Non-Polynomial Spline Method for the Solution of the System of two Nonlinear Volterra Integral Equations

Abstract

In this paper, using cubic non-polynomial spline method to solve the system of two nonlinear volterra integral equations of the second kind, we have used a matlab14 program to solve the system. Finally, several illustrative examples to show the effectiveness and accuracy of this method.

Numerical Study of the System of Nonlinear Volterra Integral Equations by Using Spline Method

“…Table (1) provides a comparison between the Absolute Error of linear non-polynomial spline(LNPS) [10] and absolute error of (QNPS) for Example (1)…”

“…Absolute Error of linear nonpolynomial [10] Absolute Error of quadratic nonpolynomial Table (2) show that comparison between absolute error of (LNPS) [10] and absolute error of (QNPS) for Example Consider the following System of two nonlinear (VIEs) 2-kind:…”

“…x Absolute Error of linear non-polynomial [10] Absolute Error of quadratic non-polynomial With the exact solution (𝜂(𝑥), 𝜇(𝑥)) = (𝑥, 𝑥 2 ). .…”

“…Costarelli and Spigler [9] in 2013 studied solving Volterra integral equations of the second kind (VIEs) 2-kind by sigmoidal functions approximation. Jumaa and Taqi [10] in 2016 use first order (NPS) method for the solution of the system of two nonlinear (VIEs). Kumar et.al [12] in 2014 used new homotopy analysis transform algorithm to solve (VIE).…”

Algorithms for Solving Volterra Integral Equations Using Non-Polynomial Spline Function

“…A. Non-Polynomial Linear Spline Function [3,7] The following is the form of the linear non-polynomial spline function: 𝐿 𝑖 (𝑡) = 𝛼 𝑖 cos ℎ (𝑡 − 𝑡 𝑖 ) + 𝛽 𝑖 sin ℎ (𝑡 − 𝑡 𝑖 ) + 𝜎 𝑖 (𝑡 − 𝑡 𝑖 ) + 𝛿 𝑖 (1) s.t 𝑖 = 0, 1, … , 𝑚 the variables 𝛼 𝑖 , 𝛽 𝑖 , 𝜎 𝑖 , 𝑎𝑛𝑑 𝛿 𝑖 Need to be determined. We divide the equation above three times for 𝑡 to find the values of 𝛼, 𝛽, 𝜎, 𝑎𝑛𝑑 𝛿.…”