morphism from a coalgebra induced by Σ to the coalgebra of formal power series (fps). fps solutions conservatively extend the classical analytic ones. To express ivps in their general form, we then introduce stratified systems, where the specification of a function can be decomposed into distinct subsystems of pdes. We lift the existence and uniqueness result of fps solutions to stratified systems. We then give a relatively complete algorithm to compute weakest preconditions and strongest postconditions for such systems. To some extent, this result reduces equational reasoning on pde initial value problems to algebraic reasoning. We illustrate some experiments conducted with a proof-of-concept implementation of the method.

The shock formation and optimal regularities of the resulting shock curves for 1D scalar conservation laws* *Yin Huicheng and Zhu Lu are supported by the NSFC (No. 11571177, No. 11731007, No. 12001162).

The study on the shock formation and the regularities of the resulting shock surfaces for hyperbolic conservation laws is a basic problem in the nonlinear partial differential equations. In this paper, we are concerned with the shock formation and the optimal regularities of the resulting shock curves for the 1D conservation law ∂ t u + ∂ x f(u) = 0 with the smooth initial data u(0, x) = u 0(x). If and , it is well-known that the solution u will blow up on the time when min g′(x) < 0 holds for g(x) = f′(u 0(x)). Let x 0 be a local minimum point of g′(x) such that g′(x 0) = min g′(x) < 0 and g″(x 0) = 0, g (3)(x 0) > 0 (which is called the generic nondegenerate condition), then by theorem 2 of Lebaud (1994 J. Math. Pures Appl. 9 523–565), a weak entropy solution u together with the shock curve x = φ(t) ∈ C 2[T*, T* + ɛ) starting from the blowup point (T*, x* = x 0 + g(x 0)T*) can be locally constructed. When the generic nondegenerate condition is violated, namely, when x 0 is a local minimum point of g′(x) such that but for some with k 0 ⩾ 2; or g (k)(x 0) = 0 for any and k ⩾ 2, we will study the shock formation and the optimal regularity of the shock curve x = φ(t), meanwhile, some precise descriptions on the behaviors of u near the blowup point (T*, x*) are given. Our main aims are to show that: around the blowup point, the shock really appears whether the initial data are degenerate with finite orders or with infinite orders; the optimal regularities of the shock solution and the resulting shock curve have the explicit relations with the degenerate degrees of the initial data.

Determining the nonlinearity in an acoustic wave equation

We consider an undetermined coefficient inverse problem for a nonlinear partial differential equation describing high‐intensity ultrasound propagation as widely used in medical imaging and therapy. The usual nonlinear term in the standard model using the Westervelt equation in pressure formulation is of the form ppt. However, this should be considered as a low‐order approximation to a more complex physical model where higher order terms will be required. Here we assume a more general case where the form taken is f(p) pt and f is unknown and must be recovered from data measurements. Corresponding to the typical measurement setup, the overposed data consist of time trace observations of the acoustic pressure at a single point or on a one‐dimensional set Σ representing the receiving transducer array at a fixed time. Additionally to an analysis of well‐posedness of the resulting pde, we show injectivity of the linearized forward map from f to the overposed data and use this as motivation for several iterative schemes to recover f. Numerical simulations will also be shown to illustrate the efficiency of the methods.

Development of Methods of Asymptotic Analysis of Transition Layers in Reaction\u2013Diffusion\u2013Advection Equations: Theory and Applications

This work presents a review and analysis of modern asymptotic methods for analysis of singularly perturbed problems with interior and boundary layers. The central part of the work is a review of the work of the author and his colleagues and disciples. It highlights boundary and initial-boundary value problems for nonlinear elliptic and parabolic partial differential equations, as well as periodic parabolic problems, which are widely used in applications and are called reaction–diffusion and reaction–diffusion–advection equations. These problems can be interpreted as models in chemical kinetics, synergetics, astrophysics, biology, and other fields. The solutions of these problems often have both narrow boundary regions of rapid change and inner layers of various types (contrasting structures, moving interior layers: fronts), which leads to the need to develop new asymptotic methods in order to study them both formally and rigorously. A general scheme for a rigorous study of contrast structures in singularly perturbed problems for partial differential equations, based on the use of the asymptotic method of differential inequalities, is presented and illustrated on relevant problems. The main achievements of this line of studies of partial differential equations are reflected, and the key directions of its development are indicated.

An adaptive one-point second-derivative Lobatto-type hybrid method for solving efficiently differential systems

This manuscript presents a one-step method incorporating a second-derivative applied to obtain approximate solutions of first-order initial-value problems of ordinary and time-dependent partial differential equations. The new scheme is derived through interpolation and collocation approaches, and the characteristics of the obtained method are analyzed. An embedding-like technique is considered and executed in adaptive mode to get better performance of the proposed method. The new scheme is used for solving some real-life application problems to determine its efficiency and accuracy. The numerical solutions obtained show that the proposed error estimate strategy is suitable in a variable step-size mode, robust and competitive compared with the results provided by other methods in the available literature.

Adaptive time stepping in quench simulation for superconducting magnets

As an important issue of the quench simulation in large-scale applications, the effect of time step control is discussed on the well-known thermo-hydraulic model of superconducting magnets. Through our approach of model reduction, the reason of numerical anomaly is identified in terms of the time stepping around dynamical events, and just leads an effective countermeasure against the observed issue. Apart from the practical means out of such an investigation, a novel idea of non-iterative time step adaptation is introduced as a breakthrough not only being applied to the quench propagation model itself, but also to be extended to the common pitfalls of linearization scheme in general transient PDE (partial differential equation) problems.

Comparison of Finite-Difference Schemes for the Burgers Problem

A large number of problems in physics and engineering leads to boundary value or initial boundary value problems for linear and nonlinear partial differential equations. At the same time, the number of tasks with analytical solutions is limited. These are problems in canonical domains, such as, for example, a rectangle, a circle, or a ball, and usually for equations with constant coefficients. In practice, it is often necessary to solve problems in very complex areas and for equations with variable coefficients, often nonlinear. This leads to the need to search for approximate solutions using various numerical methods.

On the solvability of a boundary value problem for a quasilinear equation of mixed type with two degeneration lines

The article investigates the existence of a generalized solution to one boundary value problem for an equation of mixed type with two lines of degeneration in the weighted space of S.L. Sobolev. In proving the existence of a generalized solution, the spaces of functions U(Ω) and V (Ω) are introduced, the spaces H1(Ω) and (Ω) are defined as the completion of these spaces of functions, respectively, with respect to the weighted norms, including the functions K(y) and N(x). Using an auxiliary boundary value problem for a first order partial differential equation, Kondrashov’s theorem on the compactness of the embedding of (Ω) in L2(Ω) and Vishik’s lemma, the existence of a solution to the boundary value problem is proved.

The numerical solutions of linear semidiscrete evolution problems on the half‐line using the Unified Transform Method

AbstractWe discuss a semidiscrete analogue of the Unified Transform Method (UTM), introduced by A. S. Fokas, to solve initial‐boundary‐value problems for linear evolution partial differential equations of constant coefficients. The semidiscrete method is applied to various spacial discretizations of several first‐ and second‐order linear equations on the half‐line , producing the exact solution for the semidiscrete problem, given appropriate initial and boundary data. We additionally show how the UTM treats derivative boundary conditions and ghost points introduced by the choice of discretization stencil and propose the notion of “natural” discretizations. We consider the continuum limit of the semidiscrete solutions and provide several numerical examples.

Strong solutions to a fourth order exponential PDE describing epitaxial growth

In this paper we prove the global existence of a strong solution to the initial boundary value problem for the exponential partial differential equation ∂tu−Δe−Δu+e−Δu−1=0. The equation was proposed as a continuum model for epitaxial growth of crystal surfaces on vicinal surfaces with evaporation and deposition effects [6]. Our investigations reveal that we must control the size of both ‖e−Δu(x,0)‖W2,2(Ω) and ‖eΔu(x,0)‖∞,Ω suitably to achieve our results. Related results in [8,10] were established via the Weiner algebra framework. Here we offer a totally new approach, which seems to shed more light on the nature of exponential nonlinearity.

Convergence, stability analysis, and solvers for approximating sublinear positone and semipositone boundary value problems using finite difference methods

Positone and semipositone boundary value problems are semilinear elliptic partial differential equations (PDEs) that arise in reaction–diffusion models in mathematical biology and the theory of nonlinear heat generation. Under certain conditions, the problems may have multiple positive solutions or even nonexistence of a positive solution. We develop analytic techniques for proving admissibility, stability, and convergence results for simple finite difference approximations of positive solutions to sublinear problems. We also develop guaranteed solvers that can detect nonuniqueness for positone problems and nonexistence for semipositone problems. The admissibility and stability results are based on adapting the method of sub- and supersolutions typically used to analyze the underlying PDEs. The new convergence analysis technique directly shows that all pointwise limits of finite difference approximations are solutions to the boundary value problem eliminating the possibility of false algebraic solutions plaguing the convergence of the methods. Most known approximation methods for positone and semipositone boundary value problems rely upon shooting techniques; hence, they are restricted to one-dimensional problems and/or radial solutions. The results in this paper will serve as a foundation for approximating positone and semipositone boundary value problems in higher dimensions and on more general domains using simple approximation methods. Numerical tests for known applied problems with multiple positive solutions are provided. The tests focus on approximating certain positive solutions as well as generating discrete bifurcation curves that support the known existence and uniqueness results for the PDE problem.