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No transformation or linearization technique is used to handle nonlinearity and the obtained block tri-diagonal nonlinear system has been solved by Newton\u2019s block iteration method. It is discussed how our formulation is able to tackle linear singular problems and it is ensured that the methods retain their orders and accuracy everywhere in the solution region. The proposed two-level method is shown to be unconditionally stable for a class of two-dimensional fourth-order linear parabolic equation. We also discuss the alternating direction implicit (ADI) method for solving two-dimensional fourth-order linear parabolic equation. The proposed difference methods has been successfully tested on the two-dimensional vibration problem, Boussinesq equation, extended Fisher\u2013Kolmogorov equation and\nKuramoto\u2013Sivashinsky equation. Numerical results demonstrate that the schemes are highly accurate in solving a large class of physical problems.<\/jats:p>","DOI":"10.1515\/cmam-2016-0047","type":"journal-article","created":{"date-parts":[[2017,1,19]],"date-time":"2017-01-19T12:17:55Z","timestamp":1484828275000},"page":"617-641","source":"Crossref","is-referenced-by-count":9,"title":["High Accuracy Compact Operator Methods for Two-Dimensional Fourth Order Nonlinear Parabolic Partial Differential Equations"],"prefix":"10.1515","volume":"17","author":[{"given":"Ranjan Kumar","family":"Mohanty","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Faculty of Mathematics and Computer Science , South Asian University , Akbar Bhawan, Chanakyapuri , New Delhi 110021 , India"}]},{"given":"Deepti","family":"Kaur","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Mathematical Sciences , University of Delhi , Delhi 110007 , India"}]}],"member":"374","published-online":{"date-parts":[[2017,1,19]]},"reference":[{"key":"2023033116270865106_j_cmam-2016-0047_ref_001_w2aab3b7ab1b6b1ab1b8b1Aa","doi-asserted-by":"crossref","unstructured":"C.  Andrade and S.  McKee,\nHigh accuracy A.D.I. methods for fourth order parabolic equations with variable coefficients,\nJ. Comput. Appl. Math. 3 (1977), 11\u201314.\n10.1016\/0771-050X(77)90019-5","DOI":"10.1016\/0771-050X(77)90019-5"},{"key":"2023033116270865106_j_cmam-2016-0047_ref_002_w2aab3b7ab1b6b1ab1b8b2Aa","doi-asserted-by":"crossref","unstructured":"D. G.  Aronson and H. F.  Weinberger,\nMultidimensional nonlinear diffusion arising in population genetics,\nAdv. Math. 30 (1978), 33\u201376.\n10.1016\/0001-8708(78)90130-5","DOI":"10.1016\/0001-8708(78)90130-5"},{"key":"2023033116270865106_j_cmam-2016-0047_ref_003_w2aab3b7ab1b6b1ab1b8b3Aa","unstructured":"M. J.  Boussinesq,\nTheorie des ondes et des remous qui se propagent le long d\u2019un canal rectangular horizontal,\nJ. Math. Pures Appl. 17 (1872), 55\u2013108."},{"key":"2023033116270865106_j_cmam-2016-0047_ref_004_w2aab3b7ab1b6b1ab1b8b4Aa","doi-asserted-by":"crossref","unstructured":"S. D.  Conte,\nNumerical solution of vibration problems in two space variables,\nPacific J. Math. 7 (1957), 1535\u20131544.\n10.2140\/pjm.1957.7.1535","DOI":"10.2140\/pjm.1957.7.1535"},{"key":"2023033116270865106_j_cmam-2016-0047_ref_005_w2aab3b7ab1b6b1ab1b8b5Aa","doi-asserted-by":"crossref","unstructured":"S. H.  Crandall,\nAn optimum implicit recurrence formula for the heat conduction equation,\nQuart. Appl. Math. 13 (1955), 318\u2013320.\n10.1090\/qam\/73292","DOI":"10.1090\/qam\/73292"},{"key":"2023033116270865106_j_cmam-2016-0047_ref_006_w2aab3b7ab1b6b1ab1b8b6Aa","doi-asserted-by":"crossref","unstructured":"P.  Danumjaya and A. K.  Pani,\nOrthogonal cubic spline collocation method for the extended Fisher\u2013Kolmogorov equation,\nJ. Comput. Appl. Math. 174 (2005), 101\u2013117.\n10.1016\/j.cam.2004.04.002","DOI":"10.1016\/j.cam.2004.04.002"},{"key":"2023033116270865106_j_cmam-2016-0047_ref_007_w2aab3b7ab1b6b1ab1b8b7Aa","unstructured":"P.  Danumjaya and A. K.  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