Exponentially fitted difference scheme for singularly perturbed mixed integro-differential equations

In this study, singularly perturbed mixed integro-differential equations (SPMIDEs) are taken into account. First, the asymptotic behavior of the solution is investigated. Then, by using interpolating quadrature rules and an exponential basis function, the finite difference scheme is constructed on a uniform mesh. The stability and convergence of the proposed scheme are analyzed in the discrete maximum norm. Some numerical examples are solved, and numerical outcomes are obtained.

On exponential fitting of finite difference methods for heat equations

In this article, a new kind of finite difference scheme that is exponentially fitted, inspired from Fourier analysis, for a fourth space derivative was developed for solving diffusion problems. Dispersion relation and local truncation error of the method were discussed. Stability analysis of the method revealed that it is conditionally stable. Compared to the corresponding fourth order classical scheme in the literature, the proposed scheme is efficient and accurate. Mathematics Subject Classification (2020): 65M06, 65N06. Keywords: Exponential fitting, Finite difference, Local truncation error, Heat equations.

Exponentially fitted multisymplectic scheme for conservative Maxwell equations with oscillary solutions

Aiming at conservative Maxwell equations with periodic oscillatory solutions, we adopt exponentially fitted trapezoidal scheme to approximate the temporal and spatial derivatives. The scheme is a multisymplectic scheme. Under periodic boundary condition, the scheme satisfies two discrete energy conservation laws. The scheme also preserves two discrete divergences. To reduce computation cost, we split the original Maxwell equations into three local one-dimension (LOD) Maxwell equations. Then exponentially fitted trapezoidal scheme, applied to the resulted LOD equations, generates LOD multisymplectic scheme. We prove the unconditional stability and convergence of the LOD multisymplectic scheme. Convergence of numerical dispersion relation is also analyzed. At last, we present two numerical examples with periodic oscillatory solutions to confirm the theoretical analysis. Numerical results indicate that the LOD multisymplectic scheme is efficient, stable and conservative in solving conservative Maxwell equations with oscillatory solutions. In addition, to one-dimension Maxwell equations, we apply least square method and LOD multisymplectic scheme to fit the electric permittivity by using exact solution disturbed with small random errors as measured data. Numerical results of parameter inversion fit well with measured data, which shows that least square method combined with LOD multisymplectic scheme is efficient to estimate the model parameter under small random disturbance.

Fitted Numerical Scheme for Solving Singularly Perturbed Parabolic Delay Partial Differential Equations

In this paper, exponentially fitted finite difference scheme is developed for solving singularly perturbed parabolic delay partial differential equations having small delay on the spatial variable. The term with the delay is approximated using Taylor series approximation. The resulting singularly perturbed parabolic partial differential equation is treated using im- plicit Euler method in the temporal discretization with exponentially fitted operator finite difference method in the spatial discretization. The parameter uniform convergence analysis has been carried out with the order of convergence one. Test examples and numerical results are considered to validate the theoretical analysis of the scheme.