Singular Partial Differential Equations Research Articles

EXPANSIONS OF KAMP ́E DE F ́ERIET HYPERGEOMETRIC FUNCTIONS

When studying the properties of the hypergeometric functions in two variables, expansion formulas are very important, allowing one to represent a function of two variables in the form of an infinite sum of products of several hypergeometric Gaussian functions, and this in turn facilitates the process of studying the properties of functions in two variables. Burchnall and Chaundy, in 1940--41, using the symbolic method, obtained more than 15 pairs of expansions for the second-order double hypergeometric Appell and Humbert functions. In order to find expansion formulas for functions depending on three or more variables, Hasanov and Srivastava introduced symbolic operators, with the help of which they were able to expand a whole class of hypergeometric functions of several variables. Hasanov, Turaev and Choi defined so-called H-operators that make it possible to find expansions for generalized hypergeometric functions of one variable. In addition, applications of these H-operators to the expansion of the hypergeometric functions of two and three variables of second order are known. On the other hand, thanks to the Kampe de Feriet functions, solutions of the boundary value problems for some degenerate and singular partial differential equations can be written in explicit forms. In this paper, expansion formulae are obtained for the hypergeometric Kampe de Feriet functions of the superior order. Some Kampe de Feriet functions are expanded in terms of the Appell and Humbert functions as illustrative examples

Stochastic solutions and singular partial differential equations

Stochastic solutions is a robust technique previously used to obtain new exact solutions for deterministic nonlinear partial differential equations as well as numerical algorithms suited to parallel computing. Here it is proposed as a solution method for partial differential equations driven by distribution-valued noises. Two examples are worked out in detail.

Modification of high performance training algorithm for solving singular perturbation partial differential equations with cubic convergence

In this study, we used the update a neural network for solve singularly perturbed partial differential equations (SPPDEs) that have been singularly disturbed. We dealt with the modernization of artiftial networks by updating the most important types of high-performance training. For nonlinear equations, the modified Levenberg-Marquardt method (MLMM) is used, in which not only an LM step but also an approximation LM step is generated at each iteration. When applying the trust region technique, a new type of projected reduction for the merit function is created to assure global convergence of the new method. The modified LM method’s cubic convergence is demonstrated under the local error bound condition, which is weaker than nonsingularity. The numerical results demonstrate that the new method is quite efficient and might save numerous Jacobian calculations.

An existence result for optimal control of singular constraints involving 1-Laplacian operator

Learning-based partial differential equations (PDEs), which arise from applications in image processing and computer vision, are optimal control problems governed by PDEs. In this paper, we show the maximal monotonicity of the sum operator of the p-Laplacian and the singular 1-Laplacian. The effective domain of the sum operator is shown to be dense in Hilbert space equipped with the norm topology. As an application, we derive the existence of optimal solutions to optimal control of singular PDEs involving the sum operator of the p-Laplacian and 1-Laplacian in the continuous setting.

On the Asymptotic Expansion of Monomial in Singular Partial Differential Equations

People find that formal solutions to a type of doubly singular ordinary differential equations (systems) summable in a monomial of two variables, and that formal solutions to many singular partial differential equations were multisummable. So summability theory of formal power series in several variables is of great importance to the research of formal solutions to partial differential equations. This enriches the achievements in the research of formal solutions to differential equations. It is a application of monomial summability theory of formal power series.

Nonlocal elasticity and boundary condition paradoxes: a review

Nonclassical continuum mechanics theories have seen a rise in implementation over the past several years due to the increased research into micro-/nanoelectromechanical systems (MEMS/NEMS), micro-/nanoresonators, carbon nanotubes (CNTs), etc. Typically, these systems exist in the range of several nanometers to the micro-scale. There are several available theories that can capture phenomena inherent to nanoscale structures. Of the available theories, researchers utilize Eringen’s nonlocal theory most frequently because of its ease of implementation and seemingly accurate results for specific loading conditions and boundary conditions. Eringen’s integral nonlocal theory, which leads to a set of integro-partial differential equations, is difficult to solve; therefore, the integral form was reduced to a set of singular partial differential equations using a Green-type attenuation function. However, a so-called paradox has arisen between the integral and differential formulations of Eringen’s nonlocal elasticity. For certain boundary and loading conditions, instead of the expected softening effect inherent in nonlocal particle interactions, some researchers have found a stiffening effect. Still, others have found no variation from those results found using classical theories. As such, the discrepancies between the integral and differential forms have been the subject of debate for nearly two decades, with several proposed resolutions published in recent years. This paper serves to review and consolidate existing theories in nonlocal elasticity along with selected theories in nonclassical continuum mechanics, the utilization of Eringen’s nonlocal elasticity in beams, shells, and plates, the existing discrepancies and proposed solutions, and recommendations for future work.

On the Formal Solutions of a Type of Singular Partial Differential Equations

In this paper, we research a partial differential equation with three complex variables. According to the theory of singular differential equations, it is proved by direct calculation that it has a special form of formal power series solution.

A note on the applications of Wick products and Feynman diagrams in the study of singular partial differential equations

The study of singular partial differential equations has seen rapid significant developments very recently. In particular, the works of Hairer (2013, 2014), and Gubinelli et al. (2015) paved the way for others to follow, providing blueprints for further study. Yet, many manuscripts in this field consist of extensive applications of techniques from physics, specifically quantum field theory, such as Wick products which are best explained in terms of Feynman diagrams. The purpose of this short note is to describe how the necessity of Wick products comes about, their applications using Feynman diagrams, as well as the utility of Gaussian hypercontractivity theorem. We also conclude with a description of an open problem that seems to be very mathematically challenging and physically meaningful. The author’s intention is to make this note as accessible as possible to a wide audience by providing sufficient details, as well as relatively self-contained by including all relevant results which are necessary for our discussions.

Wavelet-Picard iterative method for solving singular fractional nonlinear partial differential equations with initial and boundary conditions

The present study applies the Picard iterative method to nonlinear singular partial fractional differential equations. The Haar and second-kind Chebyshev wavelets operational matrix of fractional integration will be used to solve problems combining linearization technique with the Picard method. The singular problem will be converted to an algebraic system of equations, which can be easily solved. Numerical examples are provided to illustrate the efficiency and accuracy of the technique.

Uniqueness of the solution of nonlinear singular first order partial differential equations

This paper deals with nonlinear singular partial differential equations of the form $t \partial u/\partial t = F(t,x,u,\partial u/\partial x)$ with independent variables $(t,x) \in \mathbb{R} \times \mathbb{C}$, where $F(t,x,u,v)$ is a function continuous in $t$ and holomorphic in the other variables. Under a very weak assumption we show the uniqueness of the solution of this equation. The results are applied to the problem of analytic continuation of local holomorphic solutions of equations of this type.

On a System of Linear Singular Partial Differential Equations with Weight Functions

Let X be a Banach space, Ω an open bounded subset of X, and Y a complex Banach space. We consider a Voleviˇc system of singular linear partial differential equations of the form t ∂ui ∂t = X N j=1 aij (t, x)uj (t, x) + X (j,k)∈N(i) bjk(t, x)((µ0(t)D) kuj (t, x) · x (k) k )(j,k) + gi(t, x), (1) 1 ≤ i ≤ N, in the unknown function u = (u1, u2, ..., uN ) ∈ Y N of t ≥ 0 and x ∈ Ω, where aij , bjk ∈ C, xk = (x, ..., x) (x is k times) D denotes the Frechet differentiation with respect to x, and N (i) = {(j, k) : j and k are integers, 1 ≤ j ≤ N, 0 < k ≤ n(i, j)}, (2) n(i, j) = n(i) − n(j) + 1, where n(i), i = 1, 2, ..., N, are nonnegative integers. The map µ0 belongs to C 0 ([0, T], C). We express growth estimates in terms of weight functions and we establish an existence and uniqueness theorem for our system in the class of ultradifferentiable maps with respect to the space variable x. Â

HIGH ORDER PARACONTROLLED CALCULUS

We develop in this work a general version of paracontrolled calculus that allows to treat analytically within this paradigm a whole class of singular partial differential equations with the same efficiency as regularity structures. This work deals with the analytic side of the story and offers a toolkit for the study of such equations, under the form of a number of continuity results for some operators, while emphasizing the simple and systematic mechanics of computations within paracontrolled calculus, via the introduction of two model operations$\mathsf{E}$and $\mathsf{F}$. We illustrate the efficiency of this elementary approach on the example of the generalized parabolic Anderson model equation$$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}+L)u=f(u)\unicode[STIX]{x1D701},\end{eqnarray}$$on a 3-dimensional closed manifold, and the generalized KPZ equation$$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}+L)u=f(u)\unicode[STIX]{x1D701}+g(u)(\unicode[STIX]{x2202}u)^{2},\end{eqnarray}$$driven by a$(1+1)$-dimensional space/time white noise.

Single-step full-state feedback control design for nonlinear hyperbolic PDEs

ABSTRACTThe present work proposes an extension of single-step formulation of full-state feedback control design to the class of distributed parameter system described by nonlinear hyperbolic partial differential equations (PDEs). Under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear state feedback law, both feedback control and stabilisation design objectives given as target stable dynamics are accomplished in one step. In particular, the mathematical formulation of the problem is realised via a system of first-order quasi-linear singular PDEs. By using Lyapunov's auxiliary theorem for singular PDEs, the necessary and sufficient conditions for solvability are utilised. The solution to the singular PDEs is locally analytic, which enables development of a PDE series solution. Finally, the theory is successfully applied to an exothermic plug-flow reactor system and a damped second-order hyperbolic PDE system demonstrating ability of in-domain nonlinear control law to achieve stabilisation.

Self–gravitating fluid flows with Gowdy symmetry near cosmological singularities

ABSTRACTWe consider self-gravitating fluids in cosmological spacetimes with Gowdy symmetry on the torus T3 and, in this class, we solve the singular initial value problem for the Einstein–Euler system of general relativity, when an initial data set is prescribed on the hypersurface of singularity. We specify initial conditions for the geometric and matter variables and identify the asymptotic behavior of these variables near the cosmological singularity. Our analysis of this class of nonlinear and singular partial differential equations exhibits a condition on the sound speed, which leads us to the notion of sub-critical, critical, and super-critical regimes. Solutions to the Einstein–Euler systems when the fluid is governed by a linear equation of state are constructed in the first two regimes, while additional difficulties arise in the latter one. All previous studies on inhomogeneous spacetimes concerned vacuum cosmological spacetimes only.

English

This paper explores some nonlinear systems of singular partial differential equations written in the form t A D t U = Λ(t, x)U + f t, x, ζU, t A D x U. Under an assumption on Λ, unique solvability theorems are provided in the space of functions that are holomorphic in x on an open set, differentiable with respect to t on a real interval ]0, r] and extending to a continuous function at t = 0. The studied systems contain Fuchsian systems.

On an efficient implementation and mass boundedness conditions for a discrete Dirichlet problem associated with a nonlinear system of singular partial differential equations

In this work, we propose an efficient implementation of a finite-difference method employed to approximate the solutions of a system of partial differential equations that appears in the investigation of the growth of biological films. The associated homogeneous Dirichlet problem is discretized using a linear approach. This discretization yields a positivity- and boundedness-preserving implicit technique which is represented in vector form through the multiplication by a sparse matrix. A straightforward implementation of this methodology would require a substantial amount of computer memory and time, but the problem is conveniently coded using a continual reduction of the zero sub-matrices of the representing matrix. In addition to the conditions that guarantee the positivity and the boundedness of the numerical approximations, we establish some parametric constraints that assure that the same properties for the discrete total mass at each point of the mesh-grid and each discrete time are actually satisfied. Some simulations are provided in order to illustrate both the performance of the implementation, and the preservation of the positivity and the boundedness of the numerical approximations.

PARACONTROLLED DISTRIBUTIONS AND SINGULAR PDES

We introduce an approach to study certain singular partial differential equations (PDEs) which is based on techniques from paradifferential calculus and on ideas from the theory of controlled rough paths. We illustrate its applicability on some model problems such as differential equations driven by fractional Brownian motion, a fractional Burgers-type stochastic PDE (SPDE) driven by space-time white noise, and a nonlinear version of the parabolic Anderson model with a white noise potential.

A minimum Sobolev norm technique for the numerical discretization of PDEs

Partial differential equations (PDEs) are discretized into an under-determined system of equations and a minimum Sobolev norm solution is shown to be efficient to compute and converge under very generic conditions. Numerical results of a single code, that can handle PDEs in first-order form on complicated polygonal geometries, are shown for a variety of PDEs: variable coefficient div–curl, scalar elliptic PDEs, elasticity equation, stationary linearized Navier–Stokes, scalar fourth-order elliptic PDEs, telegrapher's equations, singular PDEs, etc.

Improvement of the Modified Decomposition Method for Handling Third-Order Singular Nonlinear Partial Differential Equations with Applications in Physics

The modified decomposition method (MDM) is improved by introducing new inverse differential operators to adapt the MDM for handling third-order singular nonlinear partial differential equations (PDEs) arising in physics and mechanics. A few case-study singular nonlinear initial-value problems (IVPs) of third-order PDEs are presented and solved by the improved modified decomposition method (IMDM). The solutions are compared with the existing exact analytical solutions. The comparisons show that the IMDM is effectively capable of obtaining the exact solutions of the third-order singular nonlinear IVPs.

The Reduction of a Type of Singular Partial Differential Equations

Using the theory with respect to the existence of holomorphic solutions to a singular ordinary differential equation in the complex domain, this paper shows how to constructs biholomorphic trans-formations in different cases to reduce partial differential equations with singularity at the origin to more simple forms.