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charged particle motion; oscillating magnetic field; integro-differential equation; Shannon wavelet; numerical treatment
In this work, we propose the Shannon wavelets approximation for the numerical solution of a class of integro-differential equations which describe the charged particle motion for certain configurations of oscillating magnetic fields. We show that using the Galerkin method and the connection coefficients of the Shannon wavelets, the problem is transformed to an infinite algebraic system, which can be solved by fixing a finite scale of approximation. The error analysis of the method is also investigated. Finally, some numerical experiments are reported to illustrate the accuracy and applicability of the method.
[1] Agarwal, R. P.: Boundary Value Problems for Higher Order Differential Equations. World Scientific Singapore (1986). MR 1021979 | Zbl 0619.34019
[2] Akyüz-Daşcioğlu, A.: Chebyshev polynomial solutions of systems of linear integral equations. Appl. Math. Comput. 151 (2004), 221-232. DOI 10.1016/S0096-3003(03)00334-5 | MR 2037962 | Zbl 1049.65149
[3] Akyüz-Daşcioğlu, A., Sezer, M.: Chebyshev polynomial solutions of systems of higher-order linear Fredholm-Volterra integro-differential equations. J. Franklin Inst. 342 (2005), 688-701. DOI 10.1016/j.jfranklin.2005.04.001 | MR 2166749 | Zbl 1086.65121
[4] Bojeldain, A. A.: On the numerical solving of nonlinear Volterra integro-differential equations. Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput. 11 (1991), 105-125. MR 1158315 | Zbl 0755.45011
[5] Brunner, H., Makroglou, A., Miller, R. K.: Mixed interpolation collocation methods for first and second order Volterra integro-differential equations with periodic solution. Appl. Numer. Math. 23 (1997), 381-402. DOI 10.1016/S0168-9274(96)00075-X | MR 1453423 | Zbl 0876.65090
[6] Cattani, C.: Connection coefficients of Shannon wavelets. Math. Model. Anal. 11 (2006), 117-132. MR 2231204 | Zbl 1117.65179
[7] Cattani, C.: Shannon wavelets for the solution of integrodifferential equations. Math. Probl. Eng. (2010), Article ID 408418. MR 2610514 | Zbl 1191.65174
[8] Dehghan, M., Shakeri, F.: Solution of an integro-differential equation arising in oscillating magnetic fields using He's homotopy perturbation method. Progress In Electromagnetics Research 78 (2008), 361-376. DOI 10.2528/PIER07090403
[9] Domke, K., Hacia, L.: Integral equations in some thermal problems. Int. J. Math. Comput. Simulation 2 (2007), 184-189.
[10] Machado, J. M., Tsuchida, M.: Solutions for a class of integro-differential equations with time periodic coefficients. Appl. Math. E-Notes 2 (2002), 66-71. MR 1979412 | Zbl 0999.45003
[11] Maleknejad, K., Attary, M.: An efficient numerical approximation for the linear class of Fredholm integro-differential equations based on Cattani's method. Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 2672-2679. DOI 10.1016/j.cnsns.2010.09.037 | MR 2772283 | Zbl 1221.65332
[12] Mureşan, V.: Existence, uniqueness and data dependence for the solutions of some integro-differential equations of mixed type in Banach space. Z. Anal. Anwend. 23 (2004), 205-216. DOI 10.4171/ZAA/1194 | MR 2066102 | Zbl 1062.45006
[13] Wu, G.-C., Lee, E. W. M.: Fractional variational iteration method and its application. Phys. Lett., A 374 (2010), 2506-2509. DOI 10.1016/j.physleta.2010.04.034 | MR 2640023 | Zbl 1237.34007
[14] Yildirim, A.: Solution of BVPs for fourth-order integro-differential equations by using homotopy perturbation method. Comput. Math. Appl. 56 (2008), 3175-3180. DOI 10.1016/j.camwa.2008.07.020 | MR 2474572 | Zbl 1165.65377
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