Explicit high-order energy-preserving methods for general Hamiltonian partial differential equations

In this paper we shall introduce new constructions of approximate solutions of general linear partial differential equations with constant coefficients on the whole spaces, and establish fundamental estimates of the solutions depending on the inhomogeneous terms. This will be done by combining general ideas of the Tikhonov regularization and discretization of bounded linear operator equations on reproducing kernel Hilbert spaces. Furthermore, we will provide approximate solutions for the related inverse source problems.

This paper presents the applicati on of a general purpose partial differential equation solver, called PDEase2D and the associated soil property functions to the analysis of saturated-unsaturated seepage problem. Examples of steady state and transient seepage through an earth dam are used to illustrate the use of PDEase2D in solving saturated-unsaturated seepage problems.

Is the neural tangent kernel of PINNs deep learning general partial differential equations always convergent?

It is possible to construct fully implicit Runge-Kutta methods like Gaus-Legendre, Radau-IA, Radau-IIA, Lobatto-IIIA, -IIIB, and -IIIC methods of arbitrary high order of convergence. The aim of this paper is to find a new adaptive time stepping for these classes which is based on the embedding technique. Adaptive time step control with embedding is well-known for Runge-Kutta methods, and therefore new embedded methods of order s-1 for the above classes of fully implicit Runge-Kutta methods are constructed. Since these fully implicit methods need the solution of a huge non-linear system of equations different approaches for non-linear equations are discussed and compared. It can be observed that non-linear solvers like the usually used simplified Newton method have a step size restriction if they are applied on higher order methods. We apply our new methods on some lower dimensional ODEs to show that our approach leads to an efficient method.

We present a class of nonconforming virtual element methods for general fourth-order partial differential equations in two dimensions. We develop a generic approach for constructing the necessary projection operators and virtual element spaces. Optimal error estimates in the energy norm are provided for general linear fourth-order problems with varying coefficients. We also discuss fourth-order perturbation problems and present a novel nonconforming scheme which is uniformly convergent with respect to the perturbation parameter without requiring an enlargement of the space. Numerical tests are carried out to verify the theoretical results. We conclude with a brief discussion on how our approach can easily be applied to nonlinear fourth-order problems.

A very general partial differential equation in space and time satisfied by the gene frequency in a monoecious population distributed continuously over an arbitrary habitat is derived. The treatment is restricted to a single diallelic locus in the absence of mutation and random drift, and it is supposed that time is continuous, births and deaths occur at random, and migration is independent of genotype. With the further assumptions that migration is isotropic and homogeneous, the population density is constant and uniform (as permitted by the population regulation mechanism included in the formulation), and Hardy-Weinberg proportions obtain locally, this partial differential equation reduces to the simplest multidimensional generalization of the classical Fisher-Haldane cline model. The efficacy of migration and selection in maintaining genetic variability at equilibrium in this model is investigated by deducing conditions for the existence of clines under various circumstances. The effects of the degree of dominance, a neutral belt between the regions where a particular allele is advantageous and deleterious, finiteness of the habitat, and habitat dimensionality are evaluated. Provided at least one of the alleles is favored only in a finite region, excluding the special case in which its total effective selective coefficient is zero, if conditions for supporting a cline are too unfavorable because migration is too strong, selection is too weak, or both, a cline cannot exist at all. Thus, unless there is overdominance, the population must be monomorphic. It is possible for a cline which can barely exist under the prevailing ecological circumstances to show a large amount of variation in gene frequency.

In Chapter 1 we presented basic ideas for the reduction of boundary value problems of the Laplacian to various forms of boundary integral equations based on the direct approach. This reduction can be easily extended to more general partial differential equations. Here we will consider, in particular, the Helmholtz equation, the Lame system, the Stokes equations and the biharmonic equation.

The primary aim is to develop a new class of pseudo-differential operators involving Linear Canonical Transform (LCT) and some of its essential results in Schwartz space and tempered distribution space. The secondary aim is to find applications of the LCT while constructing new general partial differential equations (Heat and Wave) and derive their solutions. Further, we have discussed particular cases of these partial differential equations. Furthermore, we have also shown the resolution of the solution of these equations graphically.

usual boundary conditions, and constraints u~U(t, z)~Em, where the control space U = U(t, z) is a given subset of Era. Here z =(z 1 . . . . . : ) are state variables and u =(u 1 .. . . , u m) control variables. As stated by P. K. RASHEVSKY [6], very general partial differential equations and systems can be written locally in the form (2). For instance, second order partial differential equations such as z~,+z~s= 1 u) can be immediately written in the form f ( t , s, z 1, zlt, z~,

A system of two time‐dependent, first order differential equations of electrical membrane resistance and of hydrostatic pressure is derived. The equations are based on a set of flux equations for a 1,–1 valent electrolyte, combined with the equation of continuity, and the condition for conservation of mass. New insight into the functioning of the membrane oscillator is gained by the strict derivation of the relaxation equation of the averaged salt concentration with convection from the general partial differential equation. This relaxation equation is successfully transformed into the useful time‐dependent first order differential equation of the function of the electrical membrane resistance. Thereby the postulated relation for the concomitant changes of the electrical membrane resistance and the averaged salt concentration in the membrane phase is employed. The equations describe both the stationary and the non‐stationary states of Teorell's membrane oscillator. Particularly, the stationary pressure‐dependent, non‐monotonous current‐voltage curves and the current‐dependent, non‐monotonous pressure‐volume flux relations are calculated. These characteristics form the basis for mechano‐electrical transduction. Furthermore, the relaxation oscillations of membrane voltage, volume flux and hydrostatic pressure and the corresponding electroosmotic and hydrodynamic phase planes, representing stable limit cycles, are described. The stability criteria of the time‐dependent states are discussed.

PDE blending surfaces with C2 continuity

The standard approach to applying IRK methods in the solution of two-point boundary value problems involves the solution of a non-linear system ofn×s equations in order to calculate the stages of the method, wheren is the number of differential equations ands is the number of stages of the implicit Runge-Kutta method. For two-point boundary value problems, we can select a subset of the implicit Runge-Kutta methods that do not require us to solve a non-linear system; the calculation of the stages can be done explicitly, as is the case for explicit Runge-Kutta methods. However, these methods have better stability properties than the explicit Runge-Kutta methods. We have called these new formulas two-point explicit Runge-Kutta (TPERK) methods. Their most important property is that, because their stages can be computed explicity, the solution of a two-point boundary value problem can be computed more efficiently than is possible using an implicit Runge-Kutta method. We have also developed a symmetric subclass of the TPERK methods, called ATPERK methods, which exhibit a number of useful properties.

This article extends the theory of classical finite-difference\nsummation-by-parts (FD-SBP) time-marching methods to the generalized\nsummation-by-parts (GSBP) framework. Dual-consistent GSBP time-marching methods\nare shown to retain: A and L-stability, as well as superconvergence of integral\nfunctionals when integrated with the quadrature associated with the\ndiscretization. This also implies that the solution approximated at the end of\neach time step is superconvergent. In addition GSBP time-marching methods\nconstructed with a diagonal norm are BN-stable. This article also formalizes\nthe connection between FD-SBP/GSBP time-marching methods and implicit\nRunge-Kutta methods. Through this connection, the minimum accuracy of the\nsolution approximated at the end of a time step is extended for nonlinear\nproblems. It is also exploited to derive conditions under which nonlinearly\nstable GSBP time-marching methods can be constructed. The GSBP approach to time\nmarching can simplify the construction of high-order fully-implicit Runge-Kutta\nmethods with a particular set of properties favourable for stiff initial value\nproblems, such as L-stability. It can facilitate the analysis of fully discrete\napproximations to PDEs and is amenable to to multi-dimensional spcae-time\ndiscretizations, in which case the explicit connection to Runge-Kutta methods\nis often lost. A few examples of known and novel Runge-Kutta methods associated\nwith GSBP operators are presented. The novel methods, all of which are L-stable\nand BN-stable, include a four-stage seventh-order fully-implicit method, a\nthree-stage third-order diagonally-implicit method, and a fourth-order\nfour-stage diagonally-implicit method. The relative efficiency of the schemes\nis investigated and compared with a few popular non-GSBP Runge-Kutta methods.\n

Remarks on the Dirichlet problem for general linear partial differential equations

Given a local one parameter Lie group of transformations G, we determine the most general scalar partial differential equation in (1 + 1)‐independent variables of a given order admitting G as nonclassical symmetry in the Bluman and Cole sense.