Tension spline-quasilinearization scheme for generalized nonlinear time-fractional convection-diffusion equations

Document Type : Research Article

Authors

Alemu Senbeta Bekela ¹

Lemi Guta Enyadene ²

Mesfin Mekuria Woldaregay ²

¹ Department of Applied Mathematics, Adama Science and Technology University, Adama,, Ethiopia.

² Department of Applied Mathematics, Adama Science and Technology University, Adama, Oromia, Ethiopia.

10.22067/ijnao.2026.96568.1780

Abstract

This article presents an efficient numerical technique based on non-polynomial tension spline for solving generalized nonlinear time fractional convection-diffusion equation with polynomial nonlinear convection terms. The proposed method uses non-polynomial tension spline spatial discretization with the Crank-Nicolson time-stepping scheme, and employs the quasi-linearization technique to handle the nonlinear term $p^{v}\frac{\partial p}{\partial x}$ for $v=1,2,...,10$. The tension spline introduces a controllable smoothness parameter, which can improve accuracy and reduce spurious oscillations. Additionally, the order of convergence for the suggested scheme is determined by analyzing the truncation error, and we theoretically presented the stability of the developed scheme using the von Neumann method. The method is shown to be conditionally stable. The method is validated on two representative test problems, showing good agreement with expected solutions and improved accuracy compared with existing numerical methods. These results illustrates the potential of the approach for solving nonlinear time-fractional convection-diffusion problems.

Keywords

Caputo’s fractional derivative

Crank-Nicolson scheme

Nonlinear time-fractional convection-diffusion equation

Quasi-linearization

Non-polynomial Tension spline

Subjects

• Numerical Analysis

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References

[1] Abolhasani, M., Abbasbandy, S. and Allahviranloo, T. A new variational iteration method
for a class of fractional convection-diffusion equations in large domains, Math. 5(26), (2017),
1–15.
[2] Ahmad, H., Farooq, M., Khan, I., Nawaz, R., Fewster-Young, N. and Askar, S. Analysis of
nonlinear fractional-order Fisher equation using two reliable techniques, Open Phys. 22(1),
(2024), 1–8.
[3] Akram, T., Abbas, M., Riaz, M.B., Ismail, A.I. and Ali, N.M. An efficient numerical
technique for solving time fractional Burgers equation, Alex. Eng. J. 59(4), (2020), 2201–
2220.
[4] Alam, K.N., Ara, A. and Mahmood, A. Numerical solutions of time-fractional Burgers equa-
tions: A comparison between generalized differential transformation technique and homotopy
perturbation method, Int. J. Numer. Methods Heat Fluid Flow, 22(2), (2012), 175–193.
[5] Atangana, A. Fractal-fractional differentiation and integration: connecting fractal calculus
and fractional calculus to predict complex system, Chaos Solitons Fractals. 102 (2017), 396–
406.
[6] Bekela, A.S., Belachew, M.T. and Wole, G.A. A numerical method using Laplace-like trans-
form and variational theory for solving time-fractional nonlinear partial differential equa-
tions with proportional delay, Adv. differ. equ. 2020(1), (2020), 1–19.
[7] Bekela, A.S. and Deresse, A.T. A hybrid yang transform Adomian decomposition method
for solving time-fractional nonlinear partial differential equation, BMC Res. Notes, 17(226),
(2024), 1–20.
[8] Bekela, A.S. and Deresse, A.T. An efficient numerical method for nonlinear time-fractional
hyperbolic partial differential equations based on fractional Shehu transform iterative method,
J. Appl. Math. 2025 (2025), 1–22.

[9] Bekela, A.S., Enyadene, L.G. and Woldaregay, M. M. Analysis and application of a hybrid
tension-spline method for nonlinear time-fractional reaction–diffusion equations, Discov.
Appl. Sci. 8(457), (2026), 1–32.
[10] Bekela, A.S. and Woldaregay, M.M. Formable transform Adomian decomposition method
for solving nonlinear time-fractional diffusion equation, Partial Differ. Equ. Appl. Math. 15
(2025), 1–10.
[11] Bellman, R.E. and Kalaba, R.E. Quasilinearization and nonlinear boundary-value problems,
Amer. Elsevier Publ. Co., New York, 3, 1965.
[12] Chawla, R., Deswal, K., Kumar, D. and Baleanu, D. Numerical simulation for general-
ized time-fractional Burgers’ equation with three distinct linearization schemes, J. Comput.
Nonlinear Dyn. 18(4), (2023), 1–16.
[13] Chawla, R., Kumar, D. and Singh, S. A second-order scheme for the generalized time-
fractional Burgers’ equation, J. Comput. Nonlinear Dyn. 19(1), (2024), 1–13.
[14] Choudhary, R., Singh, S. and Kumar, D. A second-order numerical scheme for the time-
fractional partial differential equations with a time delay, Comput. Appl. Math. 41(3),
(2022), 1–28.
[15] Delkhosh, M. Introduction of derivatives and integrals of fractional order and its applications,
Appl. Math. Phys. 1(4), (2013), 103–119.
[16] Deresse, A.T. and Bekela, A.S. A deep learning approach: physics-informed neural networks
for solving a nonlinear telegraph equation with different boundary conditions, BMC Res.
Notes, 18(77), (2025), 1–21.
[17] Deresse, A.T., Bekela, A.S. and Dufera, T.T. MT-PINNs: multi-term physics-informed
neural networks for solving initial boundary value problems of 2D and 3D nonlinear telegraph
equations, Bound. Value Probl. 2025(146), (2025), 1–29.
[18] Deresse, A.T. and Bekela, A.S. and Dufera, T.T. Leveraging advanced deep neural networks
algorithm for solving multi-dimensional forward and inverse problems of Klein-Gordon equa-
tion with quadratic, cubic, and fifth-degree polynomials nonlinearity, Comput. Math. Model.
36(3), (2025), 489–516.
[19] Du, M., Wang, Z. and Hu, H. Measuring memory with the order of fractional derivative,
Sci. Rep. 3(1), (2013), 1–3.

[20] Esen, A., Bulut, F. and Oruç, O. A unified approach for the numerical solution of time-
fractional Burgers’ type equations, Eur. Phys. J. Plus, 131(4), (2016), 1–13.
[21] Esen, A. and Tasbozan, O. Numerical solution of time-fractional Burgers equation, Acta
Univ. Sapientiae Math. 7(2), (2015), 167–185.
[22] Esen, A. and Tasbozan, O. Numerical solution of time-fractional Burgers equation by cubic
B-spline finite elements, Mediterr. J. Math. 13(3), (2016), 1325–1337.
[23] Fang, J., Nadeem, M., Habib, M., Karim, S. and Wahash, H.A. A new iterative method
for the approximate solution of Klein-Gordon and Sine-Gordon equations, J. Funct. Spaces,
2022(2), (2022), 1–9.
[24] Gowrisankar, S. and Natesan, S. An efficient robust numerical method for singularly per-
turbed Burgers’ equation, Appl. Math. Comput. 346(1), (2019), 385–394.
[25] Guan, J., Rahman, K. and Guo, Z. Radial basis function finite difference method for solving
the generalized time-fractional Burgers equation with three types of boundary conditions,
Chaos Solitons Fractals. 199(3), (2025), 116754.
[26] Hashmi, M.S., Wajiha, M., Yao, S., Ghaffar, A. and Inc, M. Cubic spline based differential
quadrature method: A numerical approach for fractional Burger equation, Results Phys. 26
(2021), 1–11.
[27] Inc, M. The approximate and exact solutions of the space-and time-fractional Burgers equa-
tions with initial conditions by variational iteration method, J. Math. Anal. Appl. 345(1),
(2008), 476–484.
[28] Jacobs, B.A. and Harley, C. Application of nonlinear time-fractional partial differential
equations to image processing via hybrid Laplace transform method, J. Math. 2018 (2018),
1–9.
[29] Kanth, A.S.V.R., Aruna, K., Raghavendar, K., Rezazadeh, H. and Inc, M. Numerical solu-
tions of nonlinear time fractional Klein-Gordon equation via natural transform decomposi-
tion method and iterative Shehu transform method, J. Ocean Eng. Sci. (2021).
[30] Kanth, A.S.V.R. and Sirswal, D. Analysis and numerical simulation for a class of time-
fractional diffusion equation via tension spline, Numer. Algor. 79(2), (2018), 479–497.

[31] Kumar, R. and Jain, S. Time fractional generalized Korteweg-de Vries equation: Explicit
series solutions and exact solutions, J. Frac. Calc. Nonlinear Sys. 1(1), (2021), 62–77.
[32] Li, D., Zhang, C. and Ran, M. A linear finite difference scheme for generalized time fractional
Burgers equation, Appl. Math. Model. 40 (11-12), (2016), 6069–6081.
[33] Majeed, A., Kamran, M. and Rafique, M. An approximation to the solution of time-fractional
modified Burgers’ equation using extended cubic B-spline method, Comput. Appl. Math.
39(4), (2020), 1–21.
[34] Naeem, M., Aljahdaly, N.H., Shah, R. and Weera, W. The study of fractional-order
convection–reaction-diffusion equation via an Elzake Atangana–Baleanu operator, AIMS
Math. 7(10), (2022), 18080–18098.
[35] Onal, M. and Esen, A. A Crank–Nicolson approximation for the time-fractional Burgers
equation, Appl. math. nonlinear sci. 5(2), (2020), 177–184.
[36] Oru, O., Esen, A. and Bulut, F. A unified finite difference Chebyshev wavelet method for
numerically solving time-fractional Burgers’ equation, Discrete Contin. Dyn. Syst., Ser. S.
12(3), (2019), 533–542.
[37] Podlubny, I. Fractional differential equations: an introduction to fractional derivatives,
fractional differential equations, to methods of their solution and some of their applications,
Elsevier, 1998.
[38] Poojitha, S. and Awasthi, A. Eloquent numerical approach for solving generalized time-
fractional convection-diffusion-reaction problems, Phys. Scr. 99(12), (2024), 1–19.
[39] Priyendhu, K.S., Prakash, P. and Lakshmanan, M. Invariant subspace method to the initial
and boundary value problem of the higher dimensional nonlinear time-fractional PDEs,
Commun. Nonlinear Sci. Numer. Simul. 122 (2023), 107245.
[40] Rubin, S.G. and Graves, J.R.A. Viscous flow solutions with a cubic spline approximation,
Comput. Fluids, 3(1), (1975), 1–36.
[41] Salama, F.M. An efficient explicit group method for time-fractional Burgers equation, Front.
Phys. 13, (2025), 1–18.
[42] Sugimoto, N. Burgers equation with a fractional derivative; hereditary effects on nonlinear
acoustic waves, J. Fluid Mech. 225 (1991), 631–653

[43] Vijayaram, S. and Balasubramaniam, P. Optimal control for nonlinear time-fractional
Schrudinger equation: an application to quantum optics, Phys. Scr. 99(9), (2024), 095115.
[44] Yousif, M.A. and Hamasalh, F.K. Novel simulation of the time-fractional Burgers-Fisher
equations using a nonpolynomial spline fractional continuity method, AIP Adv. 12(11),
(2022), 1–13.