Superconvergence of the continuous Galerkin finite element method for delay-differential equation with several terms

The continuous Galerkin finite element method for linear delay-differential equation with several terms is studied. Adding some lower terms in the remainder of orthogonal expansion in an element so that the remainder satisfies more orthogonal condition in the element, and obtain a desired superclose function to finite element solution, thus the superconvergence of p  -degree finite element approximate solution on (p+1)$(p+1)$-order Lobatto points is derived.     

Higher order finite difference method for the reaction and anomalous-diffusion equation

In this paper, our aim is to study the high order finite difference method for the reaction and anomalous-diffusion equation. According to the equivalence of the Riemann–Liouville and Grünwald–Letnikov derivatives under the suitable smooth condition, a second-order difference approximation for the Riemann–Liouville fractional derivative is derived. A fourth-order compact difference approximation for second-order derivative in spatial is used. We analyze the solvability, conditional stability and convergence of the proposed scheme by using the Fourier method. Then we obtain that the convergence order is $O({\tau }^2+h^4)$, where τ is the temporal step length and h is the spatial step length. Finally, numerical experiments are presented to show that the numerical results are in good agreement with the theoretical analysis.     

Optimal point-wise error estimate of a compact difference scheme for the Klein–Gordon–Schrödinger equation

In this paper, we propose a compact finite difference scheme for computing the Klein–Gordon–Schrödinger equation (KGSE) with homogeneous Dirichlet boundary conditions. The proposed scheme not only conserves the total mass and energy in the discrete level but also is linearized in practical computation. Except for the standard energy method, a new technique is introduced to obtain the optimal convergent rate, without any restriction on the grid ratios, at the order of O(h⁴+τ²)$O(h^4+{\tau }^2)$ in the l^∞$l^{\infty }$-norm with time step τ and mesh size h. Finally, numerical results are reported to test the theoretical results.     

Wavelet–Galerkin solutions of one dimensional elliptic problems

Daubechies wavelet bases are used for numerical solution of partial differential equations of one dimension by Galerkin method. Galerkin bases are constructed from Daubechies functions which are compactly supported and which constitute an orthonormal basis of L²(R)$L^2(\mathbb{R})$. Theoretical and numerical results are obtained for elliptic problems of second order with different types of boundary conditions. Optimal error estimates are also obtained. Comparison of solutions with simple finite difference method suggests that for this class of problems, the present method will provide a better alternative to other classical methods. The methodology can be generalized to multidimensional problems by taking care of some technical facts.     

A quasilinearization method for a class of second order singular nonlinear differential equations with nonlinear boundary conditions

We consider the differential equation $-(1/w)({\mathit{pu}}^{\prime }{)}^{\prime }+\mu u=\mathit{Fu}$, where F is a nonlinear operator, with nonlinear boundary conditions. Under appropriate assumptions on $p,w,F$ and the boundary conditions, the existence of solutions is established. If the problem has a lower solution and an upper solution, then we use a quasilinearization method to obtain two monotonic sequences of approximate solutions converging quadratically to a solution of the equation.     

Composition operators with univalent symbol in Schatten classes

We study composition operators, induced by a sub-domain of the unit disc whose boundary intersects the unit circle at 1 and which has, in a neighborhood of 1, a polar equation 1−r=γ(|θ|)$1-r=\gamma (|\theta |)$ (see Fig. 1). We obtain an explicit characterization for the membership in Schatten p-classes, in terms of γ.     

Mean field games with congestion

We consider a class of systems of time dependent partial differential equations which arise in mean field type models with congestion. The systems couple a backward viscous Hamilton–Jacobi equation and a forward Kolmogorov equation both posed in $(0,T)\times ({\mathbb{R}}^N/{\mathbb{Z}}^N)$. Because of congestion and by contrast with simpler cases, the latter system can never be seen as the optimality conditions of an optimal control problem driven by a partial differential equation. The Hamiltonian vanishes as the density tends to +∞ and may not even be defined in the regions where the density is zero. After giving a suitable definition of weak solutions, we prove the existence and uniqueness results of the latter under rather general assumptions. No restriction is made on the horizon T.     

Global existence to the initial–boundary value problem for a system of nonlinear diffusion and wave equations

We prove the global existence of weak solution pair to the initial boundary value problem for a system of m-Laplacian type diffusion equation and nonlinear wave equation. The interaction of two equations is given through nonlinear source terms $f(u,v)$ and $g(u,v)$.     

An efficient algorithm for the Schrödinger–Poisson eigenvalue problem

We present a new implementation of the two-grid method for computing extremum eigenpairs of self-adjoint partial differential operators with periodic boundary conditions. A novel two-grid centered difference method is proposed for the numerical solutions of the nonlinear Schrödinger–Poisson (SP) eigenvalue problem.We solve the Poisson equation to obtain the nonlinear potential for the nonlinear Schrödinger eigenvalue problem, and use the block Lanczos method to compute the first k   eigenpairs of the Schrödinger eigenvalue problem until they converge on the coarse grid. Then we perform a few conjugate gradient iterations to solve each symmetric positive definite linear system for the approximate eigenvector on the fine grid. The Rayleigh quotient iteration is exploited to improve the accuracy of the eigenpairs on the fine grid. Our numerical results show how the first few eigenpairs of the Schrödinger eigenvalue problem are affected by the dopant in the Schrödinger–Poisson (SP) system. Moreover, the convergence rate of eigenvalue computations on the fine grid is O(h³)$\mathrm{O}(h^3)$.     

Solving singular two-point boundary value problem in reproducing kernel space

In this paper, we present a new method for solving singular two-point boundary value problem for certain ordinary differential equation having singular coefficients. Its exact solution is represented in the form of series in reproducing kernel space. In the mean time, the n  -term approximation _un(x)$u_n(x)$ to the exact solution u(x)$u(x)$ is obtained and is proved to converge to the exact solution. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other.     

A new predictor-corrector scheme for valuing American puts

In this paper, we present a new numerical scheme, based on the finite difference method, to solve American put option pricing problems. Upon applying a Landau transform or the so-called front-fixing technique [19] to the Black-Scholes partial differential equation, a predictor-corrector finite difference scheme is proposed to numerically solve the nonlinear differential system. Through the comparison with Zhu’s analytical solution [35], we shall demonstrate that the numerical results obtained from the new scheme converge well to the exact optimal exercise boundary and option values. The results of our numerical examples suggest that this approach can be used as an accurate and efficient method even for pricing other types of financial derivative with American-style exercise.     

On the stress singularities and boundary layer in moderately thick functionally graded sectorial plates

In this paper, the stress analysis of moderately thick functionally graded sector plate is developed for studying the singularities in vicinity of the vertex and effects of boundary layer. Based on the first-order shear deformation plate theory, the governing partial differential equations are obtained. Using an analytical method, the stretching and bending equilibrium equations are decoupled. Also, introducing a function, called boundary layer function, the three bending equations are converted into two decoupled equations. These equations are solved analytically and the effects of boundary layer function on stress components are shown. Also, the singularities of shear force, moment resultants and boundary layer function are discussed for both salient (α?180)$(\alpha ?180)$ and re-entrant (α>180)$(\alpha >180)$ sectorial plates. In order to verify the accuracy of the results, the governing equations are also solved using differential quadrature method (DQM). By comparing the results of exact method with DQM, a good agreement can be seen.     

Blow-up analysis for solutions to Neumann boundary value problem

In this paper, we will analyze the blow-up behavior of solution sequences satisfying a conformal invariant equation defined on a compact 2-dimensional surface (M,g)$(M,g)$ with boundary. We will provide some accurate point-wise estimates for the profile of these sequences.     

Global uniqueness in an inverse problem for time fractional diffusion equations

Given $(M,g)$, a compact connected Riemannian manifold of dimension $d?2$, with boundary ?M, we consider an initial boundary value problem for a fractional diffusion equation on $(0,T)\times M$, $T>0$, with time-fractional Caputo derivative of order $\alpha \in (0,1)\cup (1,2)$. We prove uniqueness in the inverse problem of determining the smooth manifold $(M,g)$ (up to an isometry), and various time-independent smooth coefficients appearing in this equation, from measurements of the solutions on a subset of ?M at fixed time. In the “flat” case where M is a compact subset of ${\mathbb{R}}^d$, two out the three coefficients ρ (density), a (conductivity) and q (potential) appearing in the equation $\rho {?}_t^{\alpha }u?\mathrm{div}\left(a\mathrm{?}u\right)+qu=0$ on $(0,T)\times M$ are recovered simultaneously.     

Upper and lower solutions and quasilinearization for a class of second order singular nonlinear differential equations with nonlinear boundary conditions

We consider the differential equation $-(1/w)({\mathit{pu}}^{\prime }{)}^{\prime }=f(·,u)$, where $f$ is a nonlinear function, with nonlinear boundary conditions. Under appropriate assumptions on $p,w,f$ and the boundary conditions, the existence of solutions is established. If the problem has a lower solution and an upper solution, then we use a quasilinearization method to obtain two monotonic sequences of approximate solutions converging quadratically to a solution of the equation.     

Fractional partial differential equations with boundary conditions

We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show well-posedness of the associated Cauchy problems in $C_0(\mathrm{\Omega })$ and $L_1(\mathrm{\Omega })$. In order to do so we develop a new method of embedding finite state Markov processes into Feller processes on bounded domains and then show convergence of the respective Feller processes. This also gives a numerical approximation of the solution. The proof of well-posedness closes a gap in many numerical algorithm articles approximating solutions to fractional differential equations that use the Lax–Richtmyer Equivalence Theorem to prove convergence without checking well-posedness.