Solving Mixed Volterra-Fredholm Integro Differential Equations by Using HAM

Abstract

In this article, we discussed the semi-analytical approximated method for solving mixed Volterra-Fredholm integro-differential equations, namely, homotopy analysis method. Moreover, we prove the existence and uniqueness results and convergence of the technique. Finally, an example is included to demonstrate the validity and applicability of the proposed technique.

Keywords

• Volterra-Fredholm integro-differential equation,homotopy analysis method,approximate solution

References

1. Araghi, M.A., Behzadi, S.S., {\em Solving nonlinear Volterra-Fredholm integro-differential equations using the modified Adomian decomposition method}, Comput. Methods in Appl. Math., {\bf 9}(2009), 321--331.
2. Behiry, S.H., Mohamed, S.I., {\em Solving high-order nonlinear Volterra-Fredholm integro-differential equations by differential transform method}, Natural Science, {\bf 4(8)} (2012), 581--587.
3. Babolian, E., Masouri, Z., Hatamzadeh, S., {\em New direct method to solve nonlinear Volterra-Fredholm integral and integro differential equation using operational matrix with Block-Pulse functions}, Progress in Electromagnetic Research, {\bf B 8}(2008), 59--76.
4. El-Sayed, S.M., Kaya, D., Zarea, S., {\em The decomposition method applied to solve high-order linear Volterra-Fredholm integro-differential equations}, International Journal of Nonlinear Sciences and Numerical Simulation, {\bf 5(2)}(2004), 105--112.
5. Fadhel, F.S., Mezaal, A.O., Salih, S.H., {\em Approximate solution of the linear mixed Volterra-Fredholm integro-differential equations of second kind by using variational iteration method}, Al- Mustansiriyah, J. Sci., {\bf 24(5)}(2013), 137--146.
6. Ghasemi, M., kajani, M., Babolian, E., {\em Application of He's homotopy perturbation method to nonlinear integro differential equations}, Appl. Math. Comput., {\bf 188}(2007), 538--548.
7. He, J.H., Wang, S.Q., {\em Variational iteration method for solving integro-differential equations}, Phys. Lett., {\bf A 367}()2007), 188--191.
8. Hamoud, A.A., Ghadle, K.P., {\em Existence and uniqueness of the solution for Volterra-Fredholm integro-differential equations}, Journal of Siberian Federal University. Mathematics \& Physics, {\bf 11(6)}(2018), 692--701.

9. Hamoud, A.A., Ghadle, K.P., {\em Modified Laplace decomposition method for fractional Volterra-Fredholm integro-differential equations}, Journal of Mathematical Modeling, {\bf 6(1)}(2018), 91--104. Hamoud, A.A., Ghadle, K.P., {\em Homotopy analysis method for the first order fuzzy Volterra-Fredholm integro-differential equations}, Indonesian J. Elec. Eng. \& Comp. Sci. {\bf11(3)}(2018), 857--867.
10. Hamoud, A.A., Ghadle, K.P., {\em Some new existence, uniqueness and convergence results for fractional Volterra-Fredholm integro-differential equations}, J. Appl. Comp. Mech. {\bf 5(1)}(2019), 58--69.
11. Hamoud, A.A., Ghadle, K.P., {\em Existence and uniqueness of solutions for fractional mixed Volterra-Fredholm integro-differential equations}, Indian J. Math., {\bf 60(3)}(2018), 375--395.
12. Hamoud, A.A. , Ghadle, K.P., {\em The approximate solutions of fractional Volterra-Fredholm integro-differential equations by using analytical techniques}, Probl. Anal. Issues Anal., {\bf7(25) No.1}(2018), 41--58.
13. Hamoud, A.A. , Ghadle, K.P., Atshan, S.M., {\em The approximate solutions of fractional integro-differential equations by using modified Adomian decomposition method}, Khayyam Journal of Mathematics, {\bf 5(1)}(2019), 21--39.
14. Hamoud, A.A. , Ghadle, K.P., {\em The reliable modified of Laplace Adomian decomposition method to solve nonlinear interval Volterra-Fredholm integral equations}, Korean J. Math., {\bf 25(3)}(2017), 323--334.
15. Hamoud, A.A., Dawood, L.A., Ghadle, K.P., Atshan, S.M., {\em Usage of the modified variational iteration technique for solving Fredholm integro-differential equations}, International Journal of Mechanical and Production Engineering Research and Development, {\bf 9(2)}(2019), 895--902.
16. Hamoud, A.A., Hussain, K.H., Ghadle, K.P., {\em The reliable modified Laplace Adomian decomposition method to solve fractional Volterra-Fredholm integro-differential equations}, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications \& Algorithms, {\bf 26}(2019), 171--184.
17. Hamoud, A.A. , Ghadle, K.P., {\em Modified Adomian decomposition method for solving fuzzy Volterra-Fredholm integral equations}, J. Indian Math. Soc., {\bf 85(1-2)} (2018), 52--69.
18. Hamoud, A.A. , Ghadle, K.P., Bani Issa, M., Giniswamy, {\em Existence and uniqueness theorems for fractional Volterra-Fredholm integro-differential equations}, Int. J. Appl. Math., {\bf 31(3)}(2018), 333--348.
19. Hamoud, A.A., Azeez, A.D., Ghadle, K.P., {\em A study of some iterative methods for solving fuzzy Volterra-Fredholm integral equations}, Indonesian J. Elec. Eng. \& Comp. Sci., {\bf 11(3)}(2018), 1228--1235.
20. Hamoud, A.A., Ghadle, K.P., {\em Usage of the homotopy analysis method for solving fractional Volterra-Fredholm integro-differential equation of the second kind}, Tamkang Journal of Mathematics, {\bf 49(4)}(2018), 301--315.
21. Hamoud, A.A., Bani Issa, M., Ghadle, K.P., Abdulghani, M., {\em Existence and convergence results for caputo fractional Volterra integro-differential equations}, Journal of Mathematics and Applications, {\bf41} (2018), 109--122.
22. Hamoud, A.A., Bani Issa, M., Ghadle, K.P., {\em Existence and uniqueness results for nonlinear Volterra-Fredholm integro-differential equations}, Nonlinear Functional Analysis and Applications, {\bf 23(4)}(2018), 797--805.
23. Hamoud, A., Mohammed, N., Ghadle, K., {\em Solving FIDEs by using semi-analytical techniques}, Communications in Advanced Mathematical Sciences, {\bf II(3)}(2019), 192--198.
24. Hamoud, A., Mohammed, N., Ghadle, K., Dhondge, S., {\em Solving integro-differential equations by using numerical techniques}, International Journal of Applied Engineering Research, {\bf 14(14)}(2019), 3219--3225.
25. Hamoud, A. Ghadle, K., {\em The reliable modified of Adomian decomposition method for solving integro-differential equations}, Journal of the Chungcheong Mathematical Society, {\bf 32(4)}(2019), 409--420.
26. Hamoud, A. Ghadle, K., {\em On the numerical solution of nonlinear Volterra-Fredholm integral equations by variational iteration method}, Int. J. Adv. Sci. Tech. Research, {\bf 3}(2016), 45--51.
27. Hamoud, A. Ghadle, K., {\em The combined modified Laplace with Adomian decomposition method for solving the nonlinear Volterra-Fredholm integro-differential equations}, J. Korean Soc. Ind. Appl. Math., {\bf 21}(2017), 17--28. Hussain, K., Hamoud, A., Mohammed, N., {\em Some new uniqueness results for fractional integro-differential equations}, Nonlinear Functional Analysis and Applications, {\bf 24(4)}(2019), 827--836. Hamoud, A., Mohammed, N. , Ghadle, K., {\em A study of some effective techniques for solving Volterra-Fredholm integral equations}, { Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis}, {\bf26} (2019), 389-406. Hamoud, A., Ghadle, K., Pathade, P., {\em An existence and convergence results for Caputo fractional Volterra integro-differential equations}, Jordan Journal of Mathematics and Statistics, {\bf 12}(2019), 307-327.
28. Jerri, A.M., Introduction to Integral Equations with Applications, New York, Wiley, 1999.
29. Shadan, S.B., {\em The use of iterative method to solve two-dimensional nonlinear Volterra-Fredholm integro-differential equations}, J. of Communication in Numerical Analysis, (2012), 1--20.
30. Salih, Y., Mehmet, S., {\em The approximate solution of higher order linear Volterra-Fredholm integro-differential equations in term of Taylor polynomials}, Appl. Math. Comput., {\bf 112}(2000), 291--308.
31. Wazwaz, A.M., Linear and Nonlinear Integral Equations Methods and Applications, Springer Heidelberg Dordrecht London New York, 2011.
32. Wazwaz, A.M., {\em The combined Laplace transform-Adomian decomposition method for handling nonlinear Volterra integro-differential equations}, Appl. Math. Comput., {\bf 216}(2010), 1304--1309.
33. Wazwaz, A.M., {\em A comparison between variational iteration method and Adomian decomposition method}, J. Comput. Appl. Math., {\bf 207}(2007), 129--136.