Alternating direction method for a two‐dimensional parabolic equation with a nonlocal boundary condition

References

• Alefeld , G. 1976 . On the convergence of the Peaceman‐Rachford iterative method . Numer. Math. , 26 : 409 – 419 .
   Web of Science ®Google Scholar
• Alefeld , G. 1978 . “ Some convergence results for the Peaceman‐Rachford method in the noncommutative case ” . In Numerical Treatment of Differential Equations, Proc, Oberwolfach, 1976 , Edited by: Dold , A. and B&Eckma , B. 1 – 9 . Berlin Heidelberg New York : Lecture Notes in Mathematics, Springer‐Verlag .
   Google Scholar
• Cannon , J.R. , Lin , Y. and Matheson , A.L. 1993 . Locally explicit schemes for three‐dimensional diffusion with non‐local boundary specification . Appl. Anal. , 50 : 1 – 19 .
   Google Scholar
• Ciegis , R. 2004 . “ Finite‐difference schemes for nonlinear parabolic problem with nonlocal boundary conditions ” . In Integral Methods in Science and Engineering: Analytic and Numerical Techniques , Edited by: Ahues , M. , Constanda , C. and Largillier , A. 47 – 52 . Boston : Birkhauser .
   Google Scholar
• Čiegis , R. September 1–5 2006 . “ Parallel LOD scheme for 3D parabolic problem with nonlocal boundary condition ” . In Lecture notes in computer science, International conference EurPar2006 Edited by: Nagel , W.E. September 1–5 , Dresden, Germany Revised Papers., volume 4128. Springer, 679–688, 2006
   Google Scholar
• Čiegis , R. 2006 . Parallel numerical algorithms for 3D parabolic problem with nonlocal boundary condition . Informatica , 17 (3) : 309 – 324 .
   Web of Science ®Google Scholar
• Čiegis , R. , Štikonas , A. , Štikoniene , O. and Suboč , O. 2001 . Stationary problems with nonlocal boundary conditions . Mathematical Modelling and Analysis , 6 (2) : 178 – 191 .
   Google Scholar
• Čiegis , R. , Štikonas , A. , Štikoniene , O. and Suboč , O. 2002 . A monotonic finite‐difference scheme for a parabolic problem with nonlocal conditions . Differential Equations , 38 (7) : 1027 – 1037 .
   Web of Science ®Google Scholar
• Ciegis , R. 2005 . Economical difference schemes for the solution of a two‐dimensional parabolic problem with an integral condition . Diff. Equations , 41 (7) : 1025 – 1029 .
   Web of Science ®Google Scholar
• Čiegis , R. Numerical schemes for 3D parabolic problem with non‐local boundary condition . Proceedings of 17th IMACS World congress, Scientific Computation, Applied Mathematics and Simulation . June 11–15 2005 , Paris, France.
   Google Scholar
• Dehghan , M. 1999 . Alternating direction implicit methods for two‐dimensional diffusion with non‐local boundary condition . International Journal of Computer Mathematics , 72 (3) : 349 – 366 .
   Web of Science ®Google Scholar
• Dehghan , M. 1999 . Implicit locally one‐dimensional methods for two‐dimensional diffusion with non‐local boundary condition . Mathematics and Computers in Simulation , 4
   Google Scholar
• Dehghan , M. 2002 . A new ADI technique for two‐dimensional parabolic equation with an integral condition . Computers and Mathematics with Applications , 43 (12) : 1477 – 1488 .
   Web of Science ®Google Scholar
• Dehghan , M. 2003 . Fractional step methods for parabolic equations with a non‐standard conditions . Applied Mathematics and Computation , 142 (1) : 177 – 187 .
   Web of Science ®Google Scholar
• Dehghan , M. 2004 . Locally explicit schemes for three‐dimensional diffusion with a non‐local boundary specification . Applied Mathematics and Computation , 138 (2–3) : 489 – 501 .
   Web of Science ®Google Scholar
• Gulin , A.V. , Ionkin , N.I. and Morozova , V.A. 2001 . Stability of a nonlocal two‐dimensional finite‐difference problem . Diff. Equations , 37 (7) : 960 – 978 .
   Web of Science ®Google Scholar
• Gulin , A. V. , Ionki , N.I. and Morozova , V.A. 2006 . Study of the norm in stability problems for nonlocal difference schemes . Diff. Equations , 42 (7) : 974 – 984 .
   Web of Science ®Google Scholar
• Gulin , A.V. and Morozova , V.A. 2003 . On the stability of a nonlocal finite‐difference boundary value problem . Diff. Equations , 39 (7) : 962 – 967 .
   Web of Science ®Google Scholar
• Noye , B.J. and Dehghan , M. 1994 . A time‐splitting finite difference method for two‐dimensional diffusion with integral condition . Communications in Numerical Methods in Engineering , 10 : 649 – 660 .
   Web of Science ®Google Scholar
• Peaceman , D. and Rachford , J.H.H. 1955 . The numerical solution of parabolic and elliptic differential equations . J. Soc. Indust. Appl. Math. , 1 (3) : 28 – 41 .
   Google Scholar
• Samarskii , A. 2001 . The Theory of Difference Schemes , New York Basel : Marcel Dekker Inc. .
   Google Scholar
• Samarskii , A.A. and Nikolaev , E.S. 1989 . Numerical Methods for Grid Equations , volume 1 , Basel, Boston : Birkhauser Verlag .
   Google Scholar
• Sapagovas , M.P. 2002 . The eigenvalues of some problem with a nonlocal condition . Differential equations , 38 (7) : 1020 – 1026 .
   Web of Science ®Google Scholar
• Sapagovas , M.P. and Štikonas , A.D. 2005 . On the structure of the spectrum of a differential operator with a nonlocal condition . Diff. Equations , 41 (7) : 1010 – 1018 .
   Web of Science ®Google Scholar
• Voevodin , V.V. and Kuznecov , Ju.A. 1984 . Matricy i Vychislenija , Moskva : Nauka . (in Russian)
   Google Scholar