Кешiгуi үлкен болатын Соболевтiң сингулярлы ауытқыған бастапқы-шекаралық есебiнiң S торындағы сандық сұлбасы
SOLVING SINGULARLY PERTURBED IBV SOBOLEV PROBLEMS WITH LARGE TIME DELAY USING S-MESH METHOD
DOI:
https://doi.org/10.26577/JMMCS.2023.v117.i1.08Кілттік сөздер:
Кешiктiрiлген дербес дифференциалдық теңдеу, ақырлы айырымдық әдiсi, Шишкин торы, сингулярлық ауытқу, Соболев есебiАннотация
Бұл мақаланың мақсаты кешiгуi бар сингулярлы ауытқыған Соболев типтi теңдеулердiң сандық әдiсiн беру болып табылады. Бiрiншiден, сингулярлы ауытқыған мен кешiгу параметрлерi бар Соболев есебiн шешудiң асимптотикалық бағалаулары алынды. Бұл бағалау шешiмнiң бастапқы деректерге тәуелдi екенiн көрсеттi. Бұл есептi нақты кесiндiлiк-бiртектi торда (Шишкин торында) шектi-айырымдық әдiсiмен шешу құрастырылған және зерттелген, оның шешiмi сингулярлық күйзелiс параметрiне қарамастан нүктелiк мағынада жинақталады. Айырымдық сұлбалардың орнықтылығын зерттеу үшiн дискреттi норма қолданылды. Толық дискреттi сұлбаның кеңiстiкте де және уақыт бойынша да, сонымен қатар, сингулярлы ауытқу параметрiне қарамастан O τ 2 + N −2 ln2 Nl ретімен жинақталатыны көрсетiлген. Соңында, сынақ есебiнiң және сандық тәжiрибелердiң көмегiмен ұсынылған әдiстердiң теориялық дәлдiгi мен есептеу тиiмдiлiгi қосымша расталды.
Библиографиялық сілтемелер
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[2] G. M. Amiraliyev, 1990, Difference Method for the Solution of One Problem of the Theory of Dispersive Waves, Differential Equations, 26, 2146-2154. (Russian)
[3] G. M. Amiraliyev, 1987, Investigation of the Difference Schemes for the Quasi-Linear Sobolev Equations, Differential Equations, V.23, No. 8, 1453-1455. (Russian)
[4] A.R. Ansari, S.A. Bakr, G.I. Shishkin, 2007, A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations, J. of Comput. and App. Math., 205, No.1 , 552-566.
[5] A. Barati Chiyaneh, H. Duru, 2019, ON Adaptive Mesh for the Initial-Boundary Value Singularly Perturbed Delay Sobolev Problems, Numerical Methods for Partial Differential Equations, DOI: 10.1002/num.22417.
[6] A. Barati Chiyaneh, H. Duru, 2019, Uniform Difference Method for Singularly Perturbed Delay Sobolev Problems on a Piecewise Uniform Mesh, Quaestiones Mathematicae, DOI:10.2989/16073606.2019.1653395.
[7] R. K. Bullough and P. J. Caudrey, 1980, Solitons, Springer-Verlag, New York, 1-13.
[8] G. V. Demidenko and S. V. Uspenskii, 1998, Equations and systems unsolved with respect to the highest derivative, Nauchnaya Kniga, Novosibirsk (Russian).
[9] E. P. Doolan, J. J. Miller, W. H. A. Schilders, 1980, Uniform Numerical Methods for
Problems with Initial and Boundary Layers, Boole Press. Dublin.
[10] H. Duru, 2004, Difference schemes for the singularly perturbed Sobolev periodic
boundary problem, App. Math. And Comput. 149: 187–201.
[11] I. E. Egorov, S. G. Pyatkov, and S. V. Popov, 2000, Non-classical differential-operator
equations, Nauka, Novosibirsk (Russian).
[12] R. E. Ewing, 1978, Time Stepping Galerkin Methods for Nonlinear Sobolev Partial Differential Equations, SIAM J. Numer Anal. 15, 1125-1150.
[13] A. Favini and A. Yagi, 1999, Degenerate differential equations in Banach spaces, Marcel
Dekker, New York.
[14] W. E. Ford, T. W. Ting, 1974, Uniform Error Estimates for Difference Approximations to Nonlinear Pseudo-Parabolic Partial Differential Equations, SIAM J. Numer. Anal. 11, 155-169.
[15] H. Gajewski, K. Groeger, and K. Zacharias, 1974, Nichtlineare Operatorgleichungen und Operator differentialgleichungen, Mathematische Lehrbu ̈cher und Monographien, Band 38, Akademie-Verlag, Berlin; Russian transl., Mir, Moscow 1978.
[16] H. Ikezi, 1978, Experiments on Solitons in Plasmas, Solitons in Action, Ed. K. Lonngren and A. Scott, Academic Press, 152-170.
[17] M. K. Kadalbajoo and Y. N. Reddy, 1989, Asymptotic and Numerical Analysis of Singular Perturbation Problems, A survey, App. Math. And Comput. 30:223-259.
[18] J. L. Langnese, 1972, General Boundary-Value Problems for Differential Equations of Sobolev Type, SIAM J. Math. Anal., v.3, 105-119.
[19] V. I. Lebedev, 1957, The Method of Difference for the Equations of Sobolev Type, Dokl. Acad. Sci. USSR, V.114, No. 6, 1166-1169.
[20] K. E. Lonngren, 1978, Observation of Solitons on Nonlinear Dispersive Transmission Lines, Soliton in Action, Academic Press. 127-148.
[21] A. A. Samarskii, 1983, Theory of Difference Schemes, 2ndEd., Nauka, Moscow.
[22] C. L. Sobolev, 1954, On a New Problems of Mathematical Physics, Izv. Acad. Sci. USSR, Mathematics, 18, No.1, 3-50.
[2] G. M. Amiraliyev, 1990, Difference Method for the Solution of One Problem of the Theory of Dispersive Waves, Differential Equations, 26, 2146-2154. (Russian)
[3] G. M. Amiraliyev, 1987, Investigation of the Difference Schemes for the Quasi-Linear Sobolev Equations, Differential Equations, V.23, No. 8, 1453-1455. (Russian)
[4] A.R. Ansari, S.A. Bakr, G.I. Shishkin, 2007, A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations, J. of Comput. and App. Math., 205, No.1 , 552-566.
[5] A. Barati Chiyaneh, H. Duru, 2019, ON Adaptive Mesh for the Initial-Boundary Value Singularly Perturbed Delay Sobolev Problems, Numerical Methods for Partial Differential Equations, DOI: 10.1002/num.22417.
[6] A. Barati Chiyaneh, H. Duru, 2019, Uniform Difference Method for Singularly Perturbed Delay Sobolev Problems on a Piecewise Uniform Mesh, Quaestiones Mathematicae, DOI:10.2989/16073606.2019.1653395.
[7] R. K. Bullough and P. J. Caudrey, 1980, Solitons, Springer-Verlag, New York, 1-13.
[8] G. V. Demidenko and S. V. Uspenskii, 1998, Equations and systems unsolved with respect to the highest derivative, Nauchnaya Kniga, Novosibirsk (Russian).
[9] E. P. Doolan, J. J. Miller, W. H. A. Schilders, 1980, Uniform Numerical Methods for
Problems with Initial and Boundary Layers, Boole Press. Dublin.
[10] H. Duru, 2004, Difference schemes for the singularly perturbed Sobolev periodic
boundary problem, App. Math. And Comput. 149: 187–201.
[11] I. E. Egorov, S. G. Pyatkov, and S. V. Popov, 2000, Non-classical differential-operator
equations, Nauka, Novosibirsk (Russian).
[12] R. E. Ewing, 1978, Time Stepping Galerkin Methods for Nonlinear Sobolev Partial Differential Equations, SIAM J. Numer Anal. 15, 1125-1150.
[13] A. Favini and A. Yagi, 1999, Degenerate differential equations in Banach spaces, Marcel
Dekker, New York.
[14] W. E. Ford, T. W. Ting, 1974, Uniform Error Estimates for Difference Approximations to Nonlinear Pseudo-Parabolic Partial Differential Equations, SIAM J. Numer. Anal. 11, 155-169.
[15] H. Gajewski, K. Groeger, and K. Zacharias, 1974, Nichtlineare Operatorgleichungen und Operator differentialgleichungen, Mathematische Lehrbu ̈cher und Monographien, Band 38, Akademie-Verlag, Berlin; Russian transl., Mir, Moscow 1978.
[16] H. Ikezi, 1978, Experiments on Solitons in Plasmas, Solitons in Action, Ed. K. Lonngren and A. Scott, Academic Press, 152-170.
[17] M. K. Kadalbajoo and Y. N. Reddy, 1989, Asymptotic and Numerical Analysis of Singular Perturbation Problems, A survey, App. Math. And Comput. 30:223-259.
[18] J. L. Langnese, 1972, General Boundary-Value Problems for Differential Equations of Sobolev Type, SIAM J. Math. Anal., v.3, 105-119.
[19] V. I. Lebedev, 1957, The Method of Difference for the Equations of Sobolev Type, Dokl. Acad. Sci. USSR, V.114, No. 6, 1166-1169.
[20] K. E. Lonngren, 1978, Observation of Solitons on Nonlinear Dispersive Transmission Lines, Soliton in Action, Academic Press. 127-148.
[21] A. A. Samarskii, 1983, Theory of Difference Schemes, 2ndEd., Nauka, Moscow.
[22] C. L. Sobolev, 1954, On a New Problems of Mathematical Physics, Izv. Acad. Sci. USSR, Mathematics, 18, No.1, 3-50.
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Как цитировать
CHIYANEH, A. B., & Duru, H. (2023). Кешiгуi үлкен болатын Соболевтiң сингулярлы ауытқыған бастапқы-шекаралық есебiнiң S торындағы сандық сұлбасы: SOLVING SINGULARLY PERTURBED IBV SOBOLEV PROBLEMS WITH LARGE TIME DELAY USING S-MESH METHOD. Қазұу Хабаршысы. Математика, механика, информатика сериясы, 117(1), 93–111. https://doi.org/10.26577/JMMCS.2023.v117.i1.08
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