## Solutions of integral and integro-differential equation systems by using differential transform method.(English)Zbl 1165.45300

Summary: The differential transform method (DTM) is applied to both integro-differential and integral equation systems. The method is further expanded with a formulation to treat Fredholm integrals. If the system considered has a solution in terms of the series expansion of known functions, this powerful method catches the exact solution. So as to show this capability and robustness, some systems of integral and integro-differential equations are solved as numerical examples.

### MSC:

 45B05 Fredholm integral equations 65N99 Numerical methods for partial differential equations, boundary value problems
Full Text:

### References:

 [1] Biazar, J.; Babolian, E.; Islam, R., Solution of a system of Volterra integral equations of the first kind by Adomian method, Appl. Math. Comput., 139, 249-258 (2003) · Zbl 1027.65180 [2] Sadeghi Goghary, H.; Javadi, Sh.; Babolian, E., Restarted Adomian method for system of nonlinear Volterra integral equations, Appl. Math. Comput., 161, 745-751 (2005) · Zbl 1061.65148 [3] Maleknejad, K.; Tavassoli Kajani, M., Solving linear integro-differential equation system by Galerkin methods with hybrid functions, Appl. Math. Comput., 159, 603-612 (2004) · Zbl 1063.65145 [4] Maleknejad, K.; Mirzaee, F.; Abbasbandy, S., Solving linear integro-differential equations system by using rationalized Haar functions method, Appl. Math. Comput., 155, 317-328 (2004) · Zbl 1056.65144 [6] Yusufoglu (Agadjanov), E., An efficient algorithm for solving integro-differential equations system, Appl. Math. Comput., 192, 51-55 (2007) · Zbl 1193.65234 [7] Wang, S. Q.; He, J. H., Variational iteration method for solving integro-differential equations, Phys. Lett. A., 367, 188-191 (2007) · Zbl 1209.65152 [8] He, J. H., Homotopy perturbation technique, Comput. Method Appl. Math., 178, 257-262 (1999) · Zbl 0956.70017 [9] He, J. H., A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Int J. Non-Linear Mech., 35, 37-43 (2000) · Zbl 1068.74618 [10] He, J. H., Homotopy perturbation method: A new nonlinear analytical technique, Appl. Math. Comput., 135, 73-79 (2003) · Zbl 1030.34013 [11] He, J. H., Comparison of homotopy perturbation method and homotopy analysis method, Appl. Math. Comput., 156, 527-539 (2004) · Zbl 1062.65074 [12] Bildik, N.; Konuralp, A., Two-dimensional differential transform method, Adomian’s decomposition method, and variational iteration method for partial differential equations, Int. J. Comput. Math., 83, 973-987 (2006) · Zbl 1115.65365 [13] Bildik, N.; Konuralp, A., The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations, Int. J. Nonlinear Sci., 7, 65-70 (2006) · Zbl 1401.35010 [14] He, J. H., New interpretation of homotopy perturbation method, Internat. J. Modern Phys. B., 20, 2561-2568 (2006) [15] He, J. H., Some asymptotic methods for strongly nonlinear equations, Internat. J. Modern Phys. B., 20, 1141-1199 (2006) · Zbl 1102.34039 [16] Zhou, J. K., Differential Transformation and its Application for Electrical Circuits (1986), Huazhong University Press: Huazhong University Press Wuhan, China [17] Ozdemir, O.; Kaya, M. O., Flapwise bending vibration analysis of a rotating tapered cantilever Bernoulli-Euler beam by differential transform method, J. Sound Vib., 289, 413-420 (2006) [18] Arikoglu, A.; Ozkol, I., Solution of difference equations by using differential transform method, Appl. Math. Comput., 174, 1216-1228 (2006) · Zbl 1138.65309 [19] Arikoglu, A.; Ozkol, I., Solution of differential-difference equations by using differential transform method, Appl. Math. Comput., 181, 153-162 (2006) · Zbl 1148.65310 [20] Arikoglu, A.; Ozkol, I., Solution of fractional differential equations by using differential transform method, Chaos Soliton. Fract., 34, 1473-1481 (2007) · Zbl 1152.34306 [21] Keskin, Y.; Kurnaz, A.; Kiris, M. E.; Oturanc, G., Approximate solutions of generalized pantograph equations by the differential transform method, Int. J. Nonlinear Sci., 8, 159-164 (2007) [22] Arikoglu, A.; Ozkol, I., Solution of boundary value problems for integro-differential equations by using differential transform method, Appl. Math. Comput., 168, 1145-1158 (2005) · Zbl 1090.65145 [23] Delves, L. M.; Mohamed, J. L., Computational Methods for Integral Equations (1985), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0592.65093
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.