A convenient technique for solving linear and nonlinear Abel integral equations by the Adomian decomposition method. (English) Zbl 1252.65207

Summary: Linear and nonlinear Abel integral equations are transformed in such a manner that the Adomian decomposition method can be applied. Some examples with closed-form solutions are studied in detail to further illustrate the proposed technique, and the results obtained indicate that this approach is indeed practical and efficient.


65R20 Numerical methods for integral equations
45A05 Linear integral equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
45G05 Singular nonlinear integral equations
Full Text: DOI


[1] Wazwaz, A., A First Course in Integral Equations (1997), World Scientific Publishing: World Scientific Publishing Singapore · Zbl 0924.45001
[2] Wazwaz, A., A reliable modification of Adomian decomposition method, Appl. Math. Comput., 102, 77-86 (1999) · Zbl 0928.65083
[3] Polyanin, A. D.; Manzhirov, A. V., Handbook of Integral Equations (1998), CRC Press: CRC Press Boca Raton · Zbl 1021.45001
[4] Cherruault, Y.; Seng, V., The resolution of non-linear integral equations of the first kind using the decomposition method, Kybernetes, 26, 2, 198-206 (1997) · Zbl 0932.65142
[5] Adomian, G., Nonlinear Stochastic Operator Equations (1986), Academic Press: Academic Press Orlando, FL · Zbl 0614.35013
[6] Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0802.65122
[7] Adomian, G., Nonlinear Stochastic Systems Theory and Applications to Physics (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht
[8] Liu, Ya-ping; Tao, Lu, Mechanical quadrature methods and their extrapolation for solving first kind Abel integral equations, J. Comput. Appl. Math., 201, 300 313 (2007) · Zbl 1113.65123
[9] Liu, Ya-ping; Tao, Lu, High accuracy combination algorithm and a posteriori error estimation for solving the first kind Abel integral equations, Appl. Math. Comput., 178, 441451 (2006) · Zbl 1104.65127
[10] Gladwin, C. J.; Garey, L. E., Multi-step methods for first kind singular Volterra integral equations, J. Integ. Equat. Appl., 3, 4 (1991), Fall · Zbl 0757.65141
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.