MLPG method for two-dimensional diffusion equation with Neumann's and non-classical boundary conditions

Alves, 2003, A modified element-free Galerkin method with essential boundary conditions enforced by an extended partition of unity finite element weight function, Internat. J. Numer. Methods Engrg., 57, 1523, 10.1002/nme.730 Atluri, 2004 Atluri, 2002 Atluri, 2002, The meshless local Petrov–Galerkin (MLPG) method: a simple and less costly alternative to the finite element and boundary element methods, Comput. Modeling Engrg. Sci., 3, 11 Atluri, 1998, A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics, Comput. Mech., 22, 117, 10.1007/s004660050346 Atluri, 1998, A new meshless local Petrov–Galerkin (MLPG) approach to nonlinear problems in computer modeling and simulation, Comput. Modeling Simulation Engrg., 3, 187 Batra, 2008, Free and forced vibrations of a segmented bar by a meshless local Petrov–Galerkin (MLPG) formulation, Comput. Mech., 41, 473, 10.1007/s00466-006-0049-6 Cannon, 1990, An inverse problem of finding a parameter in a semi-linear heat equation, J. Math. Anal. Appl., 145, 470, 10.1016/0022-247X(90)90414-B Cannon, 1993, The solution of the diffusion equation in two-space variables subject to the specification of mass, Appl. Anal., 50, 1, 10.1080/00036819308840181 Cannon, 1986, Diffusion subject to specification of mass, J. Math. Anal. Appl., 115, 517, 10.1016/0022-247X(86)90012-0 Capsso, 1988, A reaction–diffusion system arising in modeling man-environment diseases, Quart. Appl. Math., 46, 431, 10.1090/qam/963580 Chen, 2003, A coupled finite element and meshless local Petrov–Galerkin method for two-dimensional potential problems, Comput. Methods Appl. Mech. Engrg., 192, 4533, 10.1016/S0045-7825(03)00421-3 Cheng, 2008, Error estimates for the finite point method, Appl. Numer. Math., 58, 884, 10.1016/j.apnum.2007.04.003 Choi, 1992, A parabolic equation with nonlocal boundary conditions arising from electrochemistry, Nonlinear Anal. Theory Methods Appl., 18, 317, 10.1016/0362-546X(92)90148-8 Day, 1982, Existence of a property of solutions of the heat equation subject to linear thermoelasticity and other theories, Quart. Appl. Math., 40, 319, 10.1090/qam/678203 Day, 1983, A decreasing property of solutions of a parabolic equation with applications to thermoelasticity and other theories, Quart. Appl. Math., XLIV, 468, 10.1090/qam/693879 Dehghan, 2002, Second-order schemes for a boundary value problem with Neumann's boundary conditions, J. Comput. Appl. Math., 138, 173, 10.1016/S0377-0427(00)00452-0 Dehghan, 2009, Meshless Local Petrov–Galerkin (MLPG) method for the unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity, Appl. Numer. Math., 59, 1043, 10.1016/j.apnum.2008.05.001 Fairweather, 1991, The reformulation and numerical solution of certain nonclassical initial–boundary value problems, SIAM J. Sci. Statist. Comput., 12, 127, 10.1137/0912007 Lin, 1991, An inverse problem for a class of quasilinear parabolic equations, SIAM J. Math. Anal., 22, 146, 10.1137/0522009 Mazzia, 2007, A comparison of numerical integration rules for the meshless local Petrov–Galerkin method, Numer. Algorithms, 45, 61, 10.1007/s11075-007-9110-6 Mirzaei, 2010, Meshless local Petrov–Galerkin (MLPG) approximation to the two dimensional sine-Gordon equation, J. Comput. Appl. Math., 233, 2737, 10.1016/j.cam.2009.11.022 Mirzaei, 2010, A meshless based method for solution of integral equations, Appl. Numer. Math., 60, 245, 10.1016/j.apnum.2009.12.003 Nguyen, 2008, Meshless methods: A review and computer implementation aspects, Math. Comput. Simulation, 79, 763, 10.1016/j.matcom.2008.01.003 Qian, 2005, Three-dimensional transient heat conduction in a functionally graded thick plate with a higher-order plate theory and a meshless local Petrov–Galerkin method, Comput. Mech., 35, 214, 10.1007/s00466-004-0617-6 Sladek, 2004, Meshless local Petrov–Galerkin method for heat conduction problem in an anisotropic medium, CMES Comput. Model. Eng. Sci., 6, 309 Sladek, 2007, Heat conduction analysis of 3D axisymmetric and anisotropic FGM bodies by meshless local Petrov–Galerkin method, Comput. Mech., 39, 323, 10.1007/s00466-006-0031-3 Sladek, 2003, Local BIEM for transient heat conduction analysis in 3-D axisymmetric functionally graded solids, Comput. Mech., 32, 169, 10.1007/s00466-003-0470-z Sladek, 2008, Analysis of transient heat conduction in 3D anisotropic functionally graded solids by the MLPG, Comput. Modeling Engrg. Sci., 32, 161 Sladek, 2005, Transient heat conduction in anisotropic and functionally graded media by local integral equations, Engrg. Anal. Bound. Elem., 29, 1047, 10.1016/j.enganabound.2005.05.011 Sladek, 2009, Inverse fracture problems in piezoelectric solids by local integral equation method, Engrg. Anal. Bound. Elem., 33, 1089, 10.1016/j.enganabound.2009.02.009 Sladek, 2003, Transient heat conduction analysis in functionally graded materials by the meshless local boundary integral equation method, Comput. Mater. Sci., 28, 494, 10.1016/j.commatsci.2003.08.006 Sladek, 2006, Meshless local Petrov–Galerkin method for continuously nonhomogeneous linear viscoelastic solids, Comput. Mech., 37, 279, 10.1007/s00466-005-0715-0 Vavourakis, 2008, A MLPG (LBIE) numerical method for solving 2D incompressible and nearly incompressible elastostatic problems, Comm. Numer. Methods Engrg., 24, 281, 10.1002/cnm.965 Wang, 1990, The numerical method for the conduction subject to moving boundary energy specification, Numer. Heat Transfer, 130, 35 Wang, 2007, Analysis of microelectromechanical systems (mems) devices by the meshless point weighted least-squares method, Comput. Mech., 40, 1, 10.1007/s00466-006-0077-2 Wang, 1989, A finite difference solution to an inverse problem determining a control function in a parabolic partial differential equations, Inverse Problems, 5, 631, 10.1088/0266-5611/5/4/013 Wang, 1990, A numerical method for the diffusion equation with nonlocal boundary specifications, Internat. J. Engrg. Sci., 28, 543, 10.1016/0020-7225(90)90056-O Zuppa, 2003, Error estimates for moving least square approximations, Bull. Braz. Math. Soc. (N.S.), 34, 231, 10.1007/s00574-003-0010-7 Zuppa, 2003, Good quality point sets and error estimates for moving least square approximations, Appl. Numer. Math., 47, 575, 10.1016/S0168-9274(03)00091-6