A novel fitted numerical scheme for time-fractional singularly perturbed convection-diffusion problems with a delay in time via cubic $B$-spline approach

[1] W.T. Aniley, G.F. Duressa, Uniformly convergent numerical method for time-fractional convection-
diffusion equation with variable coefficients, Partial Differ. Equ. Appl. 8 (2023) 100592.

[2] R. Choudhary, D. Kumar, S. Singh, Second-order convergent scheme for time-fractional partial
differential equations with a delay in time, J. Math. Chem. 61 (2023) 21–46.
[3] R. Choudhary, S. Singh, D. Kumar, A second-order numerical scheme for the time-fractional partial
differential equations with a time delay, Comput. Appl. Math. 41 (2022) 114.
[4] I.T. Daba, G.F. Duressa, Extended cubic B-spline collocation method for singularly perturbed
parabolic differential-difference equation arising in computational neuroscience, Int. J. Numer.
Method. Biomed. Eng. 37 (2021) e3418.
[5] F.W. Gelu, G.F. Duressa, A uniformly convergent collocation method for singularly perturbed delay
parabolic reaction-diffusion problem, Abstr. Appl. Anal. 2021 (2021) 8835595 .
[6] C.A. Hall, On error bounds for spline interpolation, J. Approx. Theory 1 (1968) 209–218.
[7] M.K. Kadalbajoo, P. Arora, Fitted Collocation Method for Convection-diffusion Problems with Two
Small Parameters, Neural, Parallel Sci. Comput. 20 (2012) 133–152.
[8] I. Karatay, N. Kale, S. Bayramoglu, A new difference scheme for time fractional heat equations
based on the Crank-Nicholson method, Fract. Calc. Appl. Anal. 16 (2013) 892–910.
[9] D. Kumar, P. Kumari, A parameter-uniform numerical scheme for the parabolic singularly per-
turbed initial boundary value problems with large time delay, J. Appl. Math. Comput. 59 (2019)
179–206.
[10] K.P. Kumar, J. VigoAguiar, Numerical solution of time-fractional singularly perturbed convection-
diffusion problems with a delay in time, Math. Methods Appl. Sci. 44 (2021) 3080–3097.
[11] X. Liu, M. Abbas, H. Yang, X. Qin, T. Nazir, Novel finite point approach for solving time-fractional
convection-dominated diffusion equations, Adv. Differ. Equ. 2021 (2021) 4.
[12] E.A. Megiso, M.M. Woldaregay, T.G. Dinka, Fitted Tension Spline Method for Singularly Perturbed
Time Delay Reaction Diffusion Problems, Math. Probl. Eng. 2022 (2022) 8669718 .
[13] S.T. Mohyud-Din, T. Akram, M. Abbas, A.I. Ismail, N.H. Ali, A fully implicit finite difference
scheme based on extended cubic B-splines for time fractional advection-diffusion equation, Adv.
Differ. Equ. 2018 (2018) 109 .
[14] H. Naz, T. Dumrongpokaphan, T. Sitthiwirattham, H. Alrabaiah, K.J. Ansari, A numerical scheme
for fractional order mortgage model of economics, Results Appl. Math. 18 (2023) 100367.
[15] N.T. Negero, G.F. Duressa, Uniform convergent solution of singularly perturbed parabolic differ-
ential equations with general temporal-lag, Iran. J. Sci. Technol. Trans. Sci. 46 (2022) 507–524.
[16] N.T. Negero, G.F. Duressa, A method of line with improved accuracy for singularly perturbed
parabolic convection-diffusion problems with large temporal lag, Results Appl. Math. 11 (2021)
100174.

[17] N.T. Negero, G.F. Duressa, Parameter-uniform robust scheme for singularly perturbed parabolic
convection-diffusion problems with large time-lag, Comput. Methods Differ. Equ. 10 (2022) 954–
968.
[18] F.A. Rihan, Computational methods for delay parabolic and timefractional partial differential
equations, Numer. Methods Partial Differ. Equ. 26 (2010) 1556–1571.
[19] M. Sarboland, Numerical solution of time fractional partial differential equations using multi
quadric quasi-interpolation scheme, Eur. J. Comput. Mech. 27 (2018) 89–108.
[20] V. Saw, S. Kumar, Collocation method for time fractional diffusion equation based on the Cheby-
shev polynomials of second kind, Int. J. Appl. Comput. Math. 6 (2020) 117.
[21] M. Stynes, E. O’Riordan, J.L. Gracia, Error analysis of a finite difference method on graded meshes
for a time-fractional diffusion equation, SIAM J. Numer. Anal. 55 (2017) 1057–1079.
[22] N.H. Sweilam, M.M. Khader, A.M. Mahdy, Crank-Nicolson finite difference method for solving
time-fractional diffusion equation, J. Fract. Calc. Appl. 2 (2012) 1–9.
[23] S.K. Tesfaye, M.M. Woldaregay, T.G. Dinka, G.F. Duressa, Fitted computational method for solv-
ing singularly perturbed small time lag problem, BMC Res. Notes. 15 (2022) 318.
[24] Y. Ucar, N.M. Yagmurlu, O. Tasbozan, A. Esen, Numerical solution of some fractional partial
differential equations using collocation finite element method, Prog. Fract. Differ. Appl. 1 (2015)
157–164.
[25] J.M. Varah, A lower bound for the smallest singular value of a matrix, Linear Algebra Appl. 11
(1975) 3–5.
[26] M.M. Woldaregay, W.T. Aniley, G.F. Duressa, Novel numerical scheme for singularly perturbed
time delay convection-diffusion equation, Adv. Math. Phys. 2021 (2021) 6641236 .
[27] M. Yaseen, M. Abbas, M.B. Riaz, A collocation method based on cubic trigonometric B-splines for
the numerical simulation of the time-fractional diffusion equation, Adv. Differ. Equ. 2021 (2021)
210.