Korean J. Math. Vol. 32 No. 1 (2024) pp.137-162
DOI: https://doi.org/10.11568/kjm.2024.32.1.137

# Approximation of solutions through the Fibonacci wavelets and measure of noncompactness to nonlinear Volterra-Fredholm fractional integral equations

## Abstract

This paper consists of two significant aims. The first aim of this paper is to establish the criteria for the existence of solutions to nonlinear Volterra-Fredholm (V-F) fractional integral equations on $[0, L]$, where $0<L<\infty$. The fractional integral is described here in the sense of the Katugampola fractional integral of order $\lambda>0$ and with the parameter $\beta>0$. The concepts of the fixed point theorem and the measure of noncompactness are used as the main tools to prove the existence of solutions. The second aim of this paper is to introduce a computational method to obtain approximate numerical solutions to the considered problem. This method is based on the Fibonacci wavelets with collocation technique. Besides, the results of the error analysis and discussions of the accuracy of the solutions are also presented. To the best knowledge of the authors, this is the first computational method for this generalized problem to obtain approximate solutions. Finally, two examples are discussed with the computational tables and convergence graphs to interpret the efficiency and applicability of the presented method.

## References

[1] M. A. Abdou, On a symptotic Methods for Fredholm-Volterra Integral Equation of the Second Kind in Contact Problems, J. Comput. Appl. Math. 154 (2) (2003), 431–446. https://doi.org/10.1016/S0377-0427(02)00862-2 Google Scholar

[2] M. A. Abdou, F. A. Salama, Volterra-Fredholm integral equation of the first kind and spectral relationships, Appl. Math. Comput. 153 (1) (2004), 141–153. https://doi.org/10.1016/S0096-3003(03)00619-2 Google Scholar

[3] R. Amin, H. Alrabaiah, I. Mahariq, A. Zeb, Theoretical and computational results for mixed type Volterra-Fredholm fractional integral equations, Fractals 30 (1) (2022), 2240035. https://doi.org/10.1142/S0218348X22400357 Google Scholar

[4] R. Amin, N. Senu, M. B. Hafeez, N. I. Arshad, A. Ahmadian, S. Salahshour, W. Sumelka, A computational algorithm for the numerical solution of nonlinear fractional integral equations, Fractals 30 (1) (2022), 2240030. https://doi.org/10.1142/S0218348X22400308 Google Scholar

[5] J. Bana ́s, K. Goebel, Measures of noncompactness in Banach spaces, Lecture Notes in Pure and Applied Mathematics, 60, Marcel Dekker, New York, 1980. Google Scholar

[6] I. A. Bhat, L. N. Mishra, Numerical solutions of Volterra integral equations of third kind and its convergence analysis, Symmetry, 14 (2022), 2600. https://doi.org/10.3390/sym14122600 Google Scholar

[7] I. A. Bhat, L. N. Mishra, V. N. Mishra, C. Tunc ̧, O. Tun ̧c, Precision and efficiency of an interpolation approach to weakly singular integral equations, Int. J. Numer. Method H. (2024). https://doi.org/10.1108/hff-09-2023-0553 Google Scholar

[8] A. Chandola, R. M. Pandey, R. Agarwal, L. Rathour, V. N. Mishra, On some properties and applications of the generalized m-parameter Mittag-Leffler function, Adv. Math. Models Appl. 7 (2) (2022), 130–145. Google Scholar

[9] C. Constanda, Integral equations of the first kind in plane elasticity, Quart. Appl. Math. 53 (1995), 783–793. https://api.semanticscholar.org/CorpusID:124591747 Google Scholar

[10] G. Darbo, Punti uniti in trasformazioni a codominio non compatto, Rend. Semin. Mat. Univ. Padova 24 (1955), 84–92. http://www.numdam.org/article/RSMUP_1955__24__84_0.pdf Google Scholar

[11] M. A. Darwish, K. Sadarangani, On Erd ́elyi-Kober type quadratic integral equation with linear modification of the argument, Appl. Math. Comput. 238 (2014), 30–42. https://doi.org/10.1016/j.amc.2014.04.002 Google Scholar

[12] D. Dhiman, L. N. Mishra, V. N. Mishra, Solvability of some non-linear functional integral equations via measure of noncompactness, Adv. Stud. Contemp. Math. 32 (2) (2022), 157–171. Google Scholar

[13] M. Didgar, A. R. Vahidi, Approximate solution of linear Volterra-Fredholm Integral equations and systems of Volterra-Fredholm integral equations using Taylor expansion method, Iran. J. Math. Sci. Inform. 15 (2) (2020), 31–50. http://ijmsi.ir/article-1-1131-en.html Google Scholar

[14] M. Z. Ge ̧cmen, E. C ̧ elik, Numerical solution of Volterra-Fredholm integral equations with Hosoya polynomials, Math. Methods Appl. Sci. 44 (14) (2021), 11166–11173. https://doi.org/10.1002/mma.7479 Google Scholar

[15] U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput. 218 (2011), 860–865. https://doi.org/10.1016/j.amc.2011.03.062 Google Scholar

[16] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, North-Holland, 2006. Google Scholar

[17] K. Kumar, L. Rathour, M. K. Sharma, V.N. Mishra, Fixed point approximation for suzuki generalized nonexpansive mapping using B(δ,μ) condition, Appl. Math. 13 (2) (2022), 215–227. https://doi.org/10.4236/am.2022.132017 Google Scholar

[18] K. Maleknejad, S. Sohrabi, Legendre polynomial solution of nonlinear Volterra-Fredholm integral equations, IUST Int. J. Eng. Sci. 19 (2008), 49–52. https://www.sid.ir/FileServer/JE/807200805-209 Google Scholar

[19] K. Maleknejad, M. R. F. Yami, A computational method for system of Volterra-Fredholm integral equations, Appl. Math. Comput. 183 (2006), 589–595. https://doi.org/10.1016/j.amc.2006.05.105 Google Scholar

[20] M. M. A. Metwali, V. N. Mishra, On the measure of noncompactness in Lp(R+) and applications to a product of n-integral equations, Turk. J. of Math. 47 (1) (2023), 372–386. https://doi.org/10.55730/1300-0098.3365 Google Scholar

[21] S. Micula, An iterative numerical method for Fredholm-Volterra integral equations of the second kind, Appl. Math. Comput. 270 (2015), 935–942. https://doi.org/10.1016/j.amc.2015.08.110 Google Scholar

[22] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, 1993. Google Scholar

[23] F. Mirzaee, E. Hadadiyan, Numerical solution of Volterra-Fredholm integral equations via modification of hat functions, Appl. Math. Comput. 280 (2016), 110–123. https://doi.org/10.1016/j.amc.2016.01.038 Google Scholar

[24] L. N. Mishra, V. K. Pathak, D. Baleanu, Approximation of solutions for nonlinear functional integral equations, AIMS Math. 7 (9) (2022) 17486–17506. https://doi.org/10.3934/math.2022964 Google Scholar

[25] V. N. Mishra, M. Raiz, N. Rao, Dunkl analouge of Sz ́asz Schurer Beta bivariate operators, Math. Found. Comput. 6 (4) (2023), 651–669. https://doi.org/10.3934/mfc.2022037 Google Scholar

[26] N. I. Muskhelishvili, Some basic problems of the mathematical theory of elasticity, P. Noordhoff, Groningen, Holland, 1953. Google Scholar

[27] V. K. Pathak, L. N. Mishra, On solvability and approximating the solutions for nonlinear infinite system of fractional functional integral equations in the sequence space lp, p > 1, J. Integral Equ. Appl. 35 (4) (2023), 443–458. https://doi.org/10.1216/jie.2023.35.443 Google Scholar

[28] V. K. Pathak, L. N. Mishra, Existence of solution of Erd ́elyi-Kober fractional integral equations using measure of non-compactness, Discontinuity Nonlinearity Complex. 12 (3) (2023), 701–714. https://doi.org/10.5890/dnc.2023.09.015 Google Scholar

[29] V. K. Pathak, L. N. Mishra, V. N. Mishra, On the solvability of a class of nonlinear functional integral equations involving Erd ́elyi-Kober fractional operator, Math. Methods Appl. Sci. 46 (2023), 14340–14352. https://doi.org/10.1002/mma.9322 Google Scholar

[30] V. K. Pathak, L. N. Mishra, V. N. Mishra, D. Baleanu, On the Solvability of Mixed-Type Fractional-Order Non-Linear Functional Integral Equations in the Banach Space C(I), Fractal Fract. 6 (12) (2022), 744. https://doi.org/10.3390/fractalfract6120744 Google Scholar

[31] S. K. Paul, L. N. Mishra, V. N. Mishra, D. Baleanu, An effective method for solving nonlinear integral equations involving the Riemann-Liouville fractional operator, AIMS Math. 8 (8) (2023), 17448–17469. https://doi.org/10.3934/math.2023891 Google Scholar

[32] S. K. Paul, L. N. Mishra, V. N. Mishra, Approximate numerical solutions of fractional integral equations using Laguerre and Touchard polynomials, Palestine J. Math. 12 (3) (2023), 416–431. https://tinyurl.com/5cp3x9ra Google Scholar

[33] S. K. Paul, L. N. Mishra, V. N. Mishra, D. Baleanu, Analysis of mixed type nonlinear Volterra-Fredholm integral equations involving the Erd ́elyi-Kober fractional operator, J. King Saud Univ. -Sci. 35 (10) (2023), 102949. https://doi.org/10.1016/j.jksus.2023.102949 Google Scholar

[34] M. Raiz, A. Kumar, V. N. Mishra, N. Rao, Dunkl analogue of Sz ́asz-Schurer-Beta operators and their approximation behaviour, Math. Found. Comput. 5 (4) (2022), 315–330. https://doi.org/10.3934/mfc.2022007 Google Scholar

[35] M. Raiz, R. S. Rajawat, V. N. Mishra, α-Schurer Durrmeyer operators and their approximation properties, Ann. Univ. Craiova Math. Comput. Sci. Ser. 50 (1) (2023), 189–204. https://doi.org/10.52846/ami.v50i1.1663 Google Scholar

[36] S. Sabermahani, Y. Ordokhani, S. A. Yousefi, Fibonacci wavelets and their applications for solving two classes of time-varying delay problems, Optim. Control Appl. Methods 41 (2) (2019), 395–416. https://doi.org/10.1002/oca.2549 Google Scholar

[37] S. Sabermahani, Y. Ordokhani, Fibonacci wavelets and Galerkin method to investigate fractional optimal control problems with bibliometric analysis, J. Vib. Control 27 (15-16) (2021), 1778– 1792. https://doi.org/10.1177/1077546320948346 Google Scholar

[38] A. G. Sanatee, L. Rathour, V. N. Mishra, V. Dewangan, Some fixed point theorems in regular modular metric spaces and application to Caratheodory’s type anti-periodic boundary value problem, J. Anal. 31 (2023), 619–632. https://doi.org/10.1007/s41478-022-00469-z Google Scholar

[39] P. Shahi, L. Rathour, V. N. Mishra, Expansive fixed point theorems for tri-simulation functions, J. Eng. Exact Sci. 8 (3) (2022), 14303-01e. https://doi.org/10.18540/jcecvl8iss3pp14303-01e Google Scholar

[40] H. M. Srivastava, F. A. Shah, N. A. Nayied, Fibonacci wavelet method for the solution of the non-linear Hunter-Saxton equation, Appl. Sci. 12 (15) (2022), 7738. https://doi.org/10.3390/app12157738 Google Scholar

[41] S. Verma, P. Viswanathan, Katugampola fractional integral and fractal dimension of bivariate functions, Results Math. 76 (2021), 165. https://doi.org/10.1007/s00025-021-01475-6 Google Scholar

[42] E. Yusufo ̆glu, B. Erba ̧s, Numerical expansion methods for solving Fredholm-Volterra type linear integral equations by interpolation and quadrature rules, Kybernetes 37 (6) (2008) 768–785. https://doi.org/10.1108/03684920810876972 Google Scholar