The Weighted Residual Method in Solution of More General Nonlinear Integro-Fractional Differential Equation of Volterra Hammerstein Type with Variable Coefficients

Shazad Shawki Ahmed1 &   Mariwan Rashid Ahmed2

1 Sulaimani University, College of Science, Department of Mathematics, Sulaimaniyah-Iraq 
2 Charmo University, College of Basic Education, Department of Science, Chamchamal-Iraq 

Original: 19 March 2019
Revised: 5 May 2019
Accepted: 21 May 2019
Published online:  20 June 2019


In this paper, we solve numerically Volterra-Hammerstein (V-H) Integro-Differential Equations (IDE’s) of various fractional order  in the Caputo sense by the well-known (Collocation, Sub-domain, Least-square, and Galerkin) weighted residual method. This method is utilized to converts the integro-fractional differential equation to a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown polynomial coefficients. A good algorithm for treating numerically our problem by applying the above process has been developed, in order to express these solutions, programs are written in MatLab (V.8). The validity and reliability of this method are tested by several illustrative numerical examples with the known exact solution, the behaviors of the error are examined, and the obtained results reveal that the method is more accurate and efficient.

Key Words:  Integro-Fractional Differential Equation, Caputo derivative, Collocation, Sub-domain , Least square, Galerkin


[1] Ahmed, C. S. "Computational Methods for Solving System of Non-  Linear Volterra Integral Equations of the Second Kind", M.Sc. Thesis, Salahaddin University, (2007).

[2] Ahmed, S. S. "On System of Linear Volterra Integro- Fractional Differential Equations", Ph.D. Thesis, Sulaimani University, (2009).

[3] Atkinson, K.E., "A Survey of Numerical Methods for solving Nonlinear Integral Equations", J. of Integral Equations. Vol. 4, No. 1, pp.15-46. (1992).

[4] Aziz, Shatha A. "Analytical Study of Fractional Order Differential Equations", PhD thesis, college of science, University of AL-Nahrain, Baghdad, (2006).

[5] Aziz, K. M. "An Approximation Solution for Solving the System of Non-Linear Fredholm Integral Equations of the Second Kind", M.Sc. Thesis, Salahaddin University, (2007).

[6] Boyadjiev, L. and Dobner, H.J. "On the solution of Fractional Free Electron Laser Equation", Fractional Calculus and Applied Analysis Journal, Vol. 1, No. 4 (1998).

[7] Brunner, H., "On the Numerical Solution of Nonlinear Volterra-Fredholm Integral Equations by Collocation Methods", SIAM J. Numer Anal, Vol. 27, No. 4, pp.987-1000. (1900).

[8] Burden, Richard L. and Faires, J. Douglas. "Numerical Analysis", Ninth Edition; Youngstown State University, (2010).

[9] Delves, L.M. and Walsh, J. "Numerical solution of integral equations", OXFORD university press, (1974).

[10] Gautschi, W. "Numerical Analysis", Second Edition, Springer New York, (2012).

[11] Grattan, Ellen R. Mc. "Application of Weighted Residual Methods to Dynamic Economic Models", Federal Reserve Bank of Minneapolis, Research Department Staff Report 232, Revised March (1998).

[12] Hilfer, R. "Applications of Fractional Calculus in Physics", World Scientific, (2000).

[13] Kilbas, Anatoly A. "Hari M. Srivastava and Juan J. Trujillo", Theory and Applications of Fractional Differential Equations, Elsevier B.V. Netherlands, (2006).

[14] Jain, M.K. "Numerical solution of Differential Equations", JOHN WILEY and SONS, New York, (1979).

[15] Mercier, B. "Lecture Notes in physics: An introduction to the Numerical Analysis of Spectral Methods", Springer-Verlag Berlin Heidebeg, (1989).

[16] Miller, Kenneth S. and Ross, B. "An Introduction to the Fractional Calculus and Fractional Differential Equations", John Wiley & Sons (New York), (1993).

[17] Odibat, Zaid M. and shawagfeh, Nabil T. "Generalized Taylor’s Formula", Applied Mathematics and computation, Vol.186, pp. 286-293 (2007).

[18] Podldubny, I. "Fractional Differential Equation", Academic press, San Diego, (1999).

[19] Saeed, R. K. "Computational methods for solving system of linear Volterra integral and integro-differential Equations", PhD. Thesis, college of science, university of salahaddin, Erbil, (2006).

[20] Salih, Harith M. "On Approximated Methods for Fractional Differential Equations", M.Sc. Thesis, college of Education Ibn-Al-Haitham, University of Baghdad, (2005).

[21] Weilbeer, M. "Efficient Numerical Methods for Fractional Differential Equations and their Analytical Background", US Army Medical Research and Material Command, (2005).