The stability and convergence analysis for singularly perturbed Sobolev problems with Robin type boundary condition

References

[1] E. Abreu and A. Duran, Error estimates for semidiscrete Galerkin and collocation approximations to pseudo-parabolic problems with Dirichlet conditions, preprint (2020), https://arxiv.org/abs/2002.10813. Search in Google Scholar

[2] E. Abreu and A. Duran, On the use of spectral discretizations with time strong stability preserving properties to Dirichlet pseudo-parabolic problems, preprint (2020), https://arxiv.org/abs/2002.10811. Search in Google Scholar

[3] G. M. Amiraliyev and Y. D. Mamedov, Difference schemes on the uniform mesh for singular perturbed pseudo-parabolic equations, Turkish J. Math. 19 (1995), no. 3, 207–222. Search in Google Scholar

[4] A. R. Ansari, S. A. Bakr and G. I. Shishkin, A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations, J. Comput. Appl. Math. 205 (2007), no. 1, 552–566. 10.1016/j.cam.2006.05.032Search in Google Scholar

[5] K. Bansal, P. Rai and K. K. Sharma, Numerical treatment for the class of time dependent singularly perturbed parabolic problems with general shift arguments, Differ. Equ. Dyn. Syst. 25 (2017), no. 2, 327–346. 10.1007/s12591-015-0265-7Search in Google Scholar

[6] K. Bansal and K. K. Sharma, Parameter uniform numerical scheme for time dependent singularly perturbed convection-diffusion-reaction problems with general shift arguments, Numer. Algorithms 75 (2017), no. 1, 113–145. 10.1007/s11075-016-0199-3Search in Google Scholar

[7] A. Barati Chiyaneh and H. Duru, On adaptive mesh for the initial boundary value singularly perturbed delay Sobolev problems, Numer. Methods Partial Differential Equations 36 (2020), no. 2, 228–248. 10.1002/num.22417Search in Google Scholar

[8] A. Barati Chiyaneh and H. Durus, Uniform difference method for singularly pertubated delay Sobolev problems, Quaest. Math. 43 (2020), no. 12, 1713–1736. 10.2989/16073606.2019.1653395Search in Google Scholar

[9] T. A. Bullo, G. F. Duressa and G. A. Degla, Robust finite difference method for singularly perturbed two-parameter parabolic convection-diffusion problems, Int. J. Comput. Methods 18 (2021), no. 2, Paper No. 2050034. 10.1142/S0219876220500346Search in Google Scholar

[10] M. Chandru, P. Das and H. Ramos, Numerical treatment of two-parameter singularly perturbed parabolic convection diffusion problems with non-smooth data, Math. Methods Appl. Sci. 41 (2018), no. 14, 5359–5387. 10.1002/mma.5067Search in Google Scholar

[11] C. Clavero and J. L. Gracia, A high order HODIE finite difference scheme for 1D parabolic singularly perturbed reaction-diffusion problems, Appl. Math. Comput. 218 (2012), no. 9, 5067–5080. 10.1016/j.amc.2011.10.072Search in Google Scholar

[12] C. Clavero and J. C. Jorge, Another uniform convergence analysis technique of some numerical methods for parabolic singularly perturbed problems, Comput. Math. Appl. 70 (2015), no. 3, 222–235. 10.1016/j.camwa.2015.04.006Search in Google Scholar

[13] P. Das, A higher order difference method for singularly perturbed parabolic partial differential equations, J. Difference Equ. Appl. 24 (2018), no. 3, 452–477. 10.1080/10236198.2017.1420792Search in Google Scholar

[14] H. Duru, Difference schemes for the singularly perturbed Sobolev periodic boundary problem, Appl. Math. Comput. 149 (2004), no. 1, 187–201. 10.1016/S0096-3003(02)00965-7Search in Google Scholar

[15] L. Govindarao and J. Mohapatra, A second order numerical method for singularly perturbed delay parabolic partial differential equations, Eng. Comput. 36 (2019), no. 2, 420–444. 10.1108/EC-08-2018-0337Search in Google Scholar

[16] S. Gowrisankar and S. Natesan, 𝜀-Uniformly convergent numerical scheme for singularly perturbed delay parabolic partial differential equations, Int. J. Comput. Math. 94 (2017), no. 5, 902–921. 10.1080/00207160.2016.1154948Search in Google Scholar

[17] B. Gunes and H. Duru, A second-order difference scheme for the singularly perturbed Sobolev problems with third type boundary conditions on Bakhvalov mesh, J. Difference Equ. Appl. 28 (2022), no. 3, 385–405. 10.1080/10236198.2022.2043289Search in Google Scholar

[18] V. Gupta, M. K. Kadalbajoo and R. K. Dubey, A parameter-uniform higher order finite difference scheme for singularly perturbed time-dependent parabolic problem with two small parameters, Int. J. Comput. Math. 96 (2019), no. 3, 474–499. 10.1080/00207160.2018.1432856Search in Google Scholar

[19] M. K. Kadalbajoo and A. Awasthi, A parameter uniform difference scheme for singularly perturbed parabolic problem in one space dimension, Appl. Math. Comput. 183 (2006), no. 1, 42–60. 10.1016/j.amc.2006.05.023Search in Google Scholar

[20] M. K. Kadalbajoo and A. S. Yadaw, Parameter-uniform finite element method for two-parameter singularly perturbed parabolic reaction-diffusion problems, Int. J. Comput. Methods 9 (2012), no. 4, Article ID 1250047. 10.1142/S0219876212500478Search in Google Scholar

[21] D. Kumar and P. Kumari, A parameter-uniform numerical scheme for the parabolic singularly perturbed initial boundary value problems with large time delay, J. Appl. Math. Comput. 59 (2019), no. 1–2, 179–206. 10.1007/s12190-018-1174-zSearch in Google Scholar

[22] S. Kumar and M. Kumar, High order parameter-uniform discretization for singularly perturbed parabolic partial differential equations with time delay, Comput. Math. Appl. 68 (2014), no. 10, 1355–1367. 10.1016/j.camwa.2014.09.004Search in Google Scholar

[23] J. E. Lagnese, General boundary value problems for differential equations of Sobolev type, SIAM J. Math. Anal. 3 (1972), 105–119. 10.1137/0503013Search in Google Scholar

[24] A. D. Lipitakis, The numerical solution of singularly perturbed nonlinear partial differential equations in three space variables: The adaptive explicit inverse preconditioning approach, Model. Simul. Eng. 2019 (2019), Article ID 5157145. 10.1155/2019/5157145Search in Google Scholar

[25] J. D. Mamedov, S. Aširov and S. Atdaev, Theorems on Inequalities (in Russian), “Ylym”, Ashkhabad, 1980. Search in Google Scholar

[26] N. A. Mbroh, S. C. O. Noutchime and R. Y. M. Massoukou, A robust method of lines solution for singularly perturbed delay parabolic problem, Alexandria Eng. J. 59 (2020), no. 4, 2543–2554. 10.1016/j.aej.2020.03.042Search in Google Scholar

[27] J. Mohapatra and D. Shakti, Numerical treatment for the solution of singularly perturbed pseudo-parabolic problem on an equidistributed grid, Nonlinear Eng. 9 (2020), no. 1, 169–174. 10.1515/nleng-2020-0006Search in Google Scholar

[28] K. Mukherjee and S. Natesan, Richardson extrapolation technique for singularly perturbed parabolic convection-diffusion problems, Computing 92 (2011), no. 1, 1–32. 10.1007/s00607-010-0126-8Search in Google Scholar

[29] J. B. Munyakazi, A robust finite difference method for two-parameter parabolic convection-diffusion problems, Appl. Math. Inf. Sci. 9 (2015), no. 6, 2877–2883. Search in Google Scholar

[30] J. B. Munyakazi and K. C. Patidar, A fitted numerical method for singularly perturbed parabolic reaction-diffusion problems, Comput. Appl. Math. 32 (2013), no. 3, 509–519. 10.1007/s40314-013-0033-7Search in Google Scholar

[31] E. O’Riordan, M. L. Pickett and G. I. Shishkin, Parameter-uniform finite difference schemes for singularly perturbed parabolic diffusion-convection-reaction problems, Math. Comp. 75 (2006), no. 255, 1135–1154. 10.1090/S0025-5718-06-01846-1Search in Google Scholar

[32] R. N. Rao and P. P. Chakravarthy, A fitted Numerov method for singularly perturbed parabolic partial differential equation with a small negative shift arising in control theory, Numer. Math. Theory Methods Appl. 7 (2014), no. 1, 23–40. 10.4208/nmtma.2014.1316nmSearch in Google Scholar

[33] H.-G. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion-Reaction and Flow Problems, 2nd ed., Springer Ser. Comput. Math. 24, Springer, Berlin, 2008. Search in Google Scholar

[34] A. A. Salama and H. Z. Zidan, Fourth-order schemes of exponential type for singularly perturbed parabolic partial differential equations, Rocky Mountain J. Math. 36 (2006), no. 3, 1049–1068. 10.1216/rmjm/1181069445Search in Google Scholar

[35] A. A. Samarskii, The Theory of Difference Schemes, Monogr. Textb. Pure Appl. Math. 240, Marcel Dekker, New York, 2001. 10.1201/9780203908518Search in Google Scholar

[36] M. Sharma, A robust numerical approach for singularly perturbed time delayed parabolic partial differential equations, Differ. Equ. Dyn. Syst. 25 (2017), no. 2, 287–300. 10.1007/s12591-016-0280-3Search in Google Scholar

[37] M. K. Singh and S. Natesan, Richardson extrapolation technique for singularly perturbed system of parabolic partial differential equations with exponential boundary layers, Appl. Math. Comput. 333 (2018), 254–275. 10.1016/j.amc.2018.03.059Search in Google Scholar

[38] S. L. Sobolev, On a new problem of mathematical physics, Izv. Akad. Nauk SSSR. Ser. Mat. 18 (1954), 3–50. Search in Google Scholar

[39] Y. Wang, L. Su, X. Cao and X. Li, Using reproducing kernel for solving a class of singularly perturbed problems, Comput. Math. Appl. 61 (2011), no. 2, 421–430. 10.1016/j.camwa.2010.11.019Search in Google Scholar

[40] Y. Wang, D. Tian and Z. Li, Numerical method for singularly perturbed delay parabolic partial differential equations, Thermal Sci. 21 (2017), no. 4, 1595–1599. 10.2298/TSCI160615040WSearch in Google Scholar

[41] M. M. Woldaregay and G. F. Duressa, Parameter uniform numerical method for singularly perturbed parabolic differential difference equations, J. Nigerian Math. Soc. 38 (2019), no. 2, 223–245. Search in Google Scholar

[42] S. Yadav and P. Rai, A higher order scheme for singularly perturbed delay parabolic turning point problem, Eng. Comput. 38 (2020), no. 2, 819–851. 10.1108/EC-03-2020-0172Search in Google Scholar

[43] W. K. Zahra, M. S. El-Azab and A. M. El Mhlawy, Spline difference scheme for two-parameter singularly perturbed partial differential equations, J. Appl. Math. Inform. 32 (2014), no. 1–2, 185–201. 10.14317/jami.2014.185Search in Google Scholar

[44] C. Zhang and Z. Tan, Linearized compact difference methods combined with Richardson extrapolation for nonlinear delay Sobolev equations, Commun. Nonlinear Sci. Numer. Simul. 91 (2020), Article ID 105461. 10.1016/j.cnsns.2020.105461Search in Google Scholar