Singular Partial Differential Equations Research Articles

The Improved Modified Decomposition Method for the Treatment of Systems of Singular Nonlinear Partial Differential Equations with Initial Conditions

The Improved Modified Decomposition Method for the Treatment of Systems of Singular Nonlinear Partial Differential Equations with Initial Conditions

Convergence of Variational Iteration Method for Solving Singular Partial Differential Equations of Fractional Order

We are concerned here with singular partial differential equations of fractional order (FSPDEs). The variational iteration method (VIM) is applied to obtain approximate solutions of this type of equations. Convergence analysis of the VIM is discussed. This analysis is used to estimate the maximum absolute truncated error of the series solution. A comparison between the results of VIM solutions and exact solution is given. The fractional derivatives are described in Caputo sense.

Numerical Methods for Fractional Order Singular Partial Differential Equations with Variable Coefficients

We implement relatively analytical methods, the homotopy perturbation method and the variational iteration method, for solving singular fractional partial differential equations of fractional order. The process of the methods which produce solutions in terms of convergent series is explained. The fractional derivatives are described in Caputo sense. Some examples are given to show the accurate and easily implemented of these methods even with the presence of singularities.

On the Unique Solvability of Certain Nonlinear Singular Partial Differential Equations

We study the singular nonlinear equation tu_{t}=F(t,x,u,u_{x}) , where the function F is assumed to be continuous in t and holomorphic in the other variables. Under some growth conditions on the coefficients of the partial Taylor expansion of F , we show that if F(t,x,0,0) is of order O(\mu(t)^{\alpha}) for some \alpha\in[0,1] as t\rightarrow0 uniformly in some neighborhood of x=0 , then the equation has a unique solution u(t,x) with the same growth order.

Reduced-order state reconstruction for nonlinear dynamical systems in the presence of model uncertainty

A new approach to the reduced-order state reconstruction problem for nonlinear dynamical systems in the presence of model uncertainty is proposed. The problem of interest is conveniently formulated and addressed within the context of singular first-order non-homogeneous partial differential equations (PDE) theory, leading to a reduced-order nonlinear state estimator that is constructed through the solution of a system of singular PDEs. A set of necessary and sufficient conditions is derived that ensure the existence and uniqueness of a locally analytic solution to the above system of PDEs, and a series solution method is developed that is easily programmable with the aid of a symbolic software package such as MAPLE. Furthermore, the convergence of the estimation error or the mismatch between the actual unmeasurable states and their estimates is analyzed and characterized in the presence of model uncertainty. Finally, the performance of the proposed reduced-order state reconstruction method is evaluated in a case study involving a biological reactor that exhibits nonlinear dynamic behavior.

Exponential-type Nagumo norms and summability of formal solutions of singular partial differential equations

In this paper, we study a class of first order nonlinear degenerate partial differential equations with singularity at (t,x)=(0,0)∈C 2 . Using exponential-type Nagumo norm approach, the Gevrey asymptotic analysis is extended to case of holomorphic parameters in a natural way. A sharp condition is then established to deduce the k-summability of the formal solutions. Furthermore, analytical solutions in conical domains are found for each type of these nonlinear singular PDEs.

Convergence of formal solutions of first order singular partial differential equations of nilpotent type

Convergence of formal solutions of first order singular partial differential equations of nilpotent type

Invertiblity of some linear differential operators with singular point

We study the existence and unicity of solutions to some singular partial differential equations of first and second order.The solutions are constructed by a dynamical method.

Second-order hyperbolic Fuchsian systems and applications

We introduce a new class of singular partial differential equations, referred to as the second-order hyperbolic Fuchsian systems, and we investigate the associated initial value problem when data are imposed on the singularity. First, we establish a general existence theory of solutions with asymptotic behavior prescribed on the singularity, which relies on a new approximation scheme, suitable also for numerical purposes. Second, this theory is applied to the (vacuum) Einstein equations for Gowdy spacetimes, and allows us to recover, by more direct arguments, well-posedness results established earlier by Rendall and collaborators. Another main contribution in this paper is the proposed approximation scheme, which we refer to as the Fuchsian numerical algorithm and is shown to provide highly accurate numerical approximations to the singular initial value problem. For the class of Gowdy spacetimes, the numerical experiments presented here show the interest and efficiency of the proposed method and demonstrate the existence of a class of Gowdy spacetimes containing a smooth, incomplete and non-compact Cauchy horizon.

Three-sphere inequalities for second order singular partial differential equations

In this article, we give the three-sphere inequalities and three-ball inequalities for the singular elliptic equation div( A ∇ u) – Vu = 0, and the three-ball inequalities on the characteristic plane and the three-cylinder inequalities for the singular parabolic equation ∂ t u – div( A ∇ u) + Vu = 0, where the singular potential V belonging to the Kato-Fefferman-Phong's class. Some applications are also discussed.

Characterization of the dynamic behavior of nonlinear biosystems in the presence of model uncertainty using singular invariance PDEs: Application to immobilized enzyme and cell bioreactors

A new approach to the problem of characterizing the dynamic behavior of nonlinear biosystems in the presence of model uncertainty using the notion of slow invariant manifold is proposed. The problem of interest is addressed within the context of singular partial differential equations (PDE) theory, and in particular, through a system of singular quasi-linear invariance PDEs for which a general set of conditions for solvability is provided. Within the class of analytic solutions, this set of conditions guarantees the existence and uniqueness of a locally analytic solution which represents the system's slow invariant manifold exponentially attracting all dynamic trajectories in the absence of model uncertainty. An exact reduced-order model is then obtained through the restriction of the original biosystem dynamics on the slow manifold. The analyticity property of the solution to the invariance PDEs enables the development of a series solution method that can be easily implemented using MAPLE leading to polynomial approximations up to the desired degree of accuracy. Furthermore, the aforementioned attractivity property and the transition towards the above manifold is analyzed and characterized in the presence of model uncertainty. Finally, examples of certain immobilized enzyme bioreactors are considered to elucidate aspects of the proposed context of analysis.

On the singular PDE's geometry and boundary value problems

A geometric formulation of singular partial differential equations (PDEs) is considered. Surgery techniques and integral bordism groups are utilized, following previous works by Prástaro on PDEs, in order to build global solutions crossing also singular points and to study their stability properties.

Three-dimensional solidification with two possible crystallization states: Existence of solutions with flow in the melt

In this paper we discuss the existence of solutions of a system of nonlinear and singular partial differential equations constituting a phase field model with convection for non-isothermal solidification/melting of certain metallic alloys in the case where two different kinds of crystallization are possible. Each one of these crystallization states is described by its own phase field, while the liquid phase is described by another one. The model also allows the occurrence of fluid flow in non-solid regions, which are a priori unknown, and then we have a free-boundary value problem. Thus, the model relates the evolutions of these three phase fields, the temperature of the solidification/melting process and the fluid flow in non-solid regions. Copyright © 2009 John Wiley & Sons, Ltd.

A new model reduction method for nonlinear dynamical systems

The present research work proposes a new systematic approach to the problem of model-reduction for nonlinear dynamical systems. The formulation of the problem is conveniently realized through a system of singular first-order quasi-linear invariance partial differential equations (PDEs), and a rather general explicit set of conditions for solvability is derived. In particular, within the class of analytic solutions, the aforementioned set of conditions guarantees the existence and uniqueness of a locally analytic solution. The solution to the above system of singular PDEs is then proven to represent the slow invariant manifold of the nonlinear dynamical system under consideration exponentially attracting all dynamic trajectories. As a result, an exact reduced-order model for the nonlinear system dynamics is obtained through the restriction of the original system dynamics on the aforementioned slow manifold. The local analyticity property of the solution’s graph that corresponds to the system’s slow manifold enables the development of a series solution method, which allows the polynomial approximation of the system dynamics on the slow manifold up to the desired degree of accuracy and can be easily implemented with the aid of a symbolic software package such as MAPLE. Finally, the proposed approach and method is evaluated through an illustrative biological reactor example.

A high-order integral algorithm for highly singular PDE solutions in Lipschitz domains

We present a new algorithm, based on integral equation formulations, for the solution of constant-coefficient elliptic partial differential equations (PDE) in closed two-dimensional domains with non-smooth boundaries; we focus on cases in which the integral-equation solutions as well as physically meaningful quantities (such as, stresses, electric/magnetic fields, etc.) tend to infinity at singular boundary points (corners). While, for simplicity, we restrict our discussion to integral equations associated with the Neumann problem for the Laplace equation, the proposed methodology applies to integral equations arising from other types of PDEs, including the Helmholtz, Maxwell, and linear elasticity equations. Our numerical results demonstrate excellent convergence as discretizations are refined, even around singular points at which solutions tend to infinity. We demonstrate the efficacy of this algorithm through applications to solution of Neumann problems for the Laplace operator over a variety of domains—including domains containing extremely sharp concave and convex corners, with angles as small as π/100 and as large as 199π/100.

Singular functional differential equations of neutral type in Banach spaces

The well-posedness of a large class of singular partial differential equations of neutral type is discussed. Here the term singularity means that the difference operator of such equations is nonatomic at zero. This fact offers many difficulties in applying the usual methods of perturbation theory and Laplace transform technique and thus makes the study interesting. Our approach is new and it is based on functional analysis of semigroup of operators in an essential way, and allows us to introduce a new concept of solutions for such equations. Finally, we study the well-posedness of a singular reaction–diffusion equation of neutral type in weighted Lebesgue's spaces.

On a nonlocal boundary value problem for a system of singular differential equations

This article deals with a system consisting of singular partial differential equations of the first and second order arising in the zero approximation of I. Vekua's hierarchical models of prismatic shells, when the thickness of the shell varies as a power function of one argument and vanishes at the cusped edge of the shell. For this system of special type a nonlocal boundary value problem in a half-plane is solved in the explicit form. The boundary value problem under consideration corresponds to stress–strain state of the cusped prismatic shell under the action of concentrated forces and moments.

Convergent and Divergent Solutions of Singular Partial Differential Equations with Resonance or Small Denominators

We show the solvability and nonsolvability of a singular nonlinear system of partial differential equations with resonance in a class of functions holomorphic in some neighborhood of the origin. These results are applied to the normal form theory of a singular vector field.

Nonlocal boundary value problem for a system of singular differential equations and applications to cusped prismatic shells

AbstractThe paper deals with a system of singular partial differential equations of the first and second order arising in the zero approximation of I.Vekua's hierarchical models of prismatic shells, when the thickness of the shell varies as a power function of one argument and vanishes at the cusped edge of the shell. For this system of special type a nonlocal boundary value problem in a half‐plane is solved in the explicit form. The boundary value problem under consideration corresponds to stress‐strain state of the cusped prismatic shell [1,2] under the action of concentrated forces and moments. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

About curvature, conformal metrics and warped products

We consider the curvature of a family of warped products of two pseduo-Riemannian manifolds (B, gB) and (F, gF) furnished with metrics of the form c2gB ⊕ w2gF and, in particular, of the type w2μgB ⊕ w2gF, where c, w:B → (0, ∞) are smooth functions and μ is a real parameter. We obtain suitable expressions for the Ricci tensor and scalar curvature of such products that allow us to establish results about the existence of Einstein or constant scalar curvature structures in these categories. If (B, gB) is Riemannian, the latter question involves nonlinear elliptic partial differential equations with concave–convex nonlinearities and singular partial differential equations of the Lichnerowicz–York-type among others.