New approach to solve Volterra q-integral equations by Differential transform method
Volterra q-integral equations
Abstract
In this paper, Volterra q-integral equations are solved by using the method of q-differential transformation [1]. Exact solutions of linear and nonlinear q-integral equations have been investigated. To illustrate the method, several problems are discussed for the effectiveness and
performance of the proposed method.
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References
[1] Jing, S.C., Fan, H.Y., q-Taylor’s formula with its q-remainder, Commun. Theoret. Phys. 23 (1), 117-120, (1995).
[2] Kac, V.G., Cheung, P., Quantum Calculus. Universitext, Springer-Verlag, New York, (2002).
[3] Jackson, F.H., q-Difference equation, American Journal of Mathematics, 32, 305-314, (1910).
[4] Jackson, F.H., On q-definite integrals, Quart. J. Math. 41, 193-203, (1910).
[5] Euler, L., Introductio in Analysin Infinitorum, Opera omnia series, 1, (1748).
[6] Thomas, E., A Comprehensive Treatment of q-Calculus, Springer Basel Heidelberg New York Dordrecht London (2012).
[7] Bangerezako, G., “An Introduction to q-Difference Equations” (Preprint Bujumbura, 2008).
[8] Zhou, J.K., Differential Transformation and Its Applications for Electrical Circuits, Huazhong University Press, Wuhan, China, (1986).
[9] Odibat, Z.M., Differential transform method for solving Volterra integral equation with separable kernels, Mathematical and Computer Modelling 48, 1144-1149, (2008).
[10] Biazar, J., Eslami, M.,Islam, M.R., Differential transform method for special systems of integral equations. Journal of King Saud University-Science 24, 211-214, (2012).
[11] Jafari, H., Haghbin, A., Hesam, S., and Baleanu, D., Solving Partial q-Differential Equations within Reduced q-Differential Transformation Method. Mathematical and Theoretical Physics Vol. 59, 399-407, (2014).
[12] Jackson, F.H., A q-form of Taylor’s theorem, Messenger of Mathematics 38, 62-64, (1909).
[13] Exton, H., q-Hypergeometric Functions and Applications, John Wiley & Sons, New York, (1983).
[14] Gasper, G., Rahman, M., Basic Hypergeometric Series. 2nd Ed., Cambridge Univ. Press, Cambridge, (2004).
[15] Carmichael, R.D., The general theory of linear q-Difference Equation, Am. J. Math. 34, 147- 168, (1912).
[16] Rajkovic, P.M., On q-iterative methods for solving equations and systems, Novi Sad J. Math. 33(2), 127-137, (2003).
[17] Ayaz, F., Applications of differential transform method to differential-algebraic equations. Appl. Math. Comput. 152, 649-657, (2004).
[18] Hassan, I.H., Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems. Choas Solitons Fractals 36 (1), 53-65, (2008).
[19] Ayaz, F.,, Solutions of the system of differential equations by differential transform method, Appl. Math. Comput. 147, 547-567, (2004).
[20] 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).
[21] Bildik, N., Konuralp, A., Bek, F. and Kucukarslan, S., Solution of different type of the partial differential equation by differential transform method and Adomian’s decomposition method, Appl. Math. Comput. 127, 551-567, (2006).
[22] Arikoglu, A., Ozkol, I., Solution of difference equations by using differential transform method, Appl. Math. Comput. 173 (1), 126-136, (2006).
[23] Arikoglu, A., Ozkol, I., Solution of differential–difference equations by using differential transform method, Appl. Math. Comput. 181 (1), 153-162, (2006).
[24] Liu, H., Song, Y., Differential transform method applied to high index differential-algebraic equations, Appl. Math. Comput. 184 (2), 748-753, (2007).
[2] Kac, V.G., Cheung, P., Quantum Calculus. Universitext, Springer-Verlag, New York, (2002).
[3] Jackson, F.H., q-Difference equation, American Journal of Mathematics, 32, 305-314, (1910).
[4] Jackson, F.H., On q-definite integrals, Quart. J. Math. 41, 193-203, (1910).
[5] Euler, L., Introductio in Analysin Infinitorum, Opera omnia series, 1, (1748).
[6] Thomas, E., A Comprehensive Treatment of q-Calculus, Springer Basel Heidelberg New York Dordrecht London (2012).
[7] Bangerezako, G., “An Introduction to q-Difference Equations” (Preprint Bujumbura, 2008).
[8] Zhou, J.K., Differential Transformation and Its Applications for Electrical Circuits, Huazhong University Press, Wuhan, China, (1986).
[9] Odibat, Z.M., Differential transform method for solving Volterra integral equation with separable kernels, Mathematical and Computer Modelling 48, 1144-1149, (2008).
[10] Biazar, J., Eslami, M.,Islam, M.R., Differential transform method for special systems of integral equations. Journal of King Saud University-Science 24, 211-214, (2012).
[11] Jafari, H., Haghbin, A., Hesam, S., and Baleanu, D., Solving Partial q-Differential Equations within Reduced q-Differential Transformation Method. Mathematical and Theoretical Physics Vol. 59, 399-407, (2014).
[12] Jackson, F.H., A q-form of Taylor’s theorem, Messenger of Mathematics 38, 62-64, (1909).
[13] Exton, H., q-Hypergeometric Functions and Applications, John Wiley & Sons, New York, (1983).
[14] Gasper, G., Rahman, M., Basic Hypergeometric Series. 2nd Ed., Cambridge Univ. Press, Cambridge, (2004).
[15] Carmichael, R.D., The general theory of linear q-Difference Equation, Am. J. Math. 34, 147- 168, (1912).
[16] Rajkovic, P.M., On q-iterative methods for solving equations and systems, Novi Sad J. Math. 33(2), 127-137, (2003).
[17] Ayaz, F., Applications of differential transform method to differential-algebraic equations. Appl. Math. Comput. 152, 649-657, (2004).
[18] Hassan, I.H., Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems. Choas Solitons Fractals 36 (1), 53-65, (2008).
[19] Ayaz, F.,, Solutions of the system of differential equations by differential transform method, Appl. Math. Comput. 147, 547-567, (2004).
[20] 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).
[21] Bildik, N., Konuralp, A., Bek, F. and Kucukarslan, S., Solution of different type of the partial differential equation by differential transform method and Adomian’s decomposition method, Appl. Math. Comput. 127, 551-567, (2006).
[22] Arikoglu, A., Ozkol, I., Solution of difference equations by using differential transform method, Appl. Math. Comput. 173 (1), 126-136, (2006).
[23] Arikoglu, A., Ozkol, I., Solution of differential–difference equations by using differential transform method, Appl. Math. Comput. 181 (1), 153-162, (2006).
[24] Liu, H., Song, Y., Differential transform method applied to high index differential-algebraic equations, Appl. Math. Comput. 184 (2), 748-753, (2007).
Published
2025-09-30
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