Research Article
BibTex RIS Cite

Laguerre Collocation Approach of Caputo Fractional Fredholm-Volterra Integro-Differential Equations

Year 2024, , 38 - 45, 18.03.2024

Dilek Varol , Ayşegül Daşcıoğlu

https://doi.org/10.32323/ujma.1390222

Abstract

This paper discusses the linear fractional Fredholm-Volterra integro-differential equations (IDEs) considered in the Caputo sense. For this purpose, Laguerre polynomials have been used to construct an approximation method to obtain the solutions of the linear fractional Fredholm-Volterra IDEs. By this approximation method, the IDE has been transformed into a linear algebraic equation system using appropriate collocation points. In addition, a novel and exact matrix expression for the Caputo fractional derivatives of Laguerre polynomials and an associated explicit matrix formulation has been established for the first time in the literature. Furthermore, a comparison between the results of the proposed method and those of methods in the literature has been provided by implementing the method in numerous examples.

Keywords

Caputo fractional derivatives Fredholm-Volterra integro-differential equations Laguerre polynomials

References

  • [1] M. Yi, J. Huang, CAS wavelet method for solving the fractional integro-differential equation with a weakly singular kernel, Int. J. Comput. Math., 92(8) (2015), 1715-1728.
  • [2] B. Q. Tang, X. F. Li, Solution of a class of Volterra integral equations with singular and weakly singular kernels, Appl. Math. Comput., 199 (2008), 406413 .
  • [3] P. K. Kythe, P. Puri, Computational Method for Linear Integral Equations, Birkhauser, Boston, 2002.
  • [4] V. V. Zozulya, P. I. Gonzalez-Chi, Weakly singular, singular and hypersingular integrals in 3-D elasticity and fracture mechanics, J. Chin. Inst. Eng., 22 (2002), 763775 .
  • [5] C. Li, Y. Wang, Numerical algorithm based on Adomian decomposition for fractional differential equations Comput. Math. with Appl., 57(10) (2009), 1672-1681 .
  • [6] B. Ghanbari, S. Djilali, Mathematical and numerical analysis of a three-species predator-prey model with herd behavior and time fractional-order derivative Math. Method Appl. Sci., 43(4) (2020), 1736-1752 .
  • [7] P. Veeresha, D. G. Prakasha, S. Kumar, A fractional model for propagation of classical optical solitons by using nonsingular derivative, Math. Method Appl. Sci., 2020 (2020), 1–15. https://doi.org/10.1002/mma.6335
  • [8] S. Kumar, A. Kumar, B. Samet, J. F. G´omez-Aguilar, M. S. Osman, A chaos study of tumor and effector cells in fractional tumor-immune model for cancer treatment, Chaos Solitons Fractals 141 (2020), 110321.
  • [9] S. Kumar, A. Kumar, B. Samet, H. Dutta, A study on fractional host–parasitoid population dynamical model to describe insect species, Numer. Methods Partial Differ. Equ., 37(2) (2021), 1673-1692.
  • [10] S. Kumar, S. Ghosh, B. Samet, E. F. D. Goufo, An analysis for heat equations arises in diffusion process using new Yang-Abdel-Aty-Cattani fractional operator, Math. Method Appl. Sci., 43(9) (2020), 6062-6080.
  • [11] S. Kumar, R. Kumar, R. P. Agarwal, B. Samet, A study of fractional Lotka-Volterra population model using Haar wavelet and Adams-Bashforth-Moulton methods, Math. Method Appl. Sci., 43(8), (2020) 5564-5578.
  • [12] B. Ghanbari, S. Kumar, R. Kumar, A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative, Chaos Solitons Fractals, 133 (2020), 109619.
  • [13] A. A. Hamoud, K. H. Hussain, K. P. Ghadle, 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, 26 (2019), 171-184.
  • [14] B. Li, Numerical solution of fractional Fredholm-Volterra integro-differential equations by means of generalized hat functions method, CMES Comput. Model. Eng. Sci., 99(2) (2014), 105-122.
  • [15] D. Nazari Susahab, M. Jahanshahi, Numerical solution of nonlinear fractional Volterra-Fredholm integro-differential equations with mixed boundary conditions, Int. J. Ind. Math., 7(1) (2015), 00563.
  • [16] S. T. Mohyud-Din, H. Khan, M. Arif, M. Rafiq, Chebyshev wavelet method to nonlinear fractional Volterra–Fredholm integro-differential equations with mixed boundary conditions, Adv. Mech. Eng., 9(3) (2017), 1-8.
  • [17] A. Setia, Y. Liu, A. S. Vatsala, Numerical solution of Fredholm-Volterra fractional integro-differential equations with nonlocal boundary conditions, J. Fract. Calc. Appl., 5(2) (2014), 155-165.
  • [18] Y. Wang, L. Zhu, SCW method for solving the fractional integro-differential equations with a weakly singular kernel, Appl. Math. Comput., 275 (2016), 72-80.
  • [19] F. Mohammadi, A. Ciancio, Wavelet-based numerical method for solving fractional integro-differential equation with a weakly singular kernel, Wavelets Linear Algebr., 4(1) (2017), 53-73.
  • [20] S. S. Chaharborj, S. S. Chaharborj, Y. Mahmoudi, Study of fractional order integrodifferential equations by using Chebyshev neural network, J. Math. Stat., 13(1) (2017), 1-13.
  • [21] L. Huang, X. F. Li, Y. Zhao, X. Y. Duan, Approximate solution of fractional integro-differential equations by Taylor expansion method, Comput. Math. Appl., 62 (2011), 1127–1134.
  • [22] S. Alkan, V. F. Hatipo˘glu, Approximate solutions of Volterra-Fredholm integro-differential equations of fractional order, Tbil. Math. J., 10(2) (2017), 1-13.
  • [23] Z. Meng, L. Wang, H. Li, W. Zhang, Legendre wavelets method for solving fractional integro-differential equations, Int. J. Comput. Math., 92(6) (2015), 1275-1291.
  • [24] H. Dehestani, Y. Ordokhani, M. Razzaghi, Combination of Lucas wavelets with Legendre–Gauss quadrature for fractional Fredholm–Volterra integro-differential equations, J. Comput. Appl. Math., 382 (2021), 113070.
  • [25] Y.Ordokhani, H. Dehestani, Numerical solution of linear Fredholm-Volterra integro-differential equations of fractional order, World J. Model. Simul., 12(3) (2016), 204-216.
  • [26] D. Nazari, S. Shahmorad, Application of the fractional differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions, J. Comput. Appl. Math., 234 (2010), 883-891.
  • [27] M. Jani, D. Bhatta, S. Javadi, Numerical solution of fractional integro-differential equations with nonlocal conditions, Appl. Appl. Math., 12(1) (2017), 98 – 111.
  • [28] J. R. Loh, C. Phang, A. Isah, New operational matrix via Genocchi polynomials for solving Fredholm-Volterra fractional integro-differential equations, Adv. Math. Phys., 2017 (2017), 3821870.
  • [29] Y. Yang, Y. Chen, Y. Huang, Spectral-collocation method for fractional Fredholm integro-differential equations, J. Korean Math. Soc., 51(1) (2014), 203-224.
  • [30] F. Mohammadi, Fractional integro-differential equation with a weakly singular kernel by using block pulse functions, U.P.B. Sci. Bull. Series A., 79(1) (2017).
  • [31] P. Rahimkhani, Y. Ordokhani, E. Babolian, Fractional-order Bernoulli functions and their applications in solving fractional Fredholem–Volterra integro-differential equations, Appl. Numer. Math., 122 (2017), 66–81.
  • [32] H. Dehestani, Y. Ordokhani, M. Razzaghi, Hybrid functions for numerical solution of fractional Fredholm-Volterra functional integro-differential equations with proportional delays, Int. J. Numer. Model. El., 32(5) (2019), e2606.
  • [33] E. Keshavarz, Y. Ordokhani, M. Razzaghi, Numerical solution of nonlinear mixed Fredholm-Volterra integro-differential equations of fractional order by Bernoulli wavelets, Comput. Methods Differ. Equ., 7(2) (2019), 163-176.
  • [34] M. R. Ali, A. R. Hadhoud, H. M. Srivastava, Solution of fractional Volterra–Fredholm integro-differential equations under mixed boundary conditions by using the HOBW method, Adv. Differ. Equ., 2019(1) (2019), 115.
  • [35] S. Kumano, T. H. Nagai, Comparison of numerical solutions for Q2 evolution equations, J. Comput. Phys., 201(2) (2004), 651-664.
  • [36] R. Kobayashi, M. Konuma, S. Kumano, FORTRAN program for a numerical solution of the nonsinglet Altarelli-Parisi equation, Comput. Phys. Commun., 86 (1995), 264-278.
  • [37] L. Schoeffel, An elegant and fast method to solve QCD evolution equations. Application to the determination of the gluon content of the Pomeron, Nucl. Instrum. Meth. A., 423 (1999), 439-445.
  • [38] N. Baykus Savasaneril, M. Sezer, Laguerre polynomial solution of high-order linear Fredholm integro-differential equations, New Trends in Math. Sci., 4(2) (2016), 273-284.
  • [39] B. Gürbüz, M. Sezer, C. Güler, Laguerre collocation method for solving Fredholm integro-differential equations with functional arguments, J. Appl. Math., (2014) 682398, 1-12.
  • [40] S. Yuzbası, Laguerre approach for solving pantograph-type Volterra integro-differential equations, Appl. Math. Comput., 232 (2014), 1183–1199.
  • [41] B. Gürbüz, M. Sezer, Laguerre polynomial solutions of a class of delay partial functional differential equations, Acta Phys. Polon. A., 132(3) (2017c), 558-560.
  • [42] K. A. Al-Zubaidy, A Numerical solution of parabolic-type Volterra partial integro-differential equations by Laguerre collocation method, Int. J. Sci. Technol., 8(4) (2013), 51-55.
  • [43] B. Gürbüz, M. Sezer, A numerical solution of parabolic-type Volterra partial integro-differential equations by Laguerre collocation method, Int. J. Appl. Phys. Math., 7(1) (2017a), 49-58.
  • [44] B. Gürbüz, M. Sezer, A new computational method based on Laguerre polynomials for solving certain nonlinear partial integro differential equations, Acta Phys. Polon. A., 132(3) (2017b), 561-563.
  • [45] A. M. S. Mahdy, R. T. Shwayyea, Numerical solution of fractional integro-differential equations by least squares method and shifted Laguerre polynomials pseudo-spectral method, IJSER, 7(4) (2016), 1589-1596.
  • [46] A. Daşcıoğlu, D. Varol, Laguerre polynomial solutions of linear fractional integro-differential equations, Math. Sci., 15 (2021), 47-54. https://doi.org/10.1007/s40096-020-00369-y
  • [47] A. Daşcıoğlu, D. Varol Bayram, Solving fractional Fredholm integro-differential equations by Laguerre polynomials, Sains Malaysiana, 48(1) (2019), 251–257.
  • [48] D. Varol Bayram, A. Das¸cıo˘glu, A method for fractional Volterra integro-differential equations by Laguerre polynomials, Adv. Differ. Equ., 2018 (2018), 466.
  • [49] I. Podlubny, Fractional Differential Equations, Academic Press, USA, 1999.
  • [50] W. W. Bell, Special Functions for Scientists and Engineers, D. Van Nostrand Company, London, 1968.

Year 2024, , 38 - 45, 18.03.2024

Dilek Varol , Ayşegül Daşcıoğlu

https://doi.org/10.32323/ujma.1390222

Abstract

References

  • [1] M. Yi, J. Huang, CAS wavelet method for solving the fractional integro-differential equation with a weakly singular kernel, Int. J. Comput. Math., 92(8) (2015), 1715-1728.
  • [2] B. Q. Tang, X. F. Li, Solution of a class of Volterra integral equations with singular and weakly singular kernels, Appl. Math. Comput., 199 (2008), 406413 .
  • [3] P. K. Kythe, P. Puri, Computational Method for Linear Integral Equations, Birkhauser, Boston, 2002.
  • [4] V. V. Zozulya, P. I. Gonzalez-Chi, Weakly singular, singular and hypersingular integrals in 3-D elasticity and fracture mechanics, J. Chin. Inst. Eng., 22 (2002), 763775 .
  • [5] C. Li, Y. Wang, Numerical algorithm based on Adomian decomposition for fractional differential equations Comput. Math. with Appl., 57(10) (2009), 1672-1681 .
  • [6] B. Ghanbari, S. Djilali, Mathematical and numerical analysis of a three-species predator-prey model with herd behavior and time fractional-order derivative Math. Method Appl. Sci., 43(4) (2020), 1736-1752 .
  • [7] P. Veeresha, D. G. Prakasha, S. Kumar, A fractional model for propagation of classical optical solitons by using nonsingular derivative, Math. Method Appl. Sci., 2020 (2020), 1–15. https://doi.org/10.1002/mma.6335
  • [8] S. Kumar, A. Kumar, B. Samet, J. F. G´omez-Aguilar, M. S. Osman, A chaos study of tumor and effector cells in fractional tumor-immune model for cancer treatment, Chaos Solitons Fractals 141 (2020), 110321.
  • [9] S. Kumar, A. Kumar, B. Samet, H. Dutta, A study on fractional host–parasitoid population dynamical model to describe insect species, Numer. Methods Partial Differ. Equ., 37(2) (2021), 1673-1692.
  • [10] S. Kumar, S. Ghosh, B. Samet, E. F. D. Goufo, An analysis for heat equations arises in diffusion process using new Yang-Abdel-Aty-Cattani fractional operator, Math. Method Appl. Sci., 43(9) (2020), 6062-6080.
  • [11] S. Kumar, R. Kumar, R. P. Agarwal, B. Samet, A study of fractional Lotka-Volterra population model using Haar wavelet and Adams-Bashforth-Moulton methods, Math. Method Appl. Sci., 43(8), (2020) 5564-5578.
  • [12] B. Ghanbari, S. Kumar, R. Kumar, A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative, Chaos Solitons Fractals, 133 (2020), 109619.
  • [13] A. A. Hamoud, K. H. Hussain, K. P. Ghadle, 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, 26 (2019), 171-184.
  • [14] B. Li, Numerical solution of fractional Fredholm-Volterra integro-differential equations by means of generalized hat functions method, CMES Comput. Model. Eng. Sci., 99(2) (2014), 105-122.
  • [15] D. Nazari Susahab, M. Jahanshahi, Numerical solution of nonlinear fractional Volterra-Fredholm integro-differential equations with mixed boundary conditions, Int. J. Ind. Math., 7(1) (2015), 00563.
  • [16] S. T. Mohyud-Din, H. Khan, M. Arif, M. Rafiq, Chebyshev wavelet method to nonlinear fractional Volterra–Fredholm integro-differential equations with mixed boundary conditions, Adv. Mech. Eng., 9(3) (2017), 1-8.
  • [17] A. Setia, Y. Liu, A. S. Vatsala, Numerical solution of Fredholm-Volterra fractional integro-differential equations with nonlocal boundary conditions, J. Fract. Calc. Appl., 5(2) (2014), 155-165.
  • [18] Y. Wang, L. Zhu, SCW method for solving the fractional integro-differential equations with a weakly singular kernel, Appl. Math. Comput., 275 (2016), 72-80.
  • [19] F. Mohammadi, A. Ciancio, Wavelet-based numerical method for solving fractional integro-differential equation with a weakly singular kernel, Wavelets Linear Algebr., 4(1) (2017), 53-73.
  • [20] S. S. Chaharborj, S. S. Chaharborj, Y. Mahmoudi, Study of fractional order integrodifferential equations by using Chebyshev neural network, J. Math. Stat., 13(1) (2017), 1-13.
  • [21] L. Huang, X. F. Li, Y. Zhao, X. Y. Duan, Approximate solution of fractional integro-differential equations by Taylor expansion method, Comput. Math. Appl., 62 (2011), 1127–1134.
  • [22] S. Alkan, V. F. Hatipo˘glu, Approximate solutions of Volterra-Fredholm integro-differential equations of fractional order, Tbil. Math. J., 10(2) (2017), 1-13.
  • [23] Z. Meng, L. Wang, H. Li, W. Zhang, Legendre wavelets method for solving fractional integro-differential equations, Int. J. Comput. Math., 92(6) (2015), 1275-1291.
  • [24] H. Dehestani, Y. Ordokhani, M. Razzaghi, Combination of Lucas wavelets with Legendre–Gauss quadrature for fractional Fredholm–Volterra integro-differential equations, J. Comput. Appl. Math., 382 (2021), 113070.
  • [25] Y.Ordokhani, H. Dehestani, Numerical solution of linear Fredholm-Volterra integro-differential equations of fractional order, World J. Model. Simul., 12(3) (2016), 204-216.
  • [26] D. Nazari, S. Shahmorad, Application of the fractional differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions, J. Comput. Appl. Math., 234 (2010), 883-891.
  • [27] M. Jani, D. Bhatta, S. Javadi, Numerical solution of fractional integro-differential equations with nonlocal conditions, Appl. Appl. Math., 12(1) (2017), 98 – 111.
  • [28] J. R. Loh, C. Phang, A. Isah, New operational matrix via Genocchi polynomials for solving Fredholm-Volterra fractional integro-differential equations, Adv. Math. Phys., 2017 (2017), 3821870.
  • [29] Y. Yang, Y. Chen, Y. Huang, Spectral-collocation method for fractional Fredholm integro-differential equations, J. Korean Math. Soc., 51(1) (2014), 203-224.
  • [30] F. Mohammadi, Fractional integro-differential equation with a weakly singular kernel by using block pulse functions, U.P.B. Sci. Bull. Series A., 79(1) (2017).
  • [31] P. Rahimkhani, Y. Ordokhani, E. Babolian, Fractional-order Bernoulli functions and their applications in solving fractional Fredholem–Volterra integro-differential equations, Appl. Numer. Math., 122 (2017), 66–81.
  • [32] H. Dehestani, Y. Ordokhani, M. Razzaghi, Hybrid functions for numerical solution of fractional Fredholm-Volterra functional integro-differential equations with proportional delays, Int. J. Numer. Model. El., 32(5) (2019), e2606.
  • [33] E. Keshavarz, Y. Ordokhani, M. Razzaghi, Numerical solution of nonlinear mixed Fredholm-Volterra integro-differential equations of fractional order by Bernoulli wavelets, Comput. Methods Differ. Equ., 7(2) (2019), 163-176.
  • [34] M. R. Ali, A. R. Hadhoud, H. M. Srivastava, Solution of fractional Volterra–Fredholm integro-differential equations under mixed boundary conditions by using the HOBW method, Adv. Differ. Equ., 2019(1) (2019), 115.
  • [35] S. Kumano, T. H. Nagai, Comparison of numerical solutions for Q2 evolution equations, J. Comput. Phys., 201(2) (2004), 651-664.
  • [36] R. Kobayashi, M. Konuma, S. Kumano, FORTRAN program for a numerical solution of the nonsinglet Altarelli-Parisi equation, Comput. Phys. Commun., 86 (1995), 264-278.
  • [37] L. Schoeffel, An elegant and fast method to solve QCD evolution equations. Application to the determination of the gluon content of the Pomeron, Nucl. Instrum. Meth. A., 423 (1999), 439-445.
  • [38] N. Baykus Savasaneril, M. Sezer, Laguerre polynomial solution of high-order linear Fredholm integro-differential equations, New Trends in Math. Sci., 4(2) (2016), 273-284.
  • [39] B. Gürbüz, M. Sezer, C. Güler, Laguerre collocation method for solving Fredholm integro-differential equations with functional arguments, J. Appl. Math., (2014) 682398, 1-12.
  • [40] S. Yuzbası, Laguerre approach for solving pantograph-type Volterra integro-differential equations, Appl. Math. Comput., 232 (2014), 1183–1199.
  • [41] B. Gürbüz, M. Sezer, Laguerre polynomial solutions of a class of delay partial functional differential equations, Acta Phys. Polon. A., 132(3) (2017c), 558-560.
  • [42] K. A. Al-Zubaidy, A Numerical solution of parabolic-type Volterra partial integro-differential equations by Laguerre collocation method, Int. J. Sci. Technol., 8(4) (2013), 51-55.
  • [43] B. Gürbüz, M. Sezer, A numerical solution of parabolic-type Volterra partial integro-differential equations by Laguerre collocation method, Int. J. Appl. Phys. Math., 7(1) (2017a), 49-58.
  • [44] B. Gürbüz, M. Sezer, A new computational method based on Laguerre polynomials for solving certain nonlinear partial integro differential equations, Acta Phys. Polon. A., 132(3) (2017b), 561-563.
  • [45] A. M. S. Mahdy, R. T. Shwayyea, Numerical solution of fractional integro-differential equations by least squares method and shifted Laguerre polynomials pseudo-spectral method, IJSER, 7(4) (2016), 1589-1596.
  • [46] A. Daşcıoğlu, D. Varol, Laguerre polynomial solutions of linear fractional integro-differential equations, Math. Sci., 15 (2021), 47-54. https://doi.org/10.1007/s40096-020-00369-y
  • [47] A. Daşcıoğlu, D. Varol Bayram, Solving fractional Fredholm integro-differential equations by Laguerre polynomials, Sains Malaysiana, 48(1) (2019), 251–257.
  • [48] D. Varol Bayram, A. Das¸cıo˘glu, A method for fractional Volterra integro-differential equations by Laguerre polynomials, Adv. Differ. Equ., 2018 (2018), 466.
  • [49] I. Podlubny, Fractional Differential Equations, Academic Press, USA, 1999.
  • [50] W. W. Bell, Special Functions for Scientists and Engineers, D. Van Nostrand Company, London, 1968.
There are 50 citations in total.

Details

Primary Language English
Subjects Numerical Solution of Differential and Integral Equations
Journal Section Articles
Authors

Dilek Varol 0000-0002-5158-5614

Ayşegül Daşcıoğlu 0000-0001-8931-6930

Early Pub Date February 15, 2024
Publication Date March 18, 2024
Submission Date November 13, 2023
Acceptance Date February 6, 2024
Published in Issue Year 2024

Cite

APA Varol, D., & Daşcıoğlu, A. (2024). Laguerre Collocation Approach of Caputo Fractional Fredholm-Volterra Integro-Differential Equations. Universal Journal of Mathematics and Applications, 7(1), 38-45. https://doi.org/10.32323/ujma.1390222
AMA Varol D, Daşcıoğlu A. Laguerre Collocation Approach of Caputo Fractional Fredholm-Volterra Integro-Differential Equations. Univ. J. Math. Appl. March 2024;7(1):38-45. doi:10.32323/ujma.1390222
Chicago Varol, Dilek, and Ayşegül Daşcıoğlu. “Laguerre Collocation Approach of Caputo Fractional Fredholm-Volterra Integro-Differential Equations”. Universal Journal of Mathematics and Applications 7, no. 1 (March 2024): 38-45. https://doi.org/10.32323/ujma.1390222.
EndNote Varol D, Daşcıoğlu A (March 1, 2024) Laguerre Collocation Approach of Caputo Fractional Fredholm-Volterra Integro-Differential Equations. Universal Journal of Mathematics and Applications 7 1 38–45.
IEEE D. Varol and A. Daşcıoğlu, “Laguerre Collocation Approach of Caputo Fractional Fredholm-Volterra Integro-Differential Equations”, Univ. J. Math. Appl., vol. 7, no. 1, pp. 38–45, 2024, doi: 10.32323/ujma.1390222.
ISNAD Varol, Dilek - Daşcıoğlu, Ayşegül. “Laguerre Collocation Approach of Caputo Fractional Fredholm-Volterra Integro-Differential Equations”. Universal Journal of Mathematics and Applications 7/1 (March 2024), 38-45. https://doi.org/10.32323/ujma.1390222.
JAMA Varol D, Daşcıoğlu A. Laguerre Collocation Approach of Caputo Fractional Fredholm-Volterra Integro-Differential Equations. Univ. J. Math. Appl. 2024;7:38–45.
MLA Varol, Dilek and Ayşegül Daşcıoğlu. “Laguerre Collocation Approach of Caputo Fractional Fredholm-Volterra Integro-Differential Equations”. Universal Journal of Mathematics and Applications, vol. 7, no. 1, 2024, pp. 38-45, doi:10.32323/ujma.1390222.
Vancouver Varol D, Daşcıoğlu A. Laguerre Collocation Approach of Caputo Fractional Fredholm-Volterra Integro-Differential Equations. Univ. J. Math. Appl. 2024;7(1):38-45.

 23181

Universal Journal of Mathematics and Applications 

29207              

Creative Commons License  The published articles in UJMA are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.