A hybrid method for solution of linear Volterra integro-differential equations (LVIDES) via finite difference and Simpson’s numerical methods (FDSM)

Abstract

In this paper, a hybrid of Finite difference-Simpson’s approach was applied to solve linear Volterra integro-differential equations. The method works efficiently great by reducing the problem into a system of linear algebraic equations. The numerical results shows the simplicity and effectiveness of the method, error estimation of the method is provided which shows that the method is of second order convergence.

A Polynomial Approximation for the Numerical Solution of First Order Volterra Integro-Differential Equations

“…The approximate solution of equation ( 17 = -4.091778999000 × 10 −16 + 1.000000000165𝑥 -5.712718121000 × 10-7𝑥 2 -0.166660841322𝑥 3 -0.24197157𝑒-4𝑥 4 + 0.8384018671𝑒-2𝑥 5 -0.54626726𝑒-4𝑥 6 -0.172795088𝑒-3𝑥 7 . This also confirmed that our method performed better than the method proposed by [17].…”

“…Some methods for determining the numerical solution of integro-differential equations include: Bernstein Method [14], Adomian decompositions method [2,3], Finite difference-Simpson method [17], Collocation method by [4,5,6,7,21,22], Hybrid linear multistep method [8,9], Chebyshev-Galerkin method [10], Bernoulli matrix method [11], Differential transform method [12], Lagrange Interpolation [13], Differential Transformation [15], Block pulse functions operational matrices [19] Chebyshev polynomials [16], Optimal Auxiliary Function Method (OAFM) [18] and Spectral Homotopy Analysis Method [20]. We consider first order Volterra integro-differential equation of the form 𝑦 ′ (𝑥) = 𝑔(𝑥) + ∫ 𝑘(𝑥, 𝑡)𝑦(𝑡)𝑑𝑡 𝑥 0 (1) with the initial condition 𝑦(0) = 𝑞…”