A novel parameter-uniform finite difference scheme on a Shishkin-type mesh for a singularly perturbed Volterra integro-differential equation is studied. The problem is discretized by the variable two-step backward differentiation formula (BDF2) for the first-order derivative term and the trapezoidal formula for the integral term. The stability of the proposed numerical method is carried out. It is shown from the convergence analysis that our presented method is almost second-order uniformly convergent with respect to the perturbation parameter ε in the discrete maximum norm. Numerical results are given to support our theoretical result.
In this paper, we study Hessian equations with the prescribed contact angle boundary value or oblique derivative boundary value and finally derive the a priori global gradient estimate for the admissible solutions.
The global regularity problem concerning the inviscid Boussinesq equations remains an open problem. In an attempt to understand this problem, we examine the damped Boussinesq equations and study how damping affects the regularity of solutions. In this paper, we consider the global existence to the damped Boussinesq equations with a class of large initial data, whose $B^{s}_{p,r}$ or $\dot{B}^{s}_{p,r}$ norms can be arbitrarily large. The idea is splitting the linear Boussinesq equations from the damped Boussinesq equations, the exponentially decaying solution of the former equations together with the structure of the Boussinesq equations help us to obtain the global smooth solutions.
In the present paper, we prove the existence, non-existence and multiplicity of positive normalized solutions (λc, uc) ∈ ℝ × H1 (ℝN) to the general Kirchhoff problem -M(∫_ℝ^N|∇ u|^2 dx)Δ u +λ u=g(u) in ℝ^N, u∈ H^1(ℝ^N),N≥ 1, satisfying the normalization constraint ∫_ℝ^Nu^2 dx=c , where M ∈ C([0, ∞)) is a given function satisfying some suitable assumptions. Our argument is not by the classical variational method, but by a global branch approach developed by Jeanjean et al. [J Math Pures Appl, 2024, 183: 44–75] and a direct correspondence, so we can handle in a unified way the nonlinearities g(s), which are either mass subcritical, mass critical or mass supercritical.
This research studies the inverse boundary value problem for fractional elliptic equation of Tricomi–Gellerstedt–Keldysh type and obtains a condition stability result. To recover the continuous dependence of the solution on the measurement data, a generalized Tikhonov regularization method based on ill-posedness analysis is constructed. Under the a priori and a posterior selection rules for the regularization parameter, corresponding Hölder type convergence results are obtained. On this basis, this thesis verifies the simulation effect of the generalized Tikhonov method through numerical examples. The examples show that the method performs well in dealing with the problem under consideration.
