In this paper, we introduce the Müntz--Kajani wavelet. We also study how to approximate a function using these wavelets and provide an error bound. We then use a collocation method based on the Müntz--Kajani wavelet to numerically solve a system of fractional differential equations. The accuracy of the numerical solution is also assessed. To show how well the method works in practice, we apply it to investigate how HIV infects CD4$^+$T cells. Numerical results are also presented to demonstrate the efficiency and Cauchy convergence of the proposed method.
[1] M. Aghashahi and M. R. Gandomani, Numerical solution of Fractional Differential Equation system using the Müntz-Legender Polynomials, Int. J. Pure Appl. Math., 115(3) (2017), 467–475.
[2] S. Akhlaghi, M. Tavassoli Kajani, and M. Allame, Numerical Solution of Fractional Order Integro-Differential Equations via Müntz Orthogonal Functions, J. Math., 2023 (2023), 6647128, 1–13.
[3] S. Akhlaghi, M. Tavassoli Kajani, and M. Allame, Application of Müntz Orthogonal Functions on the Solution of the Fractional Bagley-Torvik Equation Using Collocation Method with Error Estimate, Adv. Math. Phys., 2023 (2023), 5520787, 1–11.
[4] R. Amin, S. Yuzbasi, and S. Nazir, Efficient Numerical Scheme for the Solution of HIV Infection CD4+T-Cells Using Haar Wavelet Technique, CMES-Comput. Model. Eng. Sci., 130(3) (2022), 639–653.
[5] H. D. Azodi and M. R. Yaghouti, A new method based on fourth kind Chebyshev wavelets to a fractional order model of HIV infection of CD4+T cells, Comput. Meth. Differ. Equ., 6(3) (2018), 353–371.
[6] A. Babaei, H. Jafari, and S. Banihashemi, Numerical solution of variable order fractional non-linear quadratic integro-differential equations based on the sixth-kind Chebyshev collocation method, J. Comput. Appl. Math., 377 (2020), 112908.
[7] D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional calculus, in: Series on Complexity, Nonlinearity and Chaos, Vol. 3, World Scientific Publishing Co. Pvt. Ltd., Hackensack, NJ, 2012.
[8] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods. Fundamentals in Single Domains, Springer, Berlin Heidelberg New York, 2006.
[9] B. Celik, A. O. Akdemir, E. Set, and S. Aslan, Ostrowski-Mercer Type Integral Inequalities For Differentiable Convex Functions Via Atangana-Baleanu Fractional Integral Operators, TWMS J. Pure Appl. Math., 15(2) (2024), 269–285.
[10] S. Erman, A. Demir, and E. Ozbilge, Solving inverse non-linear fractional differential equations by generalized Chelyshkov wavelets, Alexandria Eng. J., 66 (2023), 947–956.
[11] N. R. Gande and H. Madduri, Higher order numerical schemes for the solution of fractional delay differential equations, J. Comput. Appl. Math., 402 (2022), 113810.
[12] M. R. Gandomani and M. T. Kajani, Numerical Solution of a Fractional Order Model of HIV Infection of CD4+T Cells Using Müntz-Legendre Polynomials, Int. J. Bioautomation, 20(2) (2016), 193–204.
[13] M. R. Gandomani and M. Tavassoli Kajani, Application of shifted Müntz-Legendre Polynomials for solving fractional differential equations, Int. J. Pure Appl. Math., 103(2) (2015), 263–279.
[14] P. Goel, D. Singh, D. Gopal, and H. Aydi, F-Kannan Mappings In Non-Triangular Metric Spaces And Applications To Integral And Fractional Calculus, TWMS J. Pure Appl. Math., 16(1) (2025), 19–43.
[15] E. Gokmen, B. Gurbuz, and M. Sezer, A numerical technique for solving functional integro-differential equations having variable bounds, Comput. Appl. Math., 37 (2018), 5609–5623.
[16] S. Irandoust-pakchin, S. Abdi-mazraeh, and H. Kheiri, Construction of new generating function based on linear barycentric rational interpolation for numerical solution of fractional differential equations, J. Comput. Appl. Math., 375 (2020), 112799.
[17] D. Jain, A. Bhargava, and S. Gupta, A New Approach to Population Growth Model Involving a Logistic Differential Equation of Fractional Order, Crit. Rev. Biomed. Eng., 53(2) (2025), 37–48.
[18] R. Katani, S. Shahmorad, and D. Conte, Approximate solution of multi-term fractional differential equations via a block-by-block method, J. Comput. Appl. Math., 453 (2025), 116135.
[19] J. Li, X. Su, and K. Zhao, Barycentric interpolation collocation algorithm to solve fractional differential equations, Math. Comput. Simul., 205 (2023), 340–367.
[20] M. R. Lemnaouar and K. Hattaf, On a class of fractal-fractional differential equations with generalized fractal derivatives and non-singular kernels: a theoretical and numerical study, Nonlinear Dyn., 113 (2025), 13061–13079.
[21] X. Liu and M. Jia, A class of iterative functional fractional differential equation on infinite interval, Appl. Math. Lett., 136 (2023), 108473.
[22] M. Maleki and M. Tavassoli Kajani, Numerical approximations for Volterra’s population growth model with fractional order via a multi-domain pseudospectral method, Appl. Math. Model., 39 (2015), 4300–4308.
[23] H. R. Marzban, A generalization of Müntz-Legendre polynomials and its implementation in optimal control of non-linear fractional delay systems, Chaos Solitons Fractals, 158 (2022), 112093.
[24] M. Merdan, A. Gokdogan, and A. Yildirim, On the Numerical Solution of the Model for HIV Infection of CD4+T Cells, Comput. Math. Appl., 62(1) (2011), 118–123.
[25] G. V. Milovanovi´c, Mu¨ntz orthogonal polynomials and their numerical evaluation, Applications and Computation of Orthogonal Polynomials, Springer, Oberwolfach, Germany, 1998, 179–194.
[26] M. Y. Ongun, The Laplace Adomian Decomposition Method for Solving a Model for HIV Infection of CD4+T Cells, Math. Comput. Model., 53(6) (2011), 597–603.
[27] O. Postavaru and A. Toma, Numerical solution of two-dimensional fractional-order partial differential equations using hybrid functions, Partial Differ. Equ. Appl. Math., 4 (2021), 100099.
[28] M. Pourbabaee and A. Saadatmandi, A new operational matrix based on Müntz-Legendre polynomials for solving distributed order fractional differential equations, Math. Comput. Simul., 194 (2022), 210–235.
[29] A. Roohollahi, B. Ghazanfari, and S. Akhavan, Numerical solution of the mixed Volterra-Fredholm integro-differential multi-term equations of fractional order, J. Comput. Appl. Math., 376 (2020), 112828.
[30] L. Sadek, S. Yuzbasi, and H. T. Alaoui, Two numerical solutions for solving linear and nonlinear systems of differential equations, Appl. Comput. Math. Int. J., 23(4) (2024), 421–436.
[31] M. Saffarian and A. Mohebbi, Numerical solution of two and three-dimensional time-fractional damped nonlinear Klein-Gordon equation using ADI spectral element method, Appl. Math. Comput., 405 (2021), 126182.
[32] D. Shirani, M. Tavassoli Kajani, and S. Salahshour, Numerical Solution of a SIR Fractional Model of the Distribution of Computer Viruses Using Dickson Polynomials, Int. J. Ind. Math., 13(3) (2021), 323–331.
[33] M. Tavassoli Kajani, Numerical solution of fractional pantograph equations via Müntz-Legendre polynomials, Math. Sci., 18 (2024), 387–395.
[34] S. N. Tural-Polat and A. T. Dincel, Numerical solution method for multi-term variable order fractional differential equations by shifted Chebyshev polynomials of the third kind, Alexandria Eng. J., 61(7) (2022), 5145–5153.
[35] Vivek, M. Kumar, and S. N. Mishra, A fast Fibonacci wavelet-based numerical algorithm for the solution of HIV-infected CD4+T cells model, Eur. Phys. J. Plus, 138(5) (2023), 458.
[36] H. Y. Xu, M. M. Khalsaraei, A. Shokri, H. Ramos, and M. Molayi, Nonstandard finite difference schemes with positivity property with application to a mathematical model of HIV infection, Appl. Comput. Math. Int. J., 23(4) (2024), 541–557.
[37] C. Yang, Improved spectral deferred correction methods for fractional differential equations, Chaos Solitons Fractals, 168 (2023), 113204.
[38] X. Ye, J. Cao, and C. Xu, A High Order Scheme for Fractional Differential Equations with the Caputo-Hadamard Derivative, J. Comput. Math., 43(3) (2025), 615–640.
[39] S. Yuzbasi, A numerical approach to solve the model for HIV infection of CD4+T cells, Appl. Math. Model., 36(12) (2012), 5876–5890.
Kareem Mnahi, A. , Tavassoli Kajani, M. , Jasim Mohammed, M. and Allame, M. (2026). Introducing the Müntz-Kajani wavelet and its application in the numerical solution of an HIV infection of $CD4^+T$ cells model. Computational Methods for Differential Equations, 14(4), 2000-2016. doi: 10.22034/cmde.2025.66682.3140
MLA
Kareem Mnahi, A. , , Tavassoli Kajani, M. , , Jasim Mohammed, M. , and Allame, M. . "Introducing the Müntz-Kajani wavelet and its application in the numerical solution of an HIV infection of $CD4^+T$ cells model", Computational Methods for Differential Equations, 14, 4, 2026, 2000-2016. doi: 10.22034/cmde.2025.66682.3140
HARVARD
Kareem Mnahi, A., Tavassoli Kajani, M., Jasim Mohammed, M., Allame, M. (2026). 'Introducing the Müntz-Kajani wavelet and its application in the numerical solution of an HIV infection of $CD4^+T$ cells model', Computational Methods for Differential Equations, 14(4), pp. 2000-2016. doi: 10.22034/cmde.2025.66682.3140
CHICAGO
A. Kareem Mnahi , M. Tavassoli Kajani , M. Jasim Mohammed and M. Allame, "Introducing the Müntz-Kajani wavelet and its application in the numerical solution of an HIV infection of $CD4^+T$ cells model," Computational Methods for Differential Equations, 14 4 (2026): 2000-2016, doi: 10.22034/cmde.2025.66682.3140
VANCOUVER
Kareem Mnahi, A., Tavassoli Kajani, M., Jasim Mohammed, M., Allame, M. Introducing the Müntz-Kajani wavelet and its application in the numerical solution of an HIV infection of $CD4^+T$ cells model. Computational Methods for Differential Equations, 2026; 14(4): 2000-2016. doi: 10.22034/cmde.2025.66682.3140
)