[1] Hernández D I, Rueda-Gómez D A, Villamizar-Roa É J. An optimal control problem for a Lotka-Volterra competition model with chemo-repulsion. Acta Math Sci, 2024, 44B(2): 721-751
[2] Zhou Z J, Wang F Y, Zheng X C.Analysis and discretization for an optimal control problem of a variable-coefficient Riesz-fractional diffusion equation with pointwise control constraints. Acta Math Sci, 2023, 43B(2): 640-654
[3] Chen B, Timoshin S A.Optimal control of a population dynamics model with hysteresis. Acta Math Sci, 2022, 42B(1): 283-298
[4] Lions J L.Optimal Control of Systems Governed by Partial Differential Equations. Berlin: Springer-Verlag, 1971
[5] Tröltzsch F.Optimal Control of Partial Differential Equations: Theory, Method and Applications. Providence, RI: American Mathematical Society, 2010
[6] Hinze M, Pinnau R, Ulbrich M, et al.Optimization with PDE Constraints. Dordrecht: Springer-Verlag, 2009
[7] Liu W B, Yan N N.Adaptive Finite Element Methods for Optimal Control Governed by PDEs. Beijing: Science Press, 2008
[8] Fu H F, Rui H X.A priori error estimates for optimal control problems governed by transient advection-diffusion equations. J Sci Comput, 2009, 38(3): 290-315
[9] Meidner D, Vexler B.A priori error estimates for space-time finite element discretization of parabolic optimal control problems part I: problems without control constraints. SIAM J Control Optim, 2008, 47(3): 1150-1177
[10] Yang C J, Sun T J.Crank-Nicolson finite difference schemes for parabolic optimal Dirichlet boundary control problems. Math Method Appl Sci, 2022, 45(12): 7346-7363
[11] Marzban H R, Hoseini S M.A composite Chebyshev finite difference method for nonlinear optimal control problems. Commun Nonlinear Sci Numer Simulat, 2013, 18(6): 1347-1361
[12] Chen Y P, Yi N Y, Liu W B.A Legendre-Galerkin spectral method for optimal control problems governed by elliptic equations. SIAM J Numer Anal, 2008, 46(5): 2254-2275
[13] Chen Y P, Huang F L, Yi N Y, et al.A Legendre-Galerkin spectral method for optimal control problems governed by Stokes equations. SIAM J Numer Anal, 2011, 49(4): 1625-1648
[14] Rost M, Krug J.Coarsening of surface structures in unstable epitaxial growth. Phys Rev E, 1997, 55(4): 3952-3957
[15] Qin D D, Du Y W, Liu B, et al. A B-spline finite element method for nonlinear differential equations describing crystal surface growth with variable coefficient. Adv Differ Equ, 2019, 2019: Art 175
[16] Duan N, Zhao X P.Optimal control for a higher-order nonlinear parabolic equation describing crystal surface growth. Nonlinear Anal-Model, 2018, 23(2): 251-268
[17] Du F F, Sun T J.A cubic B-spline finite element method for optimal control problem governed by nonlinear parabolic equation describing crystal surface growth. IAENG Int J Appl Math, 2022, 52(3): 754-759
[18] Valdman J.MATLAB implementation of C1 finite elements: Bogner-Fox-Schmit rectangle// Wyrzykowski R, Deelman E, Dongarra J, et al. Parallel Processing and Applied Mathematics. Switzerland: Springer, 2019: 256-266
[19] Wang Y, Meng X Y, Li Y H. The Bogner-Fox-Schmit element finite volume methods on the Shishkin mesh for fourth-order singularly perturbed elliptic problems. J Sci Comput, 2022, 93(4): Art 4
[20] Ciarlet P G.The Finite Element Method for Elliptic Problems. Philadelphia: SIAM, 2002
[21] Mohanty R K, Jain M K, Dhall D.High accuracy cubic spline approximation for two dimensional quasi-linear elliptic boundary value problems. Appl Math Model, 2013, 37(1): 155-171
[22] Höllig K, Hörner J.Programming finite element methods with weighted B-splines. Comput Math Appl, 2015, 70(7): 1441-1456
[23] Gardner L R T, Gardner G A. A two dimensional bi-cubic B-spline finite element: used in a study of MHD-duct flow. Comput Methods Appl Mech Engrg, 1995, 124(4): 365-375
[24] Höllig K.Finite Element Methods with B-Splines. Philadelphia: SIAM, 2003
[25] Adams R A, Fournier J J F. Sobolev Spaces. 2nd ed. New York: Academic Press, 2003 |