Numerical solution of nonlinear Volterra-Fredholm integro-differential equations via direct method using triangular functions. (English) Zbl 1189.65306

Summary: An effective direct method to determine the numerical solution of the specific nonlinear Volterra-Fredholm integro-differential equations is proposed. The method is based on new vector forms for representation of triangular functions and its operational matrix. This approach needs no integration, so all calculations can be easily implemented. Some numerical examples are provided to illustrate the accuracy and computational efficiency of the method.


65R20 Numerical methods for integral equations
45J05 Integro-ordinary differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
45B05 Fredholm integral equations
45D05 Volterra integral equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
Full Text: DOI


[1] Maleknejad, K.; Hadizadeh, M., The numerical analysis of Adomian’s decomposition method for nonlinear Volterra integral and integro-differential equations, International Journal of Engineering Science, Iran University of Science & Technology, 8, 2a, 33-48 (1997)
[2] Wazwaz, A. M., A First Course in Integral Equations (1997), World Scientific: World Scientific Singapore · Zbl 0924.45001
[3] Brunner, H., Collocation Method for Volterra Integral and Relation Functional Equations (2004), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1059.65122
[4] Çerdik-Yaslan, H.; Akyüz-Daşcioğlu, A., Chebyshev polynomial solution of nonlinear Fredholm-Volterra integro-differential equations, Journal of Arts and Sciences, Cankaya University, 5, 89-101 (2006) · Zbl 1089.65134
[5] Delves, L. M.; Mohamed, J. L., Computational Methods for Integral Equations (1985), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0592.65093
[6] Maleknejad, K.; Mahmoudi, Y., Taylor polynomial solution of high-order nonlinear Volterra-Fredholm integro-differential equations, Applied Mathematics and Computation, 145, 641-653 (2003) · Zbl 1032.65144
[7] Babolian, E.; Masouri, Z.; Hatamzadeh-Varmazyar, S., New direct method to solve nonlinear Volterra-Fredholm integral and integro differential equation using operational matrix with Block-pulse functions, Progress In Electromagnetics Research B, 8, 59-76 (2008)
[8] Sepehrian, B.; Razzaghi, M., Single-term Walsh series method for the Volterra integro-differential equations, Engineering Analysis with Boundary Elements, 28, 1315-1319 (2004) · Zbl 1081.65551
[9] Deb, A.; Dasgupta, A.; Sarkar, G., A new set of orthogonal functions and its application to the analysis of dynamic systems, Journal of the Franklin Institute, 343, 1-26 (2006) · Zbl 1173.33306
[10] Babolian, E.; Mokhtari, R.; Salmani, M., Using direct method for solving variational problems via triangular orthogonal functions, Applied Mathematics and Computation, 191, 206-217 (2007) · Zbl 1193.65196
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.