References
- Beck JV, Blackwell B, St CR. Clair, Inverse heat conduction: ill-posed problems. New York (NY): Wiley-Interscience; 1985.
- Murio DC, Paloschi JR. Combined mollification-future temperature procedure for solution of inverse heat conduction problem. J Comput Appl Math. 1988;23:235–244.
- Tadi M. Inverse heat conduction based on boundary measurement. Inverse Prob. 1997;13:1585–1605.
- Tikhonov AN, Arsenin VY. Solution of ill-posed problems. Washington (DC): V.H. Winston and Sons; 1977.
- Alifanov OM. Inverse heat transfer problems. New York (NY): Springer; 1994.
- Murio DA. The mollification method and the numerical solution of ill-posed problems. New York (NY): Wiley-Interscience; 1993.
- Pourgholi R, Rostamian M, Emamjome M. A numerical method for solving a nonlinear inverse parabolic problem. Inverse Prob Sci Eng. 2010;18(8):1151–1164.
- Beck JV, Murio DC. Combined function specification-regularization procedure for solution of inverse heat condition problem. AIAA J. 1986;24:180–185.
- Dehghan M, Yousefi SA, Rashdi K. Ritz-Galerkin method for solving an inverse heat conduction problem with a nonlinear source term via Bernstein multi-scaling functions and cubic B-spline functions. Inverse Prob Sci Eng. 2013;21:500–523.
- Mittal RC, Tripathi A. Numerical Solutions of Generalized Burgers--Fisher and Generalized Burgers--Huxley Equations using Collocation of cubic B-Splines. Int J Comput Math. 2015;92(5):1053–1077.
- Goh J, Majid AA, Ismail AIM. Numerical method using cubic B-spline for the heat and wave equation. Comput Math Appl. 2011;62:4492–4498.
- Mittal RC, Tripathi A. Numerical solutions of symmetric regularized long wave equations using collocation of cubic b-splines finite element. Int J Comput Methods Eng Sci Mech. 2015;16(2):142–150.
- Mittal RC, Tripathi A. Numerical solutions of two-dimensional Burger’s equations using modified Bi-cubic B-spline finite elements. Eng Comput. 2015;32(5):1275–1306.
- Mittal RC, Tripathi A. Numerical Solutions of Two-Dimensional Unsteady Convection-Diffusion Problems using Modified Bi-cubic B-Spline Finite Elements. Int J Comput Math. 2015:1–21. doi:10.1080/00207160.2015.1085976
- Mittal RC, Tripathi A. A Collocation method for numerical solutions of coupled Burgers equations. Int J Comput Methodsn Eng Sci Mech. 2014;15(5):457–471.
- Mittal RC, Arora G. Numerical solution of the coupled viscous Burger’s equation. Commun Nonlinear Sci Numer Simulat. 2011;16:1304–1313.
- Bothe D, Hilhorst D. A reaction-diffusion system with fast reversible reaction. J Math Anal Appl. 2003;286:125–135.
- Chipot M, Hilhorst D, Kinderlehrer D, et al. Contraction in L1 and large time behavior for a system arising in chemical reactions and molecular motors. Differ Equ Appl. 2009;1:139–151.
- Érdi P, Tóth J. Mathematical models of chemical reactions. Theory and applications of deterministic and stochastic models. Nonlinear science: theory and applications. Princeton (NJ): Princeton University Press; 1989.
- Hillen T, Painter KJ. A user’s guide to PDE models for chemotaxis. J Math Biol. 2009;58:183–217.
- Lauffenburger D, Aris R, Keller K. Effects of cell motility and chemotaxis on microbial population growth. Biophys J. 1982;40:209–219.
- Shigesada N, Kawasaki K, Teramoto E. Spatial segregation of interacting species. J Theor Biol. 1979;79:83–99.
- Ammar-Khodja F, Benabdallah A, González-Burgos M, de Teresa L. Recent results on the controllability of linear coupled parabolic problems: A survey. Math Control Related Fields. 2011;1:267–306.
- Ammar-Khodja F, Benabdallah A, Dupaix C. Null-controllability of some reaction-diffusion systems with one control force. J Math Anal Appl. 2006;320:928–943.
- Léautaud M. Spectral inequalities for non-selfadjoint elliptic operators and application to the nullcontrollability of parabolic systems. J Funct Anal. 2010;258:2739–2778.
- Fernández-Cara E, González-Burgos M, de Teresa L. Boundary controllability of parabolic coupled equations. J Funct Anal. 2010;259:1720–1758.
- Lions J-L. Exact control, disturbances and stabilization of distributed systems. Vol. 1, Research in applied mathematics. Paris: Masson; 1988.
- Lions J-C, Magenes E. Problems with non-homogeneous boundaries. II, Annals of the Fourier Institute. 1961;11:137–178.
- Lions J-C, Magenes E. Notes on boundary problems for parabolic operators. Comput Makes Acad Sci. (Paris). 1960;251:2118–2120.
- Lions J-C, Magenes E. Problems with non-homogeneous boundaries. IV, Annali della Scuola Normal Superiore di Pisa. 1961;15:311–326.
- Prenter PM. Splines and variational methods. New York (NY): JoahnWiley & Sons; 1975.
- Lawson CL, Hanson RJ. Solving least squares problems. Philadelphia (PA): SIAM; 1995.
- Tikhonov AN, Arsenin VY. On the solution of ill-posed problems. New York (NY): Wiley; 1977.
- Martin L, Elliott L, Heggs PJ, et al. Dual Reciprocity Boundary Element Method Solution of the Cauchy Problem for Helmholtz-type Equations with Variable Coefficients. J Sound Vibration. 2006;297:89–105.
- Elden L. A Note on the Computation of the generalized cross-validation function for ill-conditioned least squares problems. BIT. 1984;24:467–472.
- Golub GH, Heath M, Wahba G. Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics. 1979;21(2):215–223.
- Hall CA, Meyer WW. Optimal error bounds for cubic spline interpolation. J Approx Theory. 1976;16:105–122.
- Kadalbajoo MK, Gupta V, Awasthi A. A uniformly convergent Bspline collocation method on a nonuniform mesh for singularly perturbed one-dimensional time-dependent linear convection-diffusion problem. J Comput Appl Math. 2008;220:271–289.
- Smith GD. Numerical solution of partial differential equation. Finite difference method. Oxford: Clarendon Press; 1978.
- Atkinson KE. An introduction to numerical analysis. 2nd ed. New York (NY): Wiley; 1989.
- Limin Ma, Zongmin Wu. Radial basis functions method for parabolic inverse problem. Int J Cumput Math. 2011;88:384–395.
- Cabeza JMG, Garcia JAM, Rodriguez AC. A Sequential Algorithm of Inverse Heat Conduction Problems Using Singular Value Decomposition. Int J Therm Sci. 2005;44:235–244.
- Mei L, Chen Y. Explicit multistep method for the numerical solution of RLW equation. Appl Math Comput. 2012;218:9547–9554.