NUMERICAL QUENCHING FOR A NONLINEAR DIFFUSION EQUATION WITH SINGULAR BOUNDARY FLUX

In this paper, we study the semidiscrete approximation of the solution of a nonlinear diffusion equation with nonlinear source and singular boundary flux. We find some conditions under which the solution of the semidiscrete form quenches in a finite time and estimate its semidiscrete quenching time. We also establish the convergence of the semidiscrete quenching time to the theoretical one when the mesh size tends to zero. Finally, we give some numerical experiments for a best illustration of our analysis.

Inverse problem with final overdetermination for time-fractional differential equation in a Banach space

We consider in a Banach space E the inverse problem(\mathbf{D}_{t}^{\alpha}u)(t)=Au(t)+\mathcal{F}(t)f,\quad t\in[0,T],u(0)=u^{0}% ,u(T)=u^{T},\,0<\alpha<1with operator A, which generates the analytic and compact α-times resolvent family {\{S_{\alpha}(t,A)\}_{t\geq 0}}, the function {\mathcal{F}(\,\cdot\,)\in C^{1}[0,T]} and {u^{0},u^{T}\in D(A)} are given and {f\in E} is an unknown element. Under natural conditions we have proved the Fredholm solvability of this problem. In the special case for a self-adjoint operator A, the existence and uniqueness theorems for the solution of the inverse problem are proved. The semidiscrete approximation theorem for this inverse problem is obtained.

Error estimates for Galerkin finite element methods for the Camassa–Holm equation

© 2019, Springer-Verlag GmbH Germany, part of Springer Nature. We consider the Camassa–Holm (CH) equation, a nonlinear dispersive wave equation that models one-way propagation of long waves of moderately small amplitude. We discretize in space the periodic initial-value problem for CH (written in its original and in system form), using the standard Galerkin finite element method with smooth splines on a uniform mesh, and prove optimal-order L2-error estimates for the semidiscrete approximation. Using the fourth-order accurate, explicit, “classical” Runge–Kutta scheme for time-stepping, we construct a highly accurate, stable, fully discrete scheme that we employ in numerical experiments to approximate solutions of CH, mainly smooth travelling waves and nonsmooth solitons of the ‘peakon’ type.

Error estimates for Galerkin finite element methods for the Camassa–Holm equation

© 2019, Springer-Verlag GmbH Germany, part of Springer Nature. We consider the Camassa–Holm (CH) equation, a nonlinear dispersive wave equation that models one-way propagation of long waves of moderately small amplitude. We discretize in space the periodic initial-value problem for CH (written in its original and in system form), using the standard Galerkin finite element method with smooth splines on a uniform mesh, and prove optimal-order L2-error estimates for the semidiscrete approximation. Using the fourth-order accurate, explicit, “classical” Runge–Kutta scheme for time-stepping, we construct a highly accurate, stable, fully discrete scheme that we employ in numerical experiments to approximate solutions of CH, mainly smooth travelling waves and nonsmooth solitons of the ‘peakon’ type.

OPTIMAL ESTIMATES FOR THE SEMIDISCRETE GALERKIN METHOD APPLIED TO PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS WITH NONSMOOTH DATA

We propose and analyse an alternate approach to a priori error estimates for the semidiscrete Galerkin approximation to a time-dependent parabolic integro-differential equation with nonsmooth initial data. The method is based on energy arguments combined with repeated use of time integration, but without using parabolic-type duality techniques. An optimal$L^2$-error estimate is derived for the semidiscrete approximation when the initial data is in$L^2$. A superconvergence result is obtained and then used to prove a maximum norm estimate for parabolic integro-differential equations defined on a two-dimensional bounded domain.

Quantum breathers in ferromagnetic chains with on-site easy axis anisotropy

Quantum breathers in one-dimensional ferromagnetic chains with on-site easy axis anisotropy are investigated analytically. Based on the Hartree approximation, we can work out the case of quantum breathers with a large number of quanta. By using the multiple-scale method combined with the semidiscrete approximation, the nonlinear Schrödinger equation is derived. It is shown that quantum breathers can exist in the Heisenberg ferromagnetic chain. In addition, we obtain the energy level formula of quantum breathers, which indicates that the energy of such quantum breathers is quantized.