Analytical Construction of Uniformly Convergent Method for Convection Diffusion Problem

In this paper, we study the uniformly convergent method on equidistant meshes for the convection-diffusion problem of type; where the formal adjoint operator of L. Lu=-εu''+bu'+c u=f(x), u(0)=0, u(1)=0 At the end of the this paper we will generate the scheme; -e^(ρ_i )/(e^(ρ_i )+1) U_(i-1)+U_i-1/(e^(ρ_i )+1) U_(i+1)=(f_i-c_i U_i ) h/b ((e^(ρ_i )-1)/(e^(ρ_i )+1))

Uniformly Convergent Numerical Scheme for Singularly Perturbed Parabolic PDEs with Shift Parameters

In this article, singularly perturbed parabolic differential difference equations are considered. The solution of the equations exhibits a boundary layer on the right side of the spatial domain. The terms containing the advance and delay parameters are approximated using Taylor series approximation. The resulting singularly perturbed parabolic PDEs are solved using the Crank–Nicolson method in the temporal discretization and nonstandard finite difference method in the spatial discretization. The existence of a unique discrete solution is guaranteed using the discrete maximum principle. The uniform stability of the scheme is investigated using solution bound. The uniform convergence of the scheme is discussed and proved. The scheme converges uniformly with the order of convergence O N − 1 + Δ t 2 , where N is number of subintervals in spatial discretization and Δ t is mesh length in temporal discretization. Two test numerical examples are considered to validate the theoretical findings of the scheme.

Accurate numerical scheme for singularly perturbed parabolic delay differential equation

Objectives Numerical treatment of singularly perturbed parabolic delay differential equation is considered. Solution of the equation exhibits a boundary layer, which makes it difficult for numerical computation. Accurate numerical scheme is proposed using $$\theta$$ θ -method in time discretization and non-standard finite difference method in space discretization. Result Stability and uniform convergence of the proposed scheme is investigated. The scheme is uniformly convergent with linear order of convergence before Richardson extrapolation and second order convergent after Richardson extrapolation. Numerical examples are considered to validate the theoretical findings.

Analysis of Fractional-Order Nonlinear Dynamic Systems with General Analytic Kernels: Lyapunov Stability and Inequalities

In this paper, we study the recently proposed fractional-order operators with general analytic kernels. The kernel of these operators is a locally uniformly convergent power series that can be chosen adequately to obtain a family of fractional operators and, in particular, the main existing fractional derivatives. Based on the conditions for the Laplace transform of these operators, in this paper, some new results are obtained—for example, relationships between Riemann–Liouville and Caputo derivatives and inverse operators. Later, employing a representation for the product of two functions, we determine a form of calculating its fractional derivative; this result is essential due to its connection to the fractional derivative of Lyapunov functions. In addition, some other new results are developed, leading to Lyapunov-like theorems and a Lyapunov direct method that serves to prove asymptotic stability in the sense of the operators with general analytic kernels. The FOB-stability concept is introduced, which generalizes the classical Mittag–Leffler stability for a wide class of systems. Some inequalities are established for operators with general analytic kernels, which generalize others in the literature. Finally, some new stability results via convex Lyapunov functions are presented, whose importance lies in avoiding the calculation of fractional derivatives for the stability analysis of dynamical systems. Some illustrative examples are given.

An Efficient Computational Method for Singularly Perturbed Delay Parabolic Partial Differential Equations

In this communication, a parameter uniform numerical scheme is proposed to solve singularly perturbed delay parabolic convection-diffusion equations. Taylor’s series expansion is applied to approximate the shift term. Then the resulting singularly perturbed parabolic convection-diffusion equation is solved by utilizing the implicit Euler method for temporal discretization on uniform mesh and hybrid numerical scheme based on a midpoint upwind scheme in the coarse mesh regions and a cubic spline method in the fine mesh regions on a piecewise uniform Shishkin mesh for the spatial discretization. The proposed numerical scheme is shown to be an ε−uniformly convergent accuracy of first-order in time and almost second-order in space directions. Some test examples are considered to testify the theoretical predictions.