References
- Adams AR. Sobolev spaces. New York (NY): Academic Press; 1975.
- Grisvard P. Elliptic problems in nonsmooth domains. London: Pitman; 1985.
- Verchota G. Layer potentials and regularity for the Dirichlet problems for Laplace's equation in Lipschitz domains. J Funct Anal. 1984;59:572–611. MR 86e:35038. doi: 10.1016/0022-1236(84)90066-1
- Ding Z, Zhou J. Constrained LQR problems governed by the potential equation on Lipschitz domain with point observations. J Math Pures Appl. 1995;74:317–344.
- Lions JL. Optimal control of systems governed by partial differential equations, 170. Band: Springer-Verlag; 1971.
- Lions JL, Magenes E. Non-homogeneous boundary value problem and applications, I. New York: Springer-Verlag; 1972.
- Rosch A, Troltzsch F. Sufficient second-order optimality conditions for an elliptic optimal control problem with pointwise control-state constraints. SIAM J Optim. 2006;17(3):776–794. doi: 10.1137/050625850
- Tolstonogov DA. About the minimum of the variational elliptic problems without the assumption of the convexity. Math Note. 1999;65(1):130–142.
- Bahaa GM. Optimal control for cooperative parabolic systems governed by Schrödinger operator with control constraints. IMA J Math Control Inform. 2007;24:1–12. doi: 10.1093/imamci/dnl001
- Bahaa GM. Optimal control problems of parabolic equations with an infinite number of variables and with equality constraints. IMA J Math Control Inform. 2008;25:37–48. doi: 10.1093/imamci/dnm002
- Bahaa GM, Kotarski W. Optimality conditions for n×n infinite order parabolic coupled systems with control constraints and general performance index. IMA J Math Control Inform. 2008;25:49–57. doi: 10.1093/imamci/dnm003
- Kotarski W, El-Saify HA, Bahaa GM. Optimal control of parabolic equation with an infinite number of variables for non-standard functional and time delay. IMA J Math Control Inform. 2002;19:461–476. doi: 10.1093/imamci/19.4.461
- Canuto C, Hussaini MY, Quarteroni A, et al. Spectral methods in fluid mechanics. New York (NY): Springer-Verlag; 1988.
- Bhrawy AH, Zaky MA. A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations. J Comput Phys. 2015;281:876–895. doi: 10.1016/j.jcp.2014.10.060
- Bhrawy AH, Zaky MA. Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations. Comput Math Appl. 2017;73:1100–1117. doi: 10.1016/j.camwa.2016.11.019
- Zaky MA, Machado JAT. On the formulation and numerical simulation of distributed-order fractional optimal control problems. Commun Nonlinear Sci Numer Simulat. 2017;52:177–189. doi: 10.1016/j.cnsns.2017.04.026
- Bhrawy AH, Zaky MA. An improved collocation method for multi-dimensional space-time variable-order fractional Schr"e;odinger equations. Appl Numer Math. 2017;111:197–218. doi: 10.1016/j.apnum.2016.09.009
- Auteri F, Parolini N, Quartapelle L. Essential imposition of Neumann Galerkin-Legendre elliptic solvers. J Comput Phys. 2003;185:427–444. doi: 10.1016/S0021-9991(02)00064-5
- Bhrawy AH, Zaky MA. Numerical simulation of multi-dimensional distributed-order generalized Schrödinger equations. Nonlinear Dyn. 2017;89:1415–1432. doi: 10.1007/s11071-017-3525-y
- Bhrawy AH, Zaky MA, Van Gorder RA. A space-time Legendre spectral tau method for the two-sided space-time Caputo fractional diffusion-wave equation. Numer Algor. 2016;71:151–180. doi: 10.1007/s11075-015-9990-9
- Livermore PW. Galerkin orthogonal polynomials. J Comp Phys. 2010;229:2046–2060. doi: 10.1016/j.jcp.2009.11.022
- Livermore PW, Ierley GR. Quasi-Lp norm orthogonal Galerkin expansions in sums of Jacobi polynomials. Numer Algor. 2010;54:533–569. doi: 10.1007/s11075-009-9353-5
- Doha EH, Bhrawy AH, Hafez RM. A Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-order differential equations. Math Comput Model. 2011;53:1820–1832. doi: 10.1016/j.mcm.2011.01.002
- Gheorghiu CI. Spectral methods for differential problems. Cluj-Napoca, Romaina: “T. Popoviciu”, Institute of Numerical Analysis; 2007.
- Doha EH, Bhrawy AH. Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials. Appl Numer Math. 2008;58:1224–1244. doi: 10.1016/j.apnum.2007.07.001
- Doha EH, Bhrawy AH, Abd-Elhameed WM. Jacobi spectral Galerkin method for elliptic Neumann problems. Numer Algorithms. 2009;50:67–91. doi: 10.1007/s11075-008-9216-5
- Doha EH, Bhrawy AH. An efficient direct solver for multidimensional elliptic Robin boundary value problems using a Legendre spectral-Galerkin method. Comput Math Appl. 2012. 64:558–517 doi:10.1016/j.camwa.2011.12.050.
- Luke Y. The special functions and their approximations, 2 New York (NY): Academic Press; 1969.
- Luke Y. Mathematical functions and their approximations. New York (NY): Academic Press; 1975.
- Szegö G. Orthogonal polynomials. Amer Math Soc Colloq Pub. 1985;23:1–440.
- Doha EH, Bhrawy AH. Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomials. Numer Algorithms. 2006;42:137–164. doi: 10.1007/s11075-006-9034-6
- Auteri F, Quartapelle L. Galerkin spectral method for the vorticity and stream function equations. J Comput Phys. 1999;149:306–332. doi: 10.1006/jcph.1998.6155