Order PDE Research Articles

Adaptive Image Processing: First Order PDE Constraint Regularizers and a Bilevel Training Scheme

A bilevel training scheme is used to introduce a novel class of regularizers, providing a unified approach to standard regularizers TGV^2 and NsTGV^2. Optimal parameters and regularizers are identified, and the existence of a solution for any given set of training imaging data is proved by Gamma -convergence under a conditional uniform bound on the trace constant of the operators and a finite-null-space condition. Some first examples and numerical results are given.

Partial differential equations from matrices with orthogonal columns

We discuss a system of third order PDEs for strictly convex smooth functions on domains of Euclidean space. We argue that it may be understood as the closure of the first order prolongation of a family of PDEs. We describe explicitly its real analytic solutions and all the solutions which satisfy a genericity condition; we also describe a family of non-generic solutions which has an application to Poisson geometry and Kähler structures on toric varieties. Our methods are geometric: we use the theory of Hessian metrics and symmetric spaces to link the analysis of the system of PDEs with properties of the manifold of matrices with orthogonal columns.

New approximate solutions to time fractional order partial differential equations optimal auxilary function method

In this article, approximate solutions of some PDE of fractional order are investi?gated with the help of a new semi-analytical method called the optimal auxiliary function method. The proposed method was tested upon the time-fractional Fisher equation, the time-fractional Fornberg-Whitham equation, and the time-fractional Inviscid Burger equation. The beauty of this method is that there is no need for discretization and assumptions of small or large parameters and provides an ap?proximate solution after only one iteration. The numerical results obtained by the proposed method compared with the other existing methods used in the literature. From the numerical and graphical results, it is clear that the proposed method gives a better solution than existing methods. The MATHEMATICA software pack?age has been used for the huge computational work.

Approximate Solution of Two Dimensional Disc-like Systems by One Dimensional Reduction: An Approach through the Green Function Formalism Using the Finite Elements Method

We present a comprehensive study for common second order PDE’s in two dimensional disc-like systems and show how their solution can be approximated by finding the Green function of an effective one dimensional system. After elaborating on the formalism, we propose to secure an exact solution via a Fourier expansion of the Green function, which entails solving an infinitely countable system of differential equations for the Green–Fourier modes that in the simplest case yields the source-free Green distribution. We present results on non separable systems—or such whose solution cannot be obtained by the usual variable separation technique—on both annulus and disc geometries, and show how the resulting one dimensional Fourier modes potentially generate a near-exact solution. Numerical solutions will be obtained via finite differentiation using Finite Difference Method (FDM) or Finite Element Method (FEM) with the three-point stencil approximation to derivatives. Comparing to known exact solutions, our results achieve an estimated numerical relative error below 10−6. Solutions show the well-known presence of peaks when r=r′ and a smooth behavior otherwise, for differential equations involving well-behaved functions. We also verified how the Green functions are symmetric under the presence of a “weight function”, which is guaranteed to exist in the presence of a curl-free vector field. Solutions of non-homogeneous differential equations are also shown using the Green formalism and showing consistent results.

Symmetries and conservation laws of the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates

Symmetries of the one-dimensional shallow water magnetohydrodynamics equations (SMHD) in Gilman’s approximation are studied. The SMHD equations are considered in case of a plane and uneven bottom topography in Lagrangian and Eulerian coordinates. Symmetry classification separates out all bottom topographies which yields substantially different admitted symmetries. The SMHD equations in Lagrangian coordinates were reduced to a single second order PDE. The Lagrangian formalism and Noether’s theorem are used to construct conservation laws of the SMHD equations. Some new conservation laws for various bottom topographies are obtained. The results are also represented in Eulerian coordinates. Invariant and partially invariant solutions are constructed.

Redefined quintic B-spline collocation technique for nonlinear higher order PDEs

Redefined quintic B-spline collocation technique for nonlinear higher order PDEs

On Symmetry and Uniqueness of Ground States for Linear and Nonlinear Elliptic PDEs

We study ground state solutions for linear and nonlinear elliptic PDEs in $\mathbb{R}^n$ with (pseudo-)differential operators of arbitrary order. We prove a general symmetry result in the nonlinear case as well as a uniqueness result for ground states in the linear case. In particular, we can deal with problems (e.g., higher order PDEs) that cannot be tackled by usual methods such as maximum principles, moving planes, or Polya--Szegö inequalities. Instead, we use arguments based on the Fourier transform and we apply a rigidity result for the Hardy--Littlewood majorant problem in $\mathbb{R}^n$ recently obtained by the last two authors of the present paper.

Quasilinear Systems of First Order PDEs with Nonlocal Hamiltonian Structures

In this paper we wonder whether a quasilinear system of PDEs of first order admits Hamiltonian formulation with local and nonlocal operators. By using the theory of differential coverings, we find differential-geometric conditions necessary to write a given system with one of the three Hamiltonian operators investigated.

A novel spectral technique for 2D fractional telegraph equation models with spatial variable coefficients

In this paper, a class of the two-dimensional (2D) mathematical models of the time fractional telegraph equation with spatial variable coefficients is considered. These models include the 2D time fractional telegraph equation (2D-TFTE), the 2D time fractional diffusion wave equation with damping (2D-TFDWED) and the 2D time fractional wave equation (2D-TFWE). The fractional derivatives are described in the Caputo sense. A novel spectral technique is used to solve the aforementioned models. The technique is based on the operational matrices of the shifted Gegenbauer polynomials in conjunction with the spectral tau method. The product of operational matrices of the space vectors is constructed in a simple way to remove the obstacle that facing the use of the tau method in solving the variable coefficients fractional/integer order PDEs. Miscellaneous numerical experiments are offered to verify the accuracy and rapid rate of convergence of the presented numerical technique.

Unveiling the Physics of Partial Differential Equations with Heuristics

Heuristic arguments and order of magnitude estimates for partial differential equations highlight essential features of the physics they describe. We present order of magnitude estimates, and their limitations, for the three classic second order PDEs of mathematical physics (wave, heat, and Laplace equations), for first order transport equations, and for two nonlinear wave equations. It is beneficial to expose the beginning student to these considerations before jumping into more rigorous mathematics. Yet these simple arguments are missing from physics textbooks.

Narrow-stencil framework for convergent numerical approximations of fully nonlinear second order PDEs

This article develops a unified general framework for designing convergent finite difference and discontinuous Galerkin methods for approximating viscosity and regular solutions of fully nonlinear second order PDEs. Unlike the well-known monotone (finite difference) framework, the proposed new framework allows for the use of narrow stencils and unstructured grids which makes it possible to construct high order methods. The general framework is based on the concepts of consistency and g-monotonicity which are both defined in terms of various numerical derivative operators. Specific methods that satisfy the framework are constructed using numerical moments. Admissibility, stability, and convergence properties are proved,and numerical experiments are provided along with some computer implementation details.
 For more information see https://ejde.math.txstate.edu/conf-proc/26/f1/abstr.html

On the Eigenvalues of the Fermionic Angular Eigenfunctions in the Kerr Metric.

In view of a result recently published in the context of the deformation theory of linear Hamiltonian systems, we reconsider the eigenvalue problem associated with the angular equation arising after the separation of the Dirac equation in the Kerr metric, and we show how a quasilinear first order PDE for the angular eigenvalues can be derived efficiently. We also prove that it is not possible to obtain an ordinary differential equation for the eigenvalues when the role of the independent variable is played by the particle energy or the black hole mass. Finally, we construct new perturbative expansions for the eigenvalues in the Kerr case and obtain an asymptotic formula for the eigenvalues in the case of a Kerr naked singularity.

Computing static behavior of flexible rectangular von Kármán plates in fast and reliable way

In this paper, several modifications of the Bubnov–Galerkin method for studying flexible rectangular plates have been constructed and validated. This is to overcome the challenges of applying the existing approaches to a specific class of problems, bordering on accuracy and machine time efficiency. Von Kármán’s (elliptic) equations consisting of a system of nonlinear PDEs of order eight, with respect to two functions, i.e., the deflection and the stress functions, are considered. The problem is reduced by employing several modifications of the Bubnov–Galerkin method combined with the Fourier approach (separation of variables) to a system of nonlinear ODEs. The paper deals with many of such techniques and a procedure that refines their solutions based on known discrepancies. A theorem on the convergence of the proposed iterative processes is formulated and proved. The developed iterative methods are used to solve rectangular plates problem, subjected to distributed and local loads for several kinds of boundary conditions. A comparison of the obtained solutions by various methods is carried out. The optimal calculation method in terms of accuracy and minimum machine time computation is revealed. In addition, we show that the proposed novel and efficient approaches can be used to estimate the static behavior of plates of arbitrary geometry.

Image denoising model based on improved fractional calculus mathematical equation

Abstract Aiming at the problems of image edge blur and detail loss in most image denoising models, an image denoising model based on improved fractional calculus is proposed. Through the classification of image noise, the integer order PDE model is discussed. Based on the mathematical theory of fractional calculus, the fractional PDE model is discussed. The model is improved by combining texture detection operator, gray value detection operator, smoothing factor and fast algorithm. The simulation experiment of the model is carried out based on MATLAB simulation platform. The results show that the improved model is significantly better than other comparison models in image denoising effect, and the operation efficiency is higher.

Reducing Nonlinear Partial Differential Equation Using Lie Infinitesimals Method

Theoretical underpinnings of differential equations have advanced greatly, particularly in the twentieth century. This expansion is due to the rapid and effective development of supporting mathematical fields (such as functional analysis, measure theory, and function spaces), as well as an ever-increasing need for applications, particularly in engineering, science, and medicine. The Lie infinitesimals method was employed to reduce the nonlinear fourth order PDE into an ordinary differential equation then the resulting ODE solved by the finite difference method. The Lie infinitesimals method is used to solve significantly more complex problems which used in manufacture.

On the sharp interface limit of a phase field model for near spherical two phase biomembranes

We consider sharp interface asymptotics for a phase field model motivated by lipid raft formation on near spherical biomembranes involving a coupling between the local mean curvature and the local composition. A reduced diffuse interface energy depending only on the membrane composition is introduced and a \Gamma -limit is derived. It is shown that the Euler–Lagrange equations for the limiting functional and the sharp interface energy coincide. Finally, we consider a system of gradient flow equations with conserved Allen–Cahn dynamics for the phase field model. Performing a formal asymptotic analysis, we obtain a system of gradient flow equations for the sharp interface energy coupling geodesic curvature flow for the phase interface to a fourth order PDE free boundary problem for the surface deformation.

Can a single scalar second order PDE govern well the propagation of the electric wave field in a heterogeneous 3D medium?

Abstract It is demonstrated in this paper that the propagation of the electric wave field in a heterogeneous medium in 3D can sometimes be governed well by a single scalar second order PDE, which is derived from Maxwell’s equations. The corresponding component of the electric field dominates two other components. This justifies some past results of the second author with coauthors about numerical solutions of coefficient inverse problems with experimental electromagnetic data. In addition, since it is simpler to work in applications with a single scalar second order PDE rather than with the complete Maxwell system, the result of this paper might be useful to researchers working on applied issues of the propagation of electromagnetic waves in inhomogeneous media.

Some problems for Petrovskii parabolic systems in generalized Sobolev spaces

We consider an inhomogeneous initial-boundary value problem for a Petrovskii parabolic system of second order PDEs. We prove that this problem induces isomorphisms between appropriate anisotropic generalized Sobolev spaces. The regularity of these spaces are given by a pair of real numbers and by a function parameter. The latter allows us to characterize the regularity of solutions to the problem more finely as compared with anisotropic Sobolev spaces.

A new clique polynomial approach for fractional partial differential equations

Abstract This paper generates a novel approach called the clique polynomial method (CPM) using the clique polynomials raised in graph theory and used for solving the fractional order PDE. The fractional derivative is defined in terms of the Caputo fractional sense and the fractional partial differential equations (FPDE) are converted into nonlinear algebraic equations and collocated with suitable grid points in the current approach. The convergence analysis for the proposed scheme is constructed and the technique proved to be uniformly convegant. We applied the method for solving four problems to justify the proposed technique. Tables and graphs reveal that this new approach yield better results. Some theorems are discussed with proof.

On solutions of PDEs by using algebras

The components of complex analytic functions define solutions for the Laplace's equation, and in a simply connected domain, each solution of this equation is the first component of a complex analytic function. In this paper, we generalize this result; for each PDE of the form , and for each affine planar vector field φ, we give an algebra with unit e = e1, with respect to which the components of all functions of the form are all the solutions for this PDE, where is differentiable in the sense of Lorch with respect to . Solutions are also constructed for the following equations: , ‐order PDEs, and ‐order PDEs; among these are the bi‐harmonic, the bi‐wave, and the bi‐telegraph equations.