Abstract
Studying convection–diffusion problems of delayed type in physics helps us to understand transport phenomena and has practical applications in various fields. The mathematical analysis of this model has practical applications in various fields, such as flow dynamics, material science, and environmental modeling. In this paper, the theory of reproducing kernel spaces (RKS) is utilized to solve singularly perturbed 2D parabolic convection–diffusion problems of delayed type. To this end, a series form for the solution is first constructed in reproducing kernel Hilbert space, and then, the approximate solution is given as an \(\mathcal {N}\)-term summation. The main contribution of the present research is that, for the first time, a novel formula is found for the homogenization of 2D initial-boundary-value problems. Furthermore, a semi-analytical RKS method is employed without employing the Gram–Schmidt orthogonalization algorithm. We derive theorems to reveal stability and convergence properties which are examined by numerical experiments. The technique is especially suited for problems having boundary-layer behavior. Numerical results are provided to demonstrate the efficiency, stability, and superiority of the proposed technique.
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References
Ahmad KN, Sulaiman M (2022) Heat transfer and thermal conductivity of magneto micropolar fluid with thermal non-equilibrium condition passing through the vertical porous medium. Waves in Random and Complex Media. https://doi.org/10.1080/17455030.2022.2108161
Ahmad KN, Sulaiman M, Alshammari FS (2022) Heat transfer analysis of an inclined longitudinal porous fin of trapezoidal, rectangular and dovetail profiles using cascade neural networks. Struct Multidiscip Optim 65:251
Ahmadi BP, Khoshsiar GR, Fardi M, Tavassoli KM (2024) Analysis of a kernel-based method for some pricing financial options. Comput Methods Differ Equ 12(1):16–30
Alarfaj FK, Ahmad KN, Sulaiman M, Alomair AM (2022) Application of a machine learning algorithm for evaluation of stiff fractional modeling of polytropic gas spheres and electric circuits. Symmetry. 14(12):2482
Ansari AR, Bakr SA, Shishkin GI (2007) A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations. J Comput Appl Math 205:552–566
Aronszajn N (1950) Theory of reproducing kernels. Trans Am Math Soc 68:337–404
Avijit D, Natesan S (2022) A novel two-step streamline-diffusion FEM for singularly perturbed 2D parabolic PDEs. Appl Numer Math 172:259–278
Azarnavid B (2023) The Bernoulli polynomials reproducing kernel method for nonlinear Volterra integro-differential equations of fractional order with convergence analysis. Computational and Applied Mathematics. 42(8), https://doi.org/10.1007/s40314-022-02148-y
Azarnavid B (2024) A kernel-based method for fractional integro-differential equations with a weakly singular kernel in multi-dimensional complex domains. Eng Anal Bound Elem 159:1–10
Babolian E, Javadi S, Moradi E (2016) Error analysis of reproducing kernel Hilbert space method for solving functional integral equations. J Comput Appl Math 300:300–311
Babu G, Prithvi M, Sharma KK, Ramesh VP (2022) A robust numerical algorithm on harmonic mesh for parabolic singularly perturbed convection-diffusion problems with time delay. Numerical Algorithms. 91(2):615–634
Bashier EB, Patidar KC (2011) A novel fitted operator finite difference method for a singularly perturbed delay parabolic partial differential equation. Appl Math Comput 217:4728–4739
Bertsimas D, Koduri N (2022) Data-driven optimization: a reproducing kernel Hilbert space approach. Oper Res 70(1):454–471
Bisiacco M, Pillonetto G (2020a) On the mathematical foundations of stable RKHSs. Automatica 118:109038
Bisiacco M, Pillonetto G (2020b) Kernel absolute summability is sufficient but not necessary for RKHS stability. SIAM J Control Optim. https://doi.org/10.1137/19M1278442
Chen T, Pillonetto G (2018) On the stability of reproducing kernel Hilbert spaces of discrete-time impulse responses. Automatica 95:529–533
Christmann A, Xiang D, Zhou D-X (2018) Total stability of kernel methods. Neurocomputing 289(10):101–118
Clavero C, Jorge JC, Lisbona F, Shishkin GI (1998) A fractional step method on a special mesh for the resolution of multidimensional evolutionary convection-diffusion problems. Appl Numer Math 27(3):211–231
Cui M, Lin Y (2009) Nonlinear Numerical Analysis in Reproducing Kernel Space. Nova Science Publishers, Inc
Das A, Natesan S (2015) Uniformly convergent hybrid numerical scheme for singularly perturbed delay parabolic convection-diffusion problems on Shishkin mesh. Appl Math Comput 271:168–186
Das A, Natesan S (2018) Second-order uniformly convergent numerical method for singularly perturbed delay parabolic partial differential equations. Int J Comput Math 95:490–510
Das A, Natesan S (2018) Fractional step method for singularly perturbed 2D delay parabolic convection diffusion problems on Shishkin mesh. Int J Appl Comput Math 4:1–23
Das A, Natesan S (2019) Parameter-uniform numerical method for singularly perturbed 2D delay parabolic convection-diffusion problems on Shishkin mesh. J Appl Math Comput 59:207–225
Falkovich G (2018) Fluid Mechanics. Cambridge University Press. ISBN 978-1-107-12956-6
Fardi M (2023) A kernel-based method for solving the time-fractional diffusion equation. Numerical Methods Partial Differential Equations. 39:2719–2733
Fardi M (2023) A kernel-based pseudo-spectral method for multi-term and distributed order time-fractional diffusion equations. Numer Methods Partial Differ Equ 39:2630–2651
Fardi M, Ghasemi M (2022) Numerical solution of singularly perturbed 2D parabolic initial - boundary - value problems based on reproducing kernel theory: Error and stability analysis. Numer Methods Partial Differ Equ 38:876–903
Fardi M, Al-Omari SKQ, Araci S (2022) A pseudo-spectral method based on reproducing kernel for solving the time-fractional diffusion-wave equation. Adv Contin Discr Model 254:1–14
Foroutan M, Ebadian A, Asadi R (2017) Reproducing kernel method in Hilbert spaces for solving the linear and nonlinear four-point boundary value problems. Int J Comput Math 95(10):1–15
Geng FZ, Qian SP (2014) Solving singularly perturbed multipantograph delay equations based on the reproducing kernel method. Abstract and Applied Analysis, Article ID, p 794716
Geng FZ, Qian SP (2015) Modified reproducing kernel method for singularly perturbed boundary value problems with a delay. Appl Math Model 39:5592–5597
Ghasemi M, Fardi M, Moradi E (2021) A reproducing kernel method for solving systems of integro-differential equations with nonlocal boundary conditions. Iran J Sci Technol Trans A: Sci 45:1375–1382
Gowrisankar S, Natesan S (2014) A robust numerical scheme for singularly perturbed delay parabolic initial-boundary-value problems on equidistributed grids. Electron Trans Numer Anal 41:376–395
Gowrisankar S, Natesan S (2017) \( \varepsilon \)-Uniformly convergent numerical scheme for singularly perturbed delay parabolic partial differential equations. Int J Comput Math 94:902–921
Khoroshun AS (2014) Using multicomponent Lyapunov functions to analyze the absolute parametric stability of singularly perturbed uncertain mechanical systems. Int Appl Mech 50:206–221
Kuang Y (ed) (1993) Delay differential equations: with applications in population dynamics. Academic Press, New York
Kumar D, Kumari P (2019) A parameter-uniform numerical scheme for the parabolic singularly perturbed initial boundary value problems with large time delay. J Appl Math Comput 59:179–206
Lü X, Cui M (2008) Analytic solutions to a class of nonlinear infinite-delay-differential equations. J Math Anal Appl 343:724–732
Mei L, Jia Y, Lin Y (2018) Simplified reproducing kernel method for impulsive delay differential equations. Appl Math Lett 83:123–129
Naidu DS, Calise AJ (2001) Singular perturbations and time scales in guidance and control of aerospace systems: a survey. J Guid Control Dyn 24:1057–1078
Natesan S, Ramanujam N (1999) A Booster method for singular perturbation problems arising in chemical reactor theory. Appl Math Comput 100:27–48
Nelson PW, Perelson AS (2002) Mathematical analysis of delay differential equation models of HIV-1 infection. Math Biosci 179:73–94
Priyadarshana S, Mohapatra J, Pattanaik SR (2023) A second order fractional step hybrid numerical algorithm for time delayed singularly perturbed 2D convection-diffusion problems. Appl Numer Math 189:107–129
Rai P, Yadav S (2021) Robust numerical schemes for singularly perturbed delay parabolic convection-diffusion problems with degenerate coefficient. Int J Comput Math 98:195–221
Shivhare M, Podila PC, Kumar D (2021) A uniformly convergent quadratic B-spline collocation method for singularly perturbed parabolic partial differential equations with two small parameters. J Math Chem 59:186–215
Sulaiman M, Ahmad KN (2023) Predictive modeling of oil and water saturation during secondary recovery with supervised learning. Phys Fluids 35:064110
Sulaiman M, Ahmad KN, Alshammari FS, Laouini G (2023) Performance of heat transfer in micropolar fluid with isothermal and isoflux boundary conditions using supervised neural networks. Mathematics 11(5):1173
Tuo R, He S, Pourhabib A, Ding Y, Huang JZ (2023) A reproducing kernel Hilbert space approach to functional calibration of computer models. J Am Stat Assoc 118(542):883–897
Villasana M, Radunskaya A (2003) A delay differential equation model for tumor growth. J Math Biol 47:270–294
Wang Y, Tian D, Li Z (2017) Numerical method for singularly perturbed delay parabolic partial differential equations. Therm Sci 21:1595–1599
Zhao T (1995) Global periodic-solutions for a differential delay system modeling a microbial population in the chemostat. J Math Anal Appl 193:329–352
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Balootaki, P.A., Ghaziani, R.K., Fardi, M. et al. A numerical investigation of singularly perturbed 2D parabolic convection–diffusion problems of delayed type based on the theory of reproducing kernels. Soft Comput (2024). https://doi.org/10.1007/s00500-023-09573-z
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DOI: https://doi.org/10.1007/s00500-023-09573-z