Non-polynomial fractional spline method for solving Fredholm integral equations

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Published: Dec 14, 2022
DOI: https://doi.org/10.58205/jiamcs.v2i3.51
Keywords:
spline method, fractional derivative, integral equation, Convergence analysis

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Rahel Jaza
University of Sulaimani
https://orcid.org/0000-0001-9953-9409
Faraidun Hamasalh

Abstract

 A new type of non-polynomial fractional spline function for approximating solutions of Fredholm-integral equations has been presented. For this purpose, we used a new idea of fractional continuity conditions by using the Caputo fractional derivative and the Riemann Liouville fractional integration to generate fractional spline derivatives. Moreover, the convergence analysis is studied with proven theorems. The approach is also well-explained and supported by four computational numerical findings, which show that it is both accurate and simple to apply.

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How to Cite
[1]
Jaza, R. and Hamasalh, F. 2022. Non-polynomial fractional spline method for solving Fredholm integral equations. Journal of Innovative Applied Mathematics and Computational Sciences. 2, 3 (Dec. 2022), 1–14. DOI:https://doi.org/10.58205/jiamcs.v2i3.51.
ARK
https://n2t.net/ark:/49935/jiamcs.v2i3.51
Issue
Vol. 2 No. 3 (2022): July- December
Section
Research Articles
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References

K. K. Ali, K. R. Raslan and T. S. El-Danaf, Non-polynomial spline method for solving coupled Burgers equations, Comput. methods differ. equ., 3(3) (2015), 218-230.

M. Al-Refai and T. Abdeljawad, Fundamental results of conformable Sturm-Liouville eigenvalue problems, J. Complex. 2017 (2017), 3720471.

M. Amin, M. Abbas, M. K. Iqbal and D. Baleanu, Non-polynomial quintic spline for numerical solution of fourth-order time fractional partial differential equations, Adv. Difference Equ., 2019 (2019), 1-22.

S. Amiri, M. Hajipour and D. Baleanu, A spectral collocation method with piecewise trigonometric basis functions for nonlinear Volterra-Fredholm integral equations, Appl. Math. Comput., 370 (2020), 124915.

A. Bellour, D. Sbibih and A. Zidna, Two cubic spline methods for solving Fredholm integral equations, Appl. Math. Comput., 276 (2016), 1-11.

D. Baleanu and D. Kumer, Fractional calculus and its applications in physics, Frontiers in Physics, Lausanne: Frontiers Media, 2019.

M. Benbachir and A. Boutiara, Measure of noncompactness for nonlinear Hilfer fractional differential equation with mixed fractional integral boundary conditions in Banach space, J. innov. appl. math. comput. sci., 2(1) (2022), 27-42.

N. G. Chegini, A. Salaripanah, R. Mokhtari and D. Isvand, Numerical solution of the regularized long wave equation using nonpolynomial splines, Nonlinear. Dyn, 69 (2012), 459-471.

M. Dalir and M. Bashour, Applications of fractional calculus1, Appl. Math. Sci., 4(21) (2010), 1021-1032.

D. A. Hammad, M. S. Semary and A. G. Khattab, Ten non-polynomial cubic splines for some classes of Fredholm integral equations, Ain Shams Eng. J., 13(4) (2022), 101666.

F. K. Hamasalh and M. A. Headayat, The applications of non-polynomial spline to the numerical solution for fractional differential equations, AIP Conference Proceedings., 2334(1) AIP Publishing, 2021.

F. K. Hamasalh and P. O. Muhammed, Computational Method for Fractional Differential Equations Using Nonpolynomial Fractional Spline, Math. Sci. Lett., 5(2) (2016), 131-136.

N. N. Hasan and M. R. Nasif, Cubic trigonometric spline for solving nonlinear volterra integral equations, Iraqi J. Sci., 60(12) (2019), 2697-2705.

R. T. Hernandez, V. R. Ramirez, G. A. I. Silve and U. M. Diwekar, A fractional calculus approach to the dynamic optimization of biological reactive systems. Part I: Fractional models for biological reactions, Chem. Eng. Sci., 117 (2014), 217-228.

C. Ionescua, A. Lopesb, D. Copota, J. A. T. Machadoc and J. H. T. Batesa, The role of fractional calculus in modeling biological phenomena: A review, Commun. Nonlinear Sci. Numer. Simul., 51 (2017), 141-159.

A. Khalid, A. Ghaffar, M. N. Naeem, K. S. Nisar and D. Baleanu, Solutions of BVPs arising in hydrodynamic and magnetohydro-dynamic stability theory using polynomial and nonpolynomial splines, Alex. Eng. J., 60(1) (2021), 941-953.

R. Khalila, M. Al Horania, A. Yousefa and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.

A. Kurt, Y. Çenesiz and O. Tasbozan, On the solution of Burgers equation with the new fractional derivative, Open Physics, 13(1) (2015), 355-360.

Z. Laadjal, T. Abdeljawad and F. Jarad, Sharp estimates of the unique solution for twopoint fractional boundary value problems with conformable derivative, Numer. Methods Partial

Differ. Equ. (2021).

A. Pal Singh, D. Deb, H. Agrawal, K. Bingi and S. Ozana, Modeling and control of robotic manipulators: A fractional calculus point of view, Arab J Sci Eng., 46(10) (2021), 9541-9552.

R. J. Qadir and F. Hamasalh, Fractional Spline Model for Computing Fredholm Integral Equations, 2022 International Conference on Computer Science and Software Engineering (CSASE). IEEE, 2022.

K. Maleknejad, J. Rashidinia and H. Jalilian, Quintic Spline functions and Fredholm integral equation, Numer. Methods Partial Differ. Equ., 9(1) (2021), 211-224.

K. Maleknejad, J. Rashidinia and H. Jalilian, Non-polynomial spline functions and Quasi-linearization to approximate nonlinear Volterra integral equation, Filomat 32(11) (2018), 3947-3956.

C. Milici, G. Draganescu and J. T. Machado, Introduction to fractional differential equations, Springer International Publishing, 2018.

M. Mutaz and A. Trounev, Fractional nonlinear Volterra-Fredholm integral equations involving Atangana-Baleanu fractional derivative: framelet applications, Adv. Difference Equ., 2020 (2020), 1-15.

J. Rashidinia, K. Maleknejad and H. Jalilian, Convergence analysis of non-polynomial spline functions for the Fredholm integral equation, Int. J. Comput. Math., 97(6) (2020), 1197-1211.

S. S. Ray and P. K. Sahu, Numerical methods for solving Fredholm integral equations of second kind, Abstr. Appl. Anal., 2013 (2013),1-17.

J. Sabatier, O. P. Agrawal and J. A. T. Machado, Advances in fractional calculus, Springer Dordrecht, 2007.

B. Safa, M. S. Abdelouahab and R. Lozi, On periodic solutions of fractional-order differential systems with a fixed length of sliding memory, J. innov. appl. math. comput. sci., 1(1) (2021), 64-78.

P. K. Srivastava, Study of differential equations with their polynomial and nonpolynomial spline based approximation, Acta Technica Corviniensis-Bulletin of Engineering, 7(3) (2014),1-12.

H. G. Sun, Y. Zhang, D. Baleanu, W. Chen and Y.Q. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul., 64 (2018), 213-231.

T. Tahernezhad and R. Jalilian, Exponential spline for the numerical solutions of linear Fredholm integro-differential equations, Adv. Difference Equ., 2020 (2020), 1-15.

A. M. Wazwaz, Linear and nonlinear integral equations, Springer Berlin, Heidelberg, 2011.