[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Discontinuity, Nonlinearity and Complexity
ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

On Time Scales Fractional Volterra-Fredholm Integro-Differential Equation

Discontinuity, Nonlinearity, and Complexity 12(3) (2023) 615--630 | DOI:10.5890/DNC.2023.09.009

Ahmed A. Hamoud$^{1}$, Amol D. Khandagale$^2$ and Kirtiwant P. Ghadle$^2$

$^1$ Department of Mathematics, Taiz University, Taiz-380 015, Yemen

$^2$ Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad-(431004), India

Download Full Text PDF

 

Abstract

The existence, uniqueness, and Ulam-Hyers stability of the Volterra-Fredholm integro-differential equation with non-instantaneous impulses and periodic boundary conditions over time scales are investigated in this paper using Banach fixed point theorems and Caputo delta fractional derivative. Finally, we present an example to confirm our main findings.

References

  1. [1]  Kumar, P., Haloi, R., Bahuguna, D., and Pandey, D.N. (2016), Existence of solutions to a new class of abstract non-instantaneous impulsive fractional integrodifferential equations, Nonlinear Dynamics and Systems Theory, 16, 73-85.
  2. [2]  Balachandran K., Kiruthika S., and Trujillo, J.J. (2011), Existence results for fractional impulsive integrodifferential equations in Banach spaces, Communications in Nonlinear Science and Numerical Simulation, 16, 1970-1977.
  3. [3]  Balachandran K., Kiruthika S., and Trujillo, J.J. (2012), Remark on the existence results for fractional impulsive integro-differential equations in Banach spaces, Communications in Nonlinear Science and Numerical Simulation, 17, 2244-2247.
  4. [4]  Hamoud, A., Mohammed, N., and Ghadle, K. (2020), Existence and uniqueness results for Volterra-Fredholm integro differential equations, Advances in the Theory of Nonlinear Analysis and its Application, 4(4), 361-372.
  5. [5]  Hamoud, A., Azeez, A., and Ghadle, K. (2018), A study of some iterative methods for solving fuzzy Volterra-Fredholm integral equations, Indonesian Journal of Electrical Engineering and Computer Science, 11, 1228-1235.
  6. [6]  Hamoud, A., Dawood, L., Ghadle, K. and Atshan, S. (2019), Usage of the modified variational iteration technique for solving Fredholm integro-differential equations, International Journal of Mechanical and Production Engineering Research and Development, 9(2), 895-902.
  7. [7]  Hamoud, A., Mohammed, N., and Ghadle, K. (2019), A study of some effective techniques for solving Volterra-Fredholm integral equations, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 26, 389-406.
  8. [8]  Bhadane, P., Ghadle, K., and Hamoud, A. (2020), Approximate solution of fractional Black-Schole's European option pricing equation by using ETHPM, Nonlinear Functional Analysis and Applications, 25(2), 331-344.
  9. [9]  Bani Issa, M., Hamoud, A., and Ghadle, K. (2021), Numerical solutions of fuzzy integro-differential equations of the second kind, Journal of Mathematics and Computer Science, 23, 67-74.
  10. [10]  Dawood, L., Sharif, A., and Hamoud, A. (2020), Solving higher-order integro differential equations by VIM and MHPM, International Journal of Applied Mathematics, 33(2), 253-264.
  11. [11]  Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and Applications of the Fractional Differential Equations, Amsterdam: Elsevier.
  12. [12]  He, J.H. (1999), Some applications of nonlinear fractional differential equations and their approximations, Bulletin of Science, Technology $\&$ Society, 15(2), 86-90.
  13. [13]  Alasadi, I. and Hamoud, A. (2021), Existence and stability results for fractional Volterra-Fredholm integro-dierential equation with mixed conditions, Advances in Dynamical Systems and Applications, 16(1), 217-236.
  14. [14]  Hamoud, A. (2021), Uniqueness and stability results for Caputo fractional Volterra-Fredholm integro-differential equations, Journal of Siberian Federal University - Mathematics and Physics, 14(3), 313-325.
  15. [15]  Hamoud, A. (2020), Existence and uniqueness of solutions for fractional neutral Volterra-Fredholm integro-differential equations, Advances in the Theory of Nonlinear Analysis and its Application, 4(4), 321-331.
  16. [16]  Hilger, S. (1990), Analysis on Measure chain-A unified approch to continuous and discrete calculus, Results in Mathematics, 18, 18-56.
  17. [17]  Bohner, M. and Peterson, A. (2003), Advances in Dynamic Equations on Time Scales, Birkhauser Boston, Boston, MA.
  18. [18]  Kulik, T. and Tisdell, C.C. (2008), Volterra integral equations on time scales: Basic qualitative and quantitative results with applications to initial value problems on unbounded domains, International Journal of Difference Equations, 3(1), 103-133.
  19. [19]  Pachpatte, D.B. (2009), On Nonstandard Volterra type dynamic integral equations on time scales, Electronic Journal of Qualitative Theory of Differential Equations, 72, 1-14.
  20. [20]  Tisdell, C.C. and Zaidi, A.H. (2008), Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling, Nonlinear Analysis: Theory, Methods $\&$ Applications, 68(11), 3504-3524.
  21. [21]  Tisdell, C.C. and Zaidi, A.H. (2009), Successive approximations to solutions of dynamic equations ontime scales, Communications on Applied Nonlinear Analysis 16(1), 61-87.
  22. [22]  Panda, R. and Dash, M. (2006), Fractional generalized splines and signal processing, Signal Process, 86, 2340-2350.
  23. [23]  Ahmadkhanlu, A. and Jahanshahi, M. (2012), On the existence and uniqueness of solution of initial value problem for fractional order differential equations on time scales, Bulletin of the Iranian Mathematical Society, 38(1), 241-252.
  24. [24]  Kumar, V. and Muslim M. (2019), Existence and stability of fractional integro differential equation with non-instantaneous integrable impulses and periodic boundary condition on time scales, Journal of King Saud University-Science, 31(4), 1311-1317.
  25. [25]  Agarwal, R.P., Hristova, S., and O'Regan, D. (2017), Caputo fractional differential equations with non-instantaneous impulses and strict stability by lyapunov functions, Filomat, 31(16), 5217-5239.
  26. [26]  Aghajani, A., Jalilian, Y., and Trujillo, J.J. (2012), On the existence of solutions of fractional integro-differential equations, Fractional Calculus and Applied Analysis, 15(1), 44-69.
  27. [27]  Feckan, M. and Wang, J. (2015), A general class of impulsive evolution equations, Topological Methods in Nonlinear Analysis, 46(2), 915-933.
  28. [28]  Gautam, G.R. and Dabas, J. (2016), Mild solution for nonlocal fractional functional differential equation with not instantaneous impulse, International Journal of Nonlinear Science, 21(3), 151-160.
  29. [29]  Hernandez, E. and O'Regan, D. (2013), On a new class of abstract impulsive differential equations, Proceedings of the American Mathematical Society, 141(5), 1641-1649.
  30. [30]  Muslim, M., Kumar, A. and Feckan, M. (2018), Existence, uniqueness and stability of solutions to second order nonlinear differential equations with non-instantaneous impulses, Journal of King Saud University-Science, 30(2), 204-213.
  31. [31]  Sousa, J., Vanterler and da C., Oliveira E. Capelas de, (2018), Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation, Applied Mathematics Letters, 81, 50-56.
  32. [32]  de Oliveira, E. C. and Sousa, J. V. D. C. (2018), Ulam-Hyers-Rassias stability for a class of fractional integro-differential equations, Results in Mathematics, 73(3), 1-16.
  33. [33]  Wang, J.R., Feckan, M. and Zhou, Y. (2012), Ulam's type stability of impulsive ordinary differential equations, Journal of Mathematical Analysis and Applications, 395(1), 258-264.
  34. [34]  Hamoud, A. and Ghadle, K. (2018), Existence and uniqueness of solutions for fractional mixed Volterra-Fredholm integro-differential equations, Indian Journal of Mathematics 60(3), 375-395.
  35. [35]  Agarwal, R.P., Bohner, M., Peterson, A., and O'Regan, D. (2003), Advances in Dynamic Equations on Time Scales, Birkhaurser, Boston.
  36. [36]  Ahmadkhanlu, A. and Jahanshahi, M. (2012), On the existence and uniqueness of solution of initial value problem for fractional order differential equations on time scales, Bulletin of the Iranian Mathematical Society, 38(1), 241-252.

Copyright © 2011-2024 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates