English version
ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Дифференциальные Уравнения
и
Процессы Управления

О журнале История Журнала Издатель Адреса Тематика Редакторский состав Рецензирование Для авторов Издательская этика Коллекция номеров English version

Singularly Perturbed Boundary-value Problems: Sundman-type Transformations, Test Problems, Exact Solutions, and Numerical Integration

Автор(ы):

Inna Konstantinovna Shingareva

Professor,
Philosophiæ Doctor (Ph.D.)
University of Sonora, Mexico

inna.shingareva@unison.mx

Andrei Dmitrievich Polyanin

Chief Researcher,
Doctor of Sciences (Phys&Math), Professor
Ishlinsky Institute for Problems in Mechanics RAS, Moscow

polyanin@ipmnet.ru

Аннотация:

Solutions of singularly perturbed boundary-value problems with a small parameter are characterized by large gradients in very narrow regions (boundary layers). This circumstance sharply limits the use of standard finite-difference methods with a fixed stepsize in such problems due to significant calculation errors or possible loss of stability. This paper presents an effective method for numerical integration of singularly perturbed boundary-value problems based on replacing the spatial variable with a new independent variable of the Sundman-type, which depends on the derivatives of the unknown function. The use of such non-local transformations, which satisfy a simple asymptotic condition, makes it possible to automatically stretch the boundary-layer region. The resulting problem turns out to be much simpler than the original one in the sense that standard (classical) numerical methods with a fixed stepsize can already be applied to solve it. Several new multiparameter nonlinear and linear singularly perturbed boundary-value problems for second-order reaction-diffusion type ODEs having monotonic and non-monotonic exact or asymptotic solutions, expressed in terms of elementary functions, are constructed. A comparison of numerical solutions with exact and asymptotic solutions is presented. The numerical results are shown that the method based on Sundman-type transformations for solving boundary-layer problems gives high accuracy. As a result of an extensive analysis of the obtained results, recommendations are given for the choice of regularizing functions that determine the most effective Sundman-type transformations. The difference between regularizing functions in boundary-layer problems and blow-up problems is discussed. The test problems formulated in this paper can be used to estimate the accuracy of any other numerical methods for solving two-point singularly perturbed boundary-value problems with a small parameter.

Ключевые слова

Ссылки:

  1. Van Dyke M. Perturbation Methods in Fluid Mechanics. Academic Press: New York, 1964.
  2. Schlichting H. Boundary Layer Theory. New York: McGraw-Hill, 1981.
  3. Polyanin, A. D., Kutepov, A. M., Vyazmin, A. V., Kazenin, D. A. Hydrodynamics, Mass and Heat Transfer in Chemical Engineering. Taylor & Francis: London, 2002.
  4. Levich V. G. Physicochemical Hydrodynamics. New Jersey: Prentice Hall, 1962.
  5. Schetz J. A. Foundations of Boundary Layer Theory for Momentum, Heat, and Mass Transfer. New York: Prentice Hall, 1984.
  6. Franz S., Roos H. G. The capriciousness of numerical methods for singular perturbations. SIAM Review, 2011, 53, 157-173.
  7. Bakhvalov N. S. On the optimization methods for solving boundary value problems with boundary layers. Zh. Vychisl. Math. Fiz., 1969, 24, 841-859 (in Russian).
  8. Il'in A. M. A difference scheme for a differential equation with a small parameter affecting the highest derivative. Mat. Zametki, 1969, 6, 237-248.
  9. Vulanovic R. A uniform numerical method for quasilinear singular perturbation problems without turning points. Computing, 1989, 41, 97-106.
  10. Jain M. K., Iyengar S. R. K., Subramanyam G. S. Variable mesh methods for the numerical solution of two-point singular perturbation problems. Comput. Methods in Appl. Mech. Eng., 1984, 42, 273-286.
  11. Shishkin G. I. Grid Approximations of Singularly Perturbed Elliptic and Parabolic Equations. Ural Branch of RAS: Ekaterinburg, 1992 (in Russian).
  12. Beckett G., Mackenzie J. A. Convergence analysis of finite difference approximations on equidistributed grids to a singularly perturbed boundary value problem. Appl. Numer. Math., 2000, 35, 87-109.
  13. Farrell P., Hegarty A., Miller J. M., O'Riordan E., Shishkin G. I. Robust Computational Techniques for Boundary Layers. Chapman & Hall/CRC Press: Boca Raton-London, 2000.
  14. Qiu Y., Sloan D. M., Tang T. Numerical solution of a singularly perturbed two-point boundary value problem using equidistribution, analysis of convergence. J. Comput. Appl. Math., 2000, 116, 121-143.
  15. Frohner A., Roos H. -G. The ε -uniform convergence of a defect correction method on a Shishkin mesh. Appl. Numerical Math., 2001, 37, 79-94.
  16. Miranker W. L. Numerical Methods for Stiff Equations and Singular Perturbation Problems. Reidel Publ. : Dordrecht, 2001.
  17. Aziz T., Khan A. A spline method for second-order singularly perturbed boundary-value problems. J. Comput. Appl. Math., 2002, 147, 445-452.
  18. Vigo-Aguiar J., Natesan S. An efficient numerical method for singular perturbation problems. J. Comput. Appl. Math., 2006, 192, 132-141.
  19. Kumar M., Kumar Mishra H., Singh P. A boundary value approach for a class of linear singularly perturbed boundary value problems. Advances Eng. Software, 2009, 40, 298-304.
  20. Rao S. C. S., Kumar M. Exponential B-spline collocation method for self-adjoint singularly perturbed boundary value problems. Appl. Numerical Math., 2008, 58, 1572-1581.
  21. Shishkin G. I., Shishkina L. P. Difference Methods for Singular Perturbation Problems. Chapman & Hall/CRC Press: Boca Raton, 2009.
  22. Kopteva N., O'Riordan E. Shishkin meshes in the numerical solution of singularly perturbed differential equations. Int. J. Numer. Analysis and Modeling, 2010, 7, 393-415.
  23. Vulkov L. G., Zadorin A. I. Two-grid algorithms for an ordinary second order equation with an exponential boundary layer in the solution. Int. J. Numer. Analysis and Modeling, 2010, 7, 580-592.
  24. Attili B. S. Numerical treatment of singularly perturbed two point boundary value problems exhibiting boundary layers. Commun. Nonlinear Sci. Numer. Simulat., 2011, 16, 3504-3511.
  25. Liu C. -S. The Lie-group shooting method for solving nonlinear singularly perturbed boundary value problems. Commun. Nonlinear~Sci. Numer. Simulat., 2012, 17, 1506-1521.
  26. Das P. Comparison of a priori and a posteriori meshes for singularly perturbed nonlinear parameterized problems. J. Comput. Appl. Math., 2015, 290, 16-25.
  27. Brdar M., Zarin H. A singularly perturbed problem with two parameters on a Bakhvalov-type mesh. J. Comput. Appl. Math., 2016, 292, 307-319.
  28. Zarin H. Exponentially graded mesh for a singularly perturbed problem with two small parameters. Appl. Numer. Math., 2017, 120, 233-242.
  29. Ahmadinia M., Safari Z. Numerical solution of singularly perturbed boundary value problems by improved least squares method. J. Comput. Appl. Math., 2018, 331, 156-165.
  30. Chakravarthya P. P., Shivharea M. Numerical study of a singularly perturbed two parameter problems on a modified Bakhvalov mesh. Comput. Math. Math. Physics, 2020, 60, 1778-1786.
  31. Alam M. J., Prasad H. S., Ranjan R. An exponentially fitted integration scheme for a class of quasilinear singular perturbation problems. J. Math. Comput. Sci., 2021, 11, 3052-3066.
  32. Cakir M., Amiraliyev G. M. A second order numerical method for singularly perturbed problem with non-local boundary condition. J. Appl. Math. Comput., 2021, doi:10. 1007/s12190-021-01506-z.
  33. Shivhare M., Chakravarthya P. P., Kumar K. Quadratic B-spline collocation method for two-parameter singularly perturbed problem on exponentially graded mesh. Int. Comput. Math., 2021, doi:10. 1080/00207160. 2021. 1901277.
  34. Liseikin V. Layer Resolving Grids and Transformations for Singular Perturbation Problems. VSP BV: Utrecht, 2001 (reprint ed., de Gruyter, 2018).
  35. Polyanin A. D., Shingareva I. K. Application of non-local transformations for numerical integration of singularly perturbed boundary-value problems with a small parameter. Int. J. Non-Linear Mechanics, 2018, 103 37-54.
  36. Polyanin A. D., Shingareva I. K. The method of nonlocal transformations: Applications to singularly perturbed boundary-value problems. J. Physics: IOP Conf. Series, 2019, 1205, 012047.
  37. Polyanin A. D., Shingareva I. K. The use of differential and non-local transformations for numerical integration of non-linear blow-up problems. Int. J. Non-Linear Mechanics, 2017, 94, 178-184.
  38. Polyanin A. D., Shingareva I. K. Non-monotonic blow-up problems: Test problems with solutions in elementary functions, numerical integration based on non-local transformations. Appl. Math. Letters, 2018, 76, 123-129.
  39. Polyanin A. D., Shingareva I. K. Non-linear problems with non-monotonic blow-up solutions: Non-local transformations, test problems, exact solutions, and numerical integration. Int. J. Non-Linear Mechanics, 2018, 99, 258-272.
  40. Polyanin A. D., Shingareva I. K. Nonlinear problems with blow-up solutions: Numerical integration based on differential and nonlocal transformations, and differential constraints. Appl. Math. Comput., 2019, 336, 107-137.
  41. Polyanin A. D., Shingareva I. K. The method of non-local transformations: Applications to blow-up problems. J. Physics: IOP Conf. Series, 2017, 937, 012042.
  42. Polyanin A. D., Shingareva I. K. Non-linear blow-up problems for systems of ODEs and PDEs: Non-local transformations, numerical and exact solutions. Int. J. Non-Linear Mechanics, 2019, 111, 28-41.
  43. Polyanin A. D., Shingareva I. K. Hypersingular nonlinear boundary-value problems with a small parameter. Appl. Math. Letters, 2018, 81, 93-98.
  44. Kudryashov N. A., Sinelshchikov D. I. On the criteria for integrability of the Lienard equation. Appl. Math. Letters, 2016, 57, 114-120.
  45. Demina M., Sinelshchikov D. Integrability properties of cubic Li\'enard oscillators with linear damping. Symmetry, 2019, 11, 1378.
  46. Muriel C., Romero J. L. Nonlocal transformations and linearization of second-order ordinary differential equations. J. Physics A, Math. Theor., 2010, 43, 434025.
  47. Moyo S., Meleshko S. V. Application of the generalised Sundman transformation to the linearisation of two second-order ordinary differential equations. J. Nonlinear Math. Physics, 2011, 18, 213-236.
  48. Meleshko S. V., Moyo S., Muriel C., Romero J. L., Guha P., Choudhury A. G. On first integrals of second-order ordinary differential equations. J. Eng. Math., 2013, 82, 17-30.
  49. Kevorkian J., Cole J. D. Perturbation Methods in Applied Mathematics. Springer: New York, 1981.
  50. Lagerstrom P. A. Matched Asymptotic Expansions. Ideas and Techniques. Springer: New York, 1988.
  51. Il'in A. M. Matching of Asymptotic Expansions of Solutions of Boundary Value Problems. American Mathematical Society: Providence, 1992.
  52. Nayfeh A. H. Perturbation Methods. Wiley-Interscience: New York, 2000.
  53. Polyanin A. D., Zaitsev V. F. Handbook of Exact Solutions for Ordinary Differential Equations, 2nd ed. Chapman & Hall/CRC Press: Boca Raton-London, 2003.
  54. Verhulst F. Methods and Applications of Singular Perturbations, Boundary Layers and Multiple Timescale Dynamics. Springer: New York, 2005.
  55. Polyanin A. D., Zaitsev V. F. Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems. CRC Press: Boca Raton-London, 2018.
  56. Keller H. B. Numerical Solution of Two Point Boundary Value Problems. SIAM: Philadelphia, 1974.
  57. Butcher J. C. The Numerical Analysis of Ordinary Differential Equations, Runge-Kutta and General Linear Methods. Wiley-Interscience: New York, 1987.
  58. Fox L., Mayers D. F. Numerical Solution of Ordinary Differential Equations for Scientists and Engineers. Chapman & Hall: London, 1987.
  59. Ascher U. M., Petzold L. R. Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM: Philadelphia, 1998.
  60. Shingareva I. K., Lizá rraga-Celaya C. Maple and Mathematica. A Problem Solving Approach for Mathematics, 2nd ed. Springer: Wien-New York, 2009.
  61. Griffiths D., Higham D. J. Numerical Methods for Ordinary Differential Equations. Springer: Wien-New York, 2010.
  62. Hairer E., Norsett S. P., Wanner G. Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd ed. Springer: Berlin, 1993.
  63. Hairer E., Wanner G. Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, 2nd ed. Springer: New York, 1996.
  64. Lambert J. D. Numerical Methods for Ordinary Differential Systems. Wiley: New York, 1991

Полный текст (pdf)