The Approximate Solutions of Fredholm Integrodifferential-Difference Equations with Variable Coefficients via Homotopy Analysis Method

Abstract

Numerical solutions of linear and nonlinear integrodifferential-difference equations are presented using homotopy analysis method. The aim of the paper is to present an efficient numerical procedure for solving linear and nonlinear integrodifferential-difference equations. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system.

Legendre spectral collocation method for Fredholm integro-differential-difference equation with variable coefficients and mixed conditions

“…Let us consider the integro-differential difference equation with variable coefficients as [7] y (x) + xy (x) + xy(x) + y (x − 1) + y(x − 1) = e −x + e + The exact solution of this problem is y(x) = e −x . The results obtained by presented method have been compared with the results obtained by homotopy analysis method (HAM) [7] and these numerical results along with the exact results are shown in Table 7.…”

“…The results obtained by presented method have been compared with the results obtained by homotopy analysis method (HAM) [7] and these numerical results along with the exact results are shown in Table 7. Table 8 cites the absolute errors.…”

“…Problems involving these equations arise frequently in many applied areas which include engineering, mechanics, physics, chemistry, astronomy, biology, economics, potential theory, electrostatics, etc. [1][2][3][4][5][6][7].…”

“…In [4], the higher-order linear Fredholm integro-differential-difference equations with variable coefficients have been solved by Legendre polynomials. Boubaker polynomial [5], Fibonacci collocation method (FCM) [6] and homotopy analysis method (HAM) [7] have been applied to solve Fredholm integro-differential-difference equations with variable coefficients. The system of nonlinear Volterra integro-differential equations has been solved by using Legendre wavelet method [8] and also two dimensional Legendre wavelets have been applied to solve fuzzy integro-differential equations [9].…”

Asymmetric Solutions of SDOF System with Wire Rope Vibration Isolator Subjected to Harmonic Excitation

“…[10][11][12] HAM still works accurately when the parameters in nonlinear ODEs and PDEs are large and has been successfully applied to a wide range of nonlinear problems. [13][14][15][16][17][18][19][20][21][22][23] However, as will be shown in the present paper, the standard HAM cannot capture asymmetric periodic solutions of nonlinear ODEs or PDEs. In the present work, a technique is put forward in the frame of HAM to capture asymmetric periodic solutions of dynamic systems with wire rope isolators which exhibit asymmetric restoring force (described in detail in the next section).…”