Numerical solutions of 3D Cauchy problems of elliptic operators in cylindrical domain using local weak equations and radial basis functions

References

• S. Atluri, The Meshless Method (MLPG) for Domain and BIE Discretizations, Tech Science Press, Encino, CA, 2004.
   Google Scholar
• S.N. Atluri and T. Zhu, A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics, Comput. Mech. 22(2) (1998), pp. 117–127. doi: 10.1007/s004660050346
   Web of Science ®Google Scholar
• S. Atluri and T. Zhu, A new meshless local Petrov–Galerkin (MLPG) approach to nonlinear problems in computer modeling and simulation, Comput. Model. Simul. Eng. 3 (1998), pp. 187–196.
   Google Scholar
• S. Atluri and S. Shen, The Meshless Local Petrov–Galerkin (MLPG) Method, Tech Science Press, Ecino, 2002.
   Google Scholar
• S.N. Atluri and S. Shen, The basis of meshless domain discretization: The meshless local Petrov–Galerkin (MLPG) method, Adv. Comput. Math. 23(12) (2005), pp. 73–93. doi: 10.1007/s10444-004-1813-9
   Web of Science ®Google Scholar
• F. Berntsson, V.A. Kozlov, L. Mpinganzima, and B.O. Turesson, An accelerated alternating procedure for the cauchy problem for the Helmholtz equation, Comput. Math. Appl. 68(1) (2014), pp. 44–60. doi: 10.1016/j.camwa.2014.05.002
   Web of Science ®Google Scholar
• A.L. Bukhgeim, J. Cheng, and M. Yamamoto, Uniqueness and stability for an inverse problem of determining a part of boundary, Inverse Probl. 15 (1999), pp. 1021–1032. doi: 10.1088/0266-5611/15/4/312
   Web of Science ®Google Scholar
• W. Chen and Z. Fu, Boundary particle method for inverse cauchy problems of inhomogeneous Helmholtz equations, J. Mar. Sci. Technol. 17(3) (2009), pp. 157–163.
   Web of Science ®Google Scholar
• M. Dehghan and R. Salehi, A method based on meshless approach for the numerical solution of the two-space dimensional hyperbolic telegraph equation, Math. Methods Appl. Sci. 35(10) (2012), pp. 1220–1233. doi: 10.1002/mma.2517
   Web of Science ®Google Scholar
• Z. Fu, W. Chen, and C. Zhang, Boundary particle method for cauchy inhomogeneous potential problems, Inv. Probl. Sci. Eng. 20(2) (2012), pp. 189–207. doi: 10.1080/17415977.2011.603085
   Web of Science ®Google Scholar
• J. Hadamard, Lectures on Cauchy Problems in Linear Partial Differential Equations, Yale University Press, New Haven, CT, 1923.
   Google Scholar
• P.C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev. 34 (1992), pp. 561–580. doi: 10.1137/1034115
   Web of Science ®Google Scholar
• P.C. Hansen, Regularization tools: A Matlab package for analysis and solution of discrete ill-posed problems, Numer. Algor. 6 (1994), pp. 1–35. doi: 10.1007/BF02149761
   Google Scholar
• Y.C. Hon and M. Li, A computational method for inverse free boundary determination problem, Int. J. Numer. Methods Eng. 73(9) (2008), pp. 1291–1309. doi: 10.1002/nme.2122
   Web of Science ®Google Scholar
• V.R. Hosseini, W. Chen, and Z. Avazzadeh, Numerical solution of fractional telegraph equation by using radial basis functions, Eng. Anal. Bound. Elem. 38 (2014), pp. 31–39. doi: 10.1016/j.enganabound.2013.10.009
   Web of Science ®Google Scholar
• S. Kazem, J.A. Rad, K. Parand, M. Shaban, and H. Saberi, The numerical study on the unsteady flow of gas in a semi-infinite porous medium using an RBF collocation method, Int. J. Comput. Math. 89(16) (2012), pp. 2240–2258. doi: 10.1080/00207160.2012.704995
   Web of Science ®Google Scholar
• C.L. Lawson and R.J. Hanson, Solving Least Squares Problems, SIAM, Philadelphia, PA, 1995.
   Google Scholar
• G.R. Liu and X. Han, Computational Inverse Techniques in Nondestructive Evaluation, CRC Press, London, 2003.
   Google Scholar
• L. Ma and Z. Wu, Radial basis functions method for parabolic inverse problem, Int. J. Comput. Math. 88(2) (2011), pp. 384–395.
   Web of Science ®Google Scholar
• R. Marklein, K. Mayer, R. Hannemann, T. Krylow, K. Balasubramanian, K.J. Langenberg, and V. Schmitz, Linear and nonlinear inversion algorithms applied in nondestructive evaluation, Inverse Probl. 18 (2002), pp. 2375–2400. doi: 10.1088/0266-5611/18/6/319
   Web of Science ®Google Scholar
• D. Mirzaei and M. Dehghan, A meshless based method for solution of integral equations, Appl. Numer. Math. 60(3) (2010), pp. 245–262. doi: 10.1016/j.apnum.2009.12.003
   Web of Science ®Google Scholar
• R. Rischette, T.N. Baranger, and N. Debit, Numerical analysis of an energy-like minimization method to solve a parabolic cauchy problem with noisy data, J. Comput. Appl. Math. 271 (2014), pp. 206–222. doi: 10.1016/j.cam.2014.03.024
   Web of Science ®Google Scholar
• R. Seieder and J. Trampert, Inverse Problems in Geophysics, Springer, New York, 1999.
   Google Scholar
• Q. Shen, Numerical solution of the Sturm–Liouville problem with local RBF-based differential quadrature collocation method, Int. J. Comput. Math. 88(2) (2011), pp. 285–295.
   Web of Science ®Google Scholar
• A. Shirzadi, Meshless local integral equations formulation for the 2d convection-diffusion equations with a nonlocal boundary condition, Comput. Model. Eng. Sci. 85(1) (2012), pp. 45–63.
   Web of Science ®Google Scholar
• A. Shirzadi and L. Ling, Convergent overdetermined-RBF-MLPG for solving second order elliptic PDEs, Adv. Appl. Math. Mech. 5(1) (2013), pp. 78–89.
   Web of Science ®Google Scholar
• A. Shirzadi, L. Ling, and S. Abbasbandy, Meshless simulations of the two-dimensional fractional-time convection-diffusion–reaction equations, Eng. Anal. Bound. Elem. 36 (2012), pp. 1522–1527. doi: 10.1016/j.enganabound.2012.05.005
   Web of Science ®Google Scholar
• A. Shirzadi, V. Sladek, and J. Sladek, A local integral equation formulation to solve coupled nonlinear reaction–diffusion equations by using moving least square approximation, Eng. Anal. Bound. Elem. 37(1) (2013), pp. 8–14. doi: 10.1016/j.enganabound.2012.08.007
   Web of Science ®Google Scholar
• A. Shirzadi, V. Sladek, and J. Sladek, A meshless simulations for 2d nonlinear reaction–diffusion Brusselator system, Comput. Model. Eng. Sci. 95(4) (2013), pp. 259–282.
   Web of Science ®Google Scholar
• J. Sladek, V. Sladek, and S.N. Atluri, Local boundary integral equation (LBIE) method for solving problem of elasticity with nonhomogeneous material properties, Comput. Mech. 24 (2000), pp. 456–462. doi: 10.1007/s004660050005
   Web of Science ®Google Scholar
• J. Sladek, V. Sladek, and C. Zhang, Stress analysis in anisotropic functionally graded materials by the MLPG method, Eng. Anal. Bound. Elem. 29(6) (2005), pp. 597–609. doi: 10.1016/j.enganabound.2005.01.011
   Web of Science ®Google Scholar
• L. Sun, W. Chen, and C. Zhang, A new formulation of regularized meshless method applied to interior and exterior anisotropic potential problems, Appl. Math. Model. 37(12–13) (2013), pp. 7452–7464. doi: 10.1016/j.apm.2013.02.036
   Web of Science ®Google Scholar
• Y. Sun, F. Ma, and D. Zhang, An integral equations method combined minimum norm solution for 3d elastostatics Cauchy problem, Comput. Methods Appl. Mech. Eng. 271 (2014), pp. 231–252. doi: 10.1016/j.cma.2013.12.013
   Web of Science ®Google Scholar
• A.N. Tikhonov and V.Y. Arsenin, On the Solution of Ill-posed Problems, Wiley, New York, 1977.
   Google Scholar
• Y. Wang and Y. Rudy, Application of the method of fundamental solution to potential-based inverse electrocardiography, Ann. Biomed. Eng. 34 (2006), pp. 1272–1288. doi: 10.1007/s10439-006-9131-7
   PubMed Web of Science ®Google Scholar
• T. Wei and Y.G. Chen, A regularization method for a cauchy problem of Laplace's equation in an annular domain, Math. Comput. Simul. 82(11) (2012), pp. 2129–2144. doi: 10.1016/j.matcom.2012.05.009
   Web of Science ®Google Scholar
• G. Yao, C.H. Tsai, and W. Chen, The comparison of three meshless methods using radial basis functions for solving fourth-order partial differential equations, Eng. Anal. Bound. Elem. 34(7) (2010), pp. 625–631. doi: 10.1016/j.enganabound.2010.03.004
   Web of Science ®Google Scholar
• T. Zhu, J.-D. Zhang, and S.N. Atluri, A local boundary integral equation (LBIE) method in computational mechanics and a meshless discretization approach, Comput. Mech. 21 (1998), pp. 223–235. doi: 10.1007/s004660050297
   Web of Science ®Google Scholar