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DOI: 10.46698/n8076-2608-1378-r
Новый численный метод решения нелинейных стохастических интегральных уравнений
Зегдане Р.
Владикавказский математический журнал. 2020. Том 22. Выпуск 4.С.68-86.
Аннотация: Цель статьи - применить кардинальные функции Чебышева к численному решению стохастических интегральных уравнений Вольтерра. Метод основан на разложении искомого приближенного решения по кардинальным функциями Чебышева. Для упомянутых базисных функций выводится новая операционная матрица интегрирования. Точнее, искомое решение разлагается в терминах кардинальных функций Чебышева с неизвестными коэффициентами. Подставляя указанное разложение в исходную задачу, операционная матрица сводит стохастическое интегральное уравнение к системе алгебраических уравнений. Исследованы сходимость и оценка погрешности в пространстве Соболева. Метод подвергнут численной оценке путем решения тестовых задач, взятых из литературы, с помощью которых демонстрируется вычислительная эффективность метода. С вычислительной точки зрения решение, полученное этим методом, отлично согласуется с решениями, полученными в других работах, и его эффективно использовать при решении различных задач.
Ключевые слова: кардинальные функции Чебышева, стохастическая операциональная матрица, броуновское движение, интеграл Ито, метод коллокации, численное решение
Образец цитирования: Zeghdane, R. New Numerical Method for Solving Nonlinear Stochastic Integral Equations // Владикавк. мат. журн. 2020. Т. 22, №4. C. 68-86 (in English). DOI 10.46698/n8076-2608-1378-r
1. Boyd, J. P. Chebyshev and Fourier Spectral Methods,
New York, Dover Publications, Inc., 2000.
2. Wazwaz, A.-M. A First Course in Integral Equations,
New Jersey, World Scientific, 1997.
DOI: 10.1142/9570.
3. Kloeden, P. E and Platen, E. Numerical Solution of Stochastic Differential Equations,
Berlin, Springer, 1992.
4. Milstein, G. N. Numerical Integration of Stochastic Differential Equations,
Mathematics and its Application, vol. 313,
Dordrecht, Kluwer, 1995.
DOI: 10.1007/978-94-015-8455-5.
5. Asgari, M., Hashemizadeh, E., Khodabin, M. and Maleknedjad, K.
Numerical Solution of Nonlinear Stochastic Integral Equation
by Stochastic Operational Matrix Based on Bernstein Polynomials,
Bull. Math. Soc. Sci. Math. Roumanie,
2014, vol. 57(105), no. 1, pp. 3-12.
6. Mirazee, F. and Samadyar, N. Application of Hat Basis Functions for Solving Two-Dimensional
Stochastic Fractional Integral Equations,
Journal of Computational and Applied Mathematics,
2018, vol. 37, pp. 4899-4916.
DOI: 10.1007/s40314-018-0608-4.
7. Mohammadi, F. A Wavalet-Based Computational Method for
Solving Stochastic Ito -Volterra Integral Equations,
Journal of Computational Physics, 2015, vol. 298, pp. 254-265.
DOI: 10.1016/J.JCP.2015.05.051.
8. Mahmoudi, Y. Wavelet Galerkin Method for Numerical Solution
of Nonlinear Integral Equation, Applied Mathematics and Computation,
2005, vol. 167, no. 2, pp. 1119-1129.
DOI: 10.1016/j.amc.2004.08.004.
9. Maleknejad, K., Mollapourasl, R. and Alizadeh, M. Numerical
Solution of Volterra Type Integral Equation
of the First Kind with Wavelet Basis,
Applied Mathematics and Computation,
2007, vol. 194, no. 2, pp. 400-405.
DOI: 10.1016/j.amc.2007.04.031.
10. Mirzaee, F., Samadyar, N. and Hoseini, S. F. Euler Polynomial Solutions of Nonlinear
Stochastic Ito -Volterra Integral Equations,
Journal of Computational and Applied Mathematics,
2018, vol. 330, pp. 574-585.
DOI: 10.1016/j.cam.2017.09.005.
11. Mirazee, F., Alipour, S. and Samadyar, N. Numerical Solution Based on Hybrid of Block-Pulse and
Parabolic Function for Solving a System of Nonlinear Stochastic
Ito -Volterra Integral Equations of Fractional Order,
Journal of Computational and Applied Mathematics,
2019, vol. 349, pp. 157-171.
DOI: 10.1016/j.cam.2018.09.040.
12. Mirazee, F. and Samadyar, N. On the Numerical Solution of Stochastic Quadratic
Integral Equations via Operational Matrix Method,
Mathematical Methods in the Applied Sciences,
2018, vol. 41, no. 12, pp. 4465-4479.
DOI: 10.1002/mma.4907.
13. Mirazee, F. and Alipour, S. Approximation Solution of Nonlinear Quadratic Integral Equations
of Fractional Order via Piecewise Linear Functions,
Journal of Computational and Applied Mathematics,
2018, vol. 331, pp. 217-227.
DOI: 10.1016/j.cam.2017.09.038.
14. Mirazee, F. and Hamzeh, A. A Computational Method for Solving Nonlinear
Stochastic Volterra Integral Equations,
Journal of Computational and Applied Mathematics,
2016, vol. 306, pp. 166-0178. DOI: 10.1016/j.cam.2016.04.012.
15. Mirazee, F. and Samadyar, N. Numerical Solutions Based on Two-Dimensional Orthonormal
Bernstein Polynomials for Solving Some Classes of Two-Dimensional
Nonlinear Integral Equations of Fractional Order,
Applied Mathematics and Computation, 2019, vol. 344, pp. 191-203.
DOI: 10.1016/j.amc.2018.10.020.
16. Mirazee, F. and Samadyar, N. Using Radial Basis Functions to Solve Two Dimensional Linear
Stochastic Integral Equations on Non-Rectangular Domains,
Engineering Analysis with Boundary Elements,
2018, vol. 92, pp. 180-195.
DOI: 10.1016/j.enganabound.2017.12.017.
17. Mirazee, F. and Samadyar, N. Numerical Solution of Nonlinear Stochastic Ito -Volterra
Integral Equations Driven by Fractional Brownian Motion,
Mathematical Methods in the Applied Sciences,
2018, vol. 41, no. 4, pp. 1410-1423.
DOI: 10.1002/mma.4671.
18. Shang, X. and Han, D. Numerical Solution of Fredholm Integral Equations
of the First Kind by Using Linear Legendre Multi-Wavelets,
Applied Mathematics and Computation,
2007, vol. 191, no. 2, pp. 440-444.
DOI: 10.1016/j.amc.2007.02.108.
19. Yousefi, S. and Razzaghi, M. Legendre Wavelets Method for the
Nonlinear Volterra-Fredholm Integral Equations,
Mathematics and Computers in Simulation,
2005, vol. 70, no. 1, pp. 1-8.
DOI: 10.1016/j.matcom. 2005. 02. 035.
20. Yousefi, S. A., Lotfi, A. and Dehghan, M. He's Variational Iteration
Method for Solving Nonlinear Mixed Volterra-Fredholm Integral Equations,
Computers and Mathematics with Applications,
2009, vol. 58, no. 11-12, pp. 2172-2176.
DOI: 10.1016/j.camwa.2009.03.083.
21. Maleknejad, K., Khodabin, M. and Rostami, M.
Numerical Method for Solving \(m\)-Dimensional Stochastic
Ito Volterra Integral Equations by Stochastic Operational
Matrix Based on Block-Pulse Functions,
Computers and Mathematics with Applications,
2012, vol. 63, no. 1, pp. 133-143.
DOI: 10.1016/j.camwa.2011.10.079.
22. Mao, X.
Approximate Solutions for a Class of Stochastic
Evolution Equations with Variable Delays. II,
Numerical Functional Analysis and Optimization,
1994, vol. 15, no. 1-2, pp. 65-76.
DOI: 10.1080/01630569408816550.
23. Mirzaee, F. and Hadadiyan, E.
A Collocation Technique for Solving Nonlinear
Stochastic Ito -Volterra Integral Equation,
Applied Mathematics and Computation,
2014, vo. 247, pp. 1011-1020.
DOI: 10.1016/j.amc.2014.09.047.
24. Heydari, M. H., Mahmoudi, M. R., Shakiba, A. and Avazzadeh, Z.
Chebyshev Cardinal Wavelets and Their Application in Solving
Nonlinear Stochastic Differential Equations with Fractional Brownian Motion, Communications in Nonlinear Science and Numerical Simulation,
2018, vol. 64, pp. 98-121.
DOI: 10.1016/j.cnsns.2018.04.018.
25. Heydari, M. H.
A New Direct Method Based on the Chebyshev Cardinal Functions
for Variable-Order Fractional Optimal Control Problems,
Journal of the Franklin Institute,
2018, vol. 355, no. 12, pp. 4970-4995 .
DOI: 10.1016/j.jfranklin.2018.05.025.
26. Heydari, M., Avazzadeh, Z. and Loghmani, G. B.
Chebyshev Cardinal Functions for Solving Volterra-Fredholm
Integro-Differential Equations Using Operational Matrices,
Iranian Journal of Science and Technology, A1,
2012, vol. 36, no. 1, pp. 13-24.
DOI: 10.22099/IJSTS.2012.2050.
27. Funaro, D. Polynomial Approximation of Differential Equations,
New York, Springer-Verlag, 1992.
28. Canuto, C., Hussaini, M., Quarteroni, A. and Zang, T.
Spectral Methods in Fluid Dynamics,
Berlin, Springer, 1988.
29. Blyth, W. F., May, R. L. and Widyaningsih, P. Volterra Integral Equations Solved
in Fredholm form Using Walsh Functions, ANZIAM Journal,
2004, vol. 45(E), pp. 269-282. DOI: 10.21914/anziamj.v45i0.887.
30. Reihani, M. H. and Abadi, Z. Rationalized Haar Functions Method for Solving Fredholm
and Volterra Integral Equations, Journal of Computational and Applied Mathematics,
2007, vol. 200, pp. 12-20. DOI: 10.1016/j.cam.2005.12.026.
31. Parand, K. and Delkhosh, M. Operational Matrix to Solve Nonlinear Riccati
Differential Equations of Arbitrary Order, St. Pertersburg Polytechnical University Journal:
Physics and Mathematics, 2007, vol. 3, no. 3, pp. 242-254.
DOI: 10.1016/j.spjpm.2017.08.001.
32. Arato, M. A Famous Nonlinear Stochastic Equation (Lotka-Volterra Model with Diffusion),
Mathematical and Computer Modelling, 2003, vol. 38, no. 7-9, pp. 709-726.
DOI: 10.1016/S0895-7177(03)90056-2.
