A common way to handle the Tikhonov regularization method for the first kind Fredholm integral equations, is first to discretize and then to work with the final linear system. This unavoidably inflicts discretization errors which may lead to disastrous results, especially when a quadrature rule is used. We propose to regularize directly the integral equation resulting in a continuous Tikhonov problem. The Tikhonov problem is reduced to a simple least squares problem by applying the Golub-Kahan bidiagonalization (GKB) directly to the integral operator. The regularization parameter and the iteration index are determined by the discrepancy principle approach. Moreover, we study the discrete version of the proposed method resulted from numerical evaluating the needed integrals. Focusing on the nodal values of the solution results in a weighted version of GKB-Tikhonov method for linear systems arisen from the Nyström discretization. Finally, we use numerical experiments on a few test problems to illustrate the performance of our algorithms.

In this paper, we study the finite element approximation for nonlinear thermal equation. Because the nonlinearity of the equation, our theoretical analysis is based on the error of temporal and spatial discretization. We consider a fully discrete second order backward difference formula based on a finite element method to approximate the temperature and electric potential, and establish optimal $L^2$ error estimates for the fully discrete finite element solution without any restriction on the time-step size. The discrete solution is bounded in infinite norm. Finally, several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

Direct and inverse problems for the scattering of cracks with mixed oblique derivative boundary conditions from the incident plane wave are considered, which describe the scattering phenomenons such as the scattering of tidal waves by spits or reefs. The solvability of the direct scattering problem is proven by using the boundary integral equation method. In order to show the equivalent boundary integral system is Fredholm of index zero, some relationships concerning the tangential potential operator is used. Due to the mixed oblique derivative boundary conditions, we cannot employ the factorization method in a usual manner to reconstruct the cracks. An alternative technique is used in the theoretical analysis such that the far field operator can be factorized in an appropriate form and fulfills the range identity theorem. Finally, we present some numerical examples to demonstrate the feasibility and effectiveness of the factorization method.

This paper revisits the classical problem “Can we hear the density of a string?”, which can be formulated as an inverse spectral problem for a Sturm-Liouville operator. Instead of inverting the map from density to spectral data directly, we propose a novel method to reconstruct the density based on inverting a sequence of trace formulas which bridge the density and its spectral data clearly in terms of a series of nonlinear integral equations. Numerical experiments are presented to verify the validity and effectiveness of the proposed numerical algorithm. The impact of different parameters involved in the algorithm is also discussed.

We study from a numerical point of view a multidimensional problem involving a viscoelastic body with two porous structures. The mechanical problem leads to a linear system of three coupled hyperbolic partial differential equations. Its corresponding variational formulation gives rise to three coupled parabolic linear equations. An existence and uniqueness result, and an energy decay property, are recalled. Then, fully discrete approximations are introduced using the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved, from which the linear convergence of the algorithm is derived under suitable additional regularity conditions. Finally, some numerical simulations are performed in one and two dimensions to show the accuracy of the approximation and the behaviour of the solution.

We consider a nonlinear stochastic Volterra integral equation with time-dependent delay and the corresponding Euler-Maruyama method in this paper. Strong convergence rate (at fixed point) of the corresponding Euler-Maruyama method is obtained when coefficients $f$ and $g$ both satisfy local Lipschitz and linear growth conditions. An example is provided to interpret our conclusions. Our result generalizes and improves the conclusion in [J. Gao, H. Liang, S. Ma, Strong convergence of the semi-implicit Euler method for nonlinear stochastic Volterra integral equations with constant delay, Appl. Math. Comput., 348 (2019) 385-398.]

A combined scheme of the improved two-grid technique with the block-centered finite difference method is constructed and analyzed to solve the nonlinear time-fractional parabolic equation. This method is considered where the nonlinear problem is solved only on a coarse grid of size $H$ and two linear problems based on the coarse-grid solutions and one Newton iteration is considered on a fine grid of size $h$. We provide the rigorous error estimate, which demonstrates that our scheme converges with order $\mathcal{O}(\Delta t^{2-\alpha}+h^2+H^4)$ on non-uniform rectangular grid. This result indicates that the improved two-grid method can obtain asymptotically optimal approximation as long as the mesh sizes satisfy $h=\mathcal{O}(H^2).$ Finally, numerical tests confirm the theoretical results of the presented method.

The present paper conjectures a topological condition which classifies a T-spline into standard, semi-standard and non-standard. We also provide the basic framework to prove the conjecture on the classification of semi-standard T-splines and give the proof for the semi-standard of bi-degree (1, $d$) and ($d$, 1) T-splines.

This paper considers the asymptotic stability of linear multistep (LM) methods for neutral systems with distributed delays. In particular, several sufficient conditions for delay-dependent stability of numerical solutions are obtained based on the argument principle. Compound quadrature formulae are used to compute the integrals. An algorithm is proposed to examine the delay-dependent stability of numerical solutions. Several numerical examples are performed to verify the theoretical results.