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Efficient algorithms to solve singular integral equations of Abel type. (English) Zbl 1165.45303

Summary: We obtain the approximate solution of Abel’s integral equation by using the following powerful, efficient but simple methods:

(i) He’s homotopy perturbation method (HPM),

(ii) Modified homotopy perturbation method (MHPM),

(iii) Adomian decomposition method (ADM) and

(iv) Modified Adomian decomposition method (MADM). The validity and applicability of these techniques are illustrated through various particular cases which demonstrate their efficiency and simplicity in solving these types of integral equations compared with the other existing methods.

45E05Integral equations with kernels of Cauchy type
45D05Volterra integral equations
[1]He, J. H.: Homotopy perturbation technique, Comput. methods appl. Mech. engrg. 178, 257-262 (1999)
[2]He, J. H.: A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int. J. Nonlinear mech. 35, 37-43 (2000) · Zbl 1068.74618 · doi:10.1016/S0020-7462(98)00085-7
[3]He, J. H.: Application of homotopy perturbation method to non linear wave equations, Chaos solitons fractls 26, 695-700 (2005) · Zbl 1072.35502 · doi:10.1016/j.chaos.2005.03.006
[4]He, J. H.: Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. Nonlinear sci. Numer. simul. 6, 207-208 (2005)
[5]Ganji, D. D.; Rafei, M.: Solitary wave solutions for a generalized Hirota–satsuma coupled KdV equation by homotopy perturbation method, Phys. lett. A 356, No. 2, 131-137 (2006) · Zbl 1160.35517 · doi:10.1016/j.physleta.2006.03.039
[6]Ganji, D. D.: The application of he’s homotopy perturbation method to nonlinear equations arising in heat transfer, Phys. lett. A 355, 337-341 (2006)
[7]Adomian, G.: Nonlinear stochastic operator equations, (1986)
[8]Adomian, G.; Rach, R.: Equality of partial solutions in the decomposition method for linear or nonlinear partial differential equations, Comput. math. Appl. 19, No. 12, 9-12 (1990) · Zbl 0702.35058 · doi:10.1016/0898-1221(90)90246-G
[9]Adomian, G.: A review of the decomposition method and some recent results for nonlinear equation, Math. comput. Modelling 13, No. 7, 17-43 (1992) · Zbl 0713.65051 · doi:10.1016/0895-7177(90)90125-7
[10]Adomian, G.; Rach, R.: Noise terms in decomposition series solution, Comput. math. Appl. 24, No. 11, 61-64 (1992) · Zbl 0777.35018 · doi:10.1016/0898-1221(92)90031-C
[11]Adomian, G.; Rach, R.: Analytic solution of nonlinear boundary-value problems in several dimensions, J. math. Anal. appl. 174, 118-127 (1993) · Zbl 0796.35017 · doi:10.1006/jmaa.1993.1105
[12]Adomian, G.: Solving frontier problems of physics: the decomposition method, (1994)
[13]Wazwaz, A. M.; Khuri, S. A.: Two methods for solving integral equations, Appl. math. Comput. 77, 79-89 (1996) · Zbl 0846.65077 · doi:10.1016/0096-3003(95)00189-1
[14]Wazwaz, A. M.: A first course in integral equations, (1997) · Zbl 0924.45001
[15]Wazwaz, A. M.: A reliable modification of Adomian decomposition method, Appl. math. Comput. 102, 77-86 (1999) · Zbl 0928.65083 · doi:10.1016/S0096-3003(98)10024-3
[16]Wazwaz, A. M.: A comparison study between the modified decomposition method and the traditional methods for solving nonlinear integral equation, Appl. math. Comput. 181, No. 2, 1703-1712 (2006) · Zbl 1105.65128 · doi:10.1016/j.amc.2006.03.023
[17]Siddiqui, A. M.; Mahmood, R.; Ghori, Q. K.: Homotopy perturbation method for thin film flow of a fourth grade fluid down a vertical cylinder, Phys. lett. A 352, 404-410 (2006) · Zbl 1187.76622 · doi:10.1016/j.physleta.2005.12.033
[18]He, J. H.: New interpretation of homotopy perturbation method, Internat. J. Modern phys. B 20, No. 18, 2561-2568 (2006)
[19]He, J. H.: Some asymptotic methods for strongly nonlinear equations, Internat. J. Modern phys. B 20, No. 10, 1141-1199 (2006) · Zbl 1102.34039 · doi:10.1142/S0217979206033796
[20]Zeilon, N.: Sur quelques points de la theorie de l’equation integrale d’abel, Arkiv. mat. Astr. fysik. 18, 1-19 (1924) · Zbl 50.0651.04
[21]Ganji, D. D.; Rajabi, A.: Assessment of homotopy-perturbation and perturbation methods in heat radiation equations, Int. commun. Heat mass transfer 33, 391-400 (2006)
[22]He, J. H.: Variational iteration method: a kind of nonlinear analytical technique: some example, Int. J. Nonlinear mech. 34, No. 4, 699-708 (1999)
[23]Liao, S. J.: An approximate solution technique not depending on small parameters: A special example, Int. J. Nonlinear mech. 30, No. 3, 371-380 (1995) · Zbl 0837.76073 · doi:10.1016/0020-7462(94)00054-E
[24]Liao, S. J.: Boundary element method for general nonlinear differential operators, Eng. anal. Boundary element 20, 91-99 (1997)
[25]Hillermeier, C.: Generalized homotopy approach to multiobjective optimization, Int. J. Optim. theory appl. 110, No. 3, 557-583 (2001) · Zbl 1064.90041 · doi:10.1023/A:1017536311488
[26]He, J. H.: An approximate solution technique depending upon an artificial parameter, Commun. nonlinear sci. Simul. 3, No. 2, 92-97 (1998) · Zbl 0921.35009 · doi:10.1016/S1007-5704(98)90070-3
[27]He, J. H.: The homotopy perturbation method for nonlinear oscillators with discontinuities, Appl. math. Comput. 151, 287-292 (2004) · Zbl 1039.65052 · doi:10.1016/S0096-3003(03)00341-2
[28]He, J. H.: Comparison of homotopy perturbation method and homotopy analysis method, Appl. math. Comput. 156, 527-539 (2004) · Zbl 1062.65074 · doi:10.1016/j.amc.2003.08.008
[29]Ganji, D. D.; Afrouzi, G. A.; Hosseinzadeh, H.; Talarposhti, R. A.: Applications of homotopy-perturbation method to the second kind of nonlinear integral equations, Phys. lett. A 371, 20-25 (2007) · Zbl 1209.65145 · doi:10.1016/j.physleta.2007.06.003
[30]He, J. H.: A review on some new recently developed nonlinear analytical techniques, Int. J. Nonlinear sci. Numer. simul. 1, No. 1, 51-70 (2000) · Zbl 0966.65056 · doi:10.1515/IJNSNS.2000.1.1.51
[31]Adomian, G.; Rach, R.: On the solution of algebraic equations by the decomposition method, Math. anal. Appl. 105, 141-166 (1985) · Zbl 0552.60060 · doi:10.1016/0022-247X(85)90102-7
[32]Yousefi, S. A.: Numerical solution of Abel’s integral equation by using Legendre wavelets, Appl. math. Comput. 175, 574-580 (2006) · Zbl 1088.65124 · doi:10.1016/j.amc.2005.07.032
[33]Baboolian, E.; Shamloo, A. S.: Numerical solution of Volterra integral and integro-differential equations of convolution type by using operational matrices of piecewise constant orthogonal functions, J. comput. Appl. math. (2007)