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A. L. Kazakov and L. F. Spevak

ANALYTICAL AND NUMERICAL SOLUTIONS TO A NONLINEAR HEAT EQUATION WITH A FRACTIONAL TIME DERIVATIVE

DOI: 10.17804/2410-9908.2025.3.016-030

The paper presents an analytical and numerical study of a degenerate nonlinear heat equation containing a conformable fractional time derivative. It briefly reviews the use of fractional calculus in mathematical models of natural-science processes and phenomena, as well as methods for solving problems containing fractional derivatives. The theorem of the existence and uniqueness of the thermal wave analytical solution at a specified zero front is proved for the equation under study. The analytical solution is constructed in the form of a power series. An incremental algorithm for constructing a numerical solution with difference approximation in time is proposed. The iteration approach based on the collocation method and approximation by radial basis functions is applied at each step. Test calculations are made in order to verify the numerical algorithm, where the numerical solutions are compared with the truncated series representing the analytical solutions.

Acknowledgement: The study was financially supported by the Russian Ministry of Science and Higher Education under the project on Studying and Modeling of the Rheology and Heat-and-Mass Transfer Phenomena in Structured Environments at Variable Initial and Boundary Conditions, No. 124020600042-9.

Keywords: parabolic equation, nonlinear heat equation, fractional derivative, conformable derivative, exact solution, existence and uniqueness theorem, analytical solution, numerical solution, collocation method, radial basis functions

References:

  1. Tarasov, V.E. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Berlin, Heidelberg, 2010, 505 p. DOI: 10.1007/978-3-642-14003-7.
  2. Povstenko, Yu., Kyrylych, T., Woźna-Szcześniak, B., and Yatsko, A. Fractional heat conduction with heat absorption in a solid with a spherical cavity under time-harmonic heat flux. Applied Sciences, 2024, 14, 1627. DOI: 10.3390/app14041627.
  3. Weia, S., Chena, W., and Zhang, J. Time-fractional derivative model for chloride ions sub-diffusion in reinforced concrete. European Journal of Environmental and Civil Engineering, 2017, 21 (3), 319–331. DOI: 10.1080/19648189.2015.1116467.
  4. Chang, A., Sun, H.G., Zheng, C., Lu, B., Lu, C., Mae, R., and Zhang, Y. A time fractional convection-diffusion equation to model gas transport through heterogeneous soil and gas reservoirs. Physica A, 2018, 502, 356–369. DOI: 10.1016/j.physa.2018.02.080.
  5. Bolster, D., Benson, D.A., and Singha, K. Upscaling chemical reactions in multicontinuum systems: when might time fractional equations work? Chaos, Solitons and Fractals, 2017, 102, 414–425. DOI: 10.1016/j.chaos.2017.04.028.
  6. Meng, R., Yin, D., Zhou, C., and Wu, H. Fractional description of time-dependent mechanical property evolution in materials with strain softening behavior. Applied Mathematical Modelling, 2016, 40, 398–406. DOI: 10.1016/j.apm.2015.04.055.
  7. Sun, H.G., Zhang, Y., Baleanu, D., Chen, W., and Chen, Y.Q. A new collection of real-world applications of fractional calculus in science and engineering. Communications in Nonlinear Science and Numerical Simulation, 2018, 64, 213–231. DOI: 10.1016/j.cnsns.2018.04.019.
  8. Khalil, R., Al Horani, M., Yousef, A., and Sababheh, M. A new definition of fractional derivative. Journal of Computational and Applied Mathematics, 2014, 264, 65–70. DOI: 10.1016/j.cam.2014.01.002.
  9. Abdeljawad, T. On conformable fractional calculus. Journal of Computational and Applied Mathematics, 2015, 279, 57–66. DOI: 10.1016/j.cam.2014.10.016.
  10. Abu Hammad, M. and Khalil, R. Conformable fractional heat differential equation. International Journal of Pure and Applied Mathematics, 2014, 94 (2), 215–221. DOI: 10.12732/ijpam.v94i2.8.
  11. Zhou, H.W., Yang, S., and Zhang, S.Q. Conformable derivative approach to anomalous diffusion. Physica A, 2018, 491, 1001–1013. DOI: 10.1016/j.physa.2017.09.101.
  12. Kazakov, A.L. and Spevak, L.F. On the construction of a heat wave generated by a boundary condition on a moving border. Diagnostics, Resource and Mechanics of materials and structures, 2021, 6, 54–67. DOI: 10.17804/2410-9908.2021.6.054-067. Available at: http://dream-journal.org/issues/2021-6/2021-6_350.html
  13. Kazakov, A.L. and Spevak, L.F. Analytical and numerical radially symmetric solutions to a heat equation with arbitrary nonlinearity. Diagnostics, Resource and Mechanics of materials and structures, 2023, 2, 49–64. DOI: 10.17804/2410-9908.2023.2.049-064. Available at: http://dream-journal.org/issues/2023-2/2023-2_400.html
  14. Kazakov, A.L. and Spevak, L.F. Self-similar solutions to a multidimensional singular heat equation with power nonlinearity. Diagnostics, Resource and Mechanics of materials and structures, 2024, 2, 6–19. DOI: 10.17804/2410-9908.2024.2.006-019. Available at: http://dream-journal.org/issues/2023-2/2023-2_400.html
  15. Kazakov, A.L. and Spevak, L.F. Constructing exact and approximate diffusion wave solutions for a quasilinear parabolic equation with power nonlinearities. Mathematics, 2022, 10 (9), 1559. DOI: 10.3390/math10091559.
  16. Kazakov, A.L. and Spevak, L.F. Exact and approximate solutions of a problem with a singularity for a convection-diffusion equation. Journal of Applied Mechanics and Technical Physics, 2021, 62 (1), 18–26. DOI: 10.1134/S002189442101003X.
  17. Kazakov, A.L. and Orlov, S.S. On some exact solutions to a nonlinear heat equation. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2016, 22 (1), 112–123. (In Russian).
  18. Kazakov, A.L., Spevak, L.F., Nefedova, O.A., and Lempert, A.A. On the analytical and numerical study of a two-dimensional nonlinear heat equation with a source term. Symmetry, 2020, 12 (6), 921. DOI: 10.3390/sym12060921.
  19. Sidorov, A.F. Izbrannye Trudy: Matematika, Mekhanika [Selected Works: Mathematics and Mechanics]. Fizmatlit Publ., Moscow, 2001, 576 p. (In Russian).
  20. Sorokin, P.N. Some generalizations of the Faa Di Bruno formula. Chebyshevskii Sbornik, 2023, 24 (5), 180–193. (In Russian). DOI: 10.22405/2226-8383-2023-24-5-180-193.
  21. Zeid, S.S. Approximation methods for solving fractional equations. Chaos, Solitons and Fractals, 2019, 125, 171–193. DOI: 10.1016/j.chaos.2019.05.008.
  22. Zhang, Y. A finite difference method for fractional partial differential equation. Applied Mathematics and Computation, 2009, 215 (2), 524–529. DOI: 10.1016/j.amc.2009.05.018.
  23. Du, R., Cao, W.R., and Sun, Z.Z. A compact difference scheme for the fractional diffusion-wave equation. Applied Mathematical Modelling, 2010, 34 (10), 2998–3007. DOI: 10.1016/j.apm.2010.01.008.
  24. Chen, M. and Deng, W. Fourth order accurate scheme for the space fractional diffusion equations. SIAM Journal on Numerical Analysis, 2014, 52 (3), 1418–1438. DOI: 10.1137/130933447.
  25. Hamou, A.A., Hammouch, Z., Azroul, E., and Agarwal, P. Monotone iterative technique for solving finite difference systems of time fractional parabolic equations with initial/periodic conditions. Applied Numerical Mathematics, 2022, 181, 561–593. DOI: 10.1016/j.apnum.2022.04.022.
  26. Jiang, Y. and Ma, J. High-order finite element methods for time-fractional partial differential equations. Journal of Computational and Applied Mathematics, 2011, 235 (11), 3285–3290. DOI: 10.1016/j.cam.2011.01.011.
  27. Liu, Y., Du, Y., Li, H., Li, J., and He, S. A two-grid mixed finite element method for a nonlinear fourth-order reaction-diffusion problem with time-fractional derivative. Computers & Mathematics with Applications, 2015, 70 (10), 2474–2492. DOI: 10.1016/j.camwa.2015.09.012.
  28. Feng, L.B., Zhuang, P., Liu, F., Turner, I., and Gu, Y.T. Finite element method for space-time fractional diffusion equation. Numerical Algorithms, 2016, 72, 749–767. DOI: 10.1007/s11075-015-0065-8.
  29. Senol, M., Tasbozan, O., and Kurt, A. Numerical solutions of fractional Burgers’ type equations with conformable derivative. Chinese Journal of Physics, 2019, 58, 75–84. DOI: 10.1016/j.cjph.2019.01.001.
  30. Bhrawy, A.H. and Baleanu, D. A spectral Legendre–Gauss–Lobatto collocation method for a space-fractional advection diffusion equations with variable coefficients. Reports on Mathematical Physics, 2013, 72 (2), 219–233. DOI: 10.1016/S0034-4877(14)60015-X.
  31. Bhrawy, A.H., Zaky, M.A., and Baleanu, D. New numerical approximations for space-time fractional Burgers’ equations via a Legendre spectral-collocation method. Romanian Reports in Physics, 2015, 67 (2), 340–349.
  32. Saad, K.M., Khader, M.M., Gomez-Aguilar, J.F., and Baleanu, D. Numerical solutions of the fractional fisher’s type equations with Atangana–Baleanu fractional derivative by using spectral collocation methods. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2019, 29 (2), 023116. DOI: 10.1063/1.5086771.
  33. Zúñiga-Aguilar, C.J., Romero-Ugalde, H.M., Gómez-Aguilar, J.F., Escobar-Jiménez, R.F., and Valtierra-Rodríguez, M. Solving fractional differential equations of variable-order involving operators with Mittag–Leffler kernel using artificial neural networks. Chaos, Solitons & Fractals, 2017, 103, 382–403. DOI: 10.1016/j.chaos.2017.06.030.
  34. Ye, Y., Fan, H., Li, Y., Liu, X., and Zhang, H. Deep neural network methods for solving forward and inverse problems of time fractional diffusion equations with conformable derivative. Neurocomputing, 2022, 509, 177–192. DOI: 10.1016/j.neucom.2022.08.030.
  35. Jafari, H., Malidareh, B.F., and Hosseini, V.R. Collocation discrete least squares meshless method for solving nonlinear multi-term time fractional differential equations. Engineering Analysis with Boundary Elements, 2024, 158, 107–120. DOI: 10.1016/j.enganabound.2023.10.014.
  36. Zhuang, P., Gu, Y., Liu, F., Turner, I., and Yarlagadda, P.K.D.V. Time-dependent fractional advection–diffusion equations by an implicit MLS meshless method. International Journal for Numerical Methods in Engineering, 2011, 88 (13), 1346–1362. DOI: 10.1002/nme.3223.
  37. Hosseini, V.R., Chen, W., and Avazzadeh, Z. Numerical solution of fractional telegraph equation by using radial basis functions. Engineering Analysis with Boundary Elements, 2014, 38, 31–39. DOI: 10.1016/j.enganabound.2013.10.009.
  38. Dehghan, M., Abbaszadeh, M., and Mohebbi, A. An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein–Gordon equations. Engineering Analysis with Boundary Elements, 2015, 50, 412–434. DOI: 10.1016/j.enganabound.2014.09.008.
  39. Chen, W., Fu, Z.-J., and Chen, C.S. Recent Advances in Radial Basis Function Collocation Methods, Springer, Berlin, Heidelberg, 2013, 90 p. DOI: 10.1007/978-3-642-39572-7.
  40. Buhmann, M.D. Radial Basis Functions: Theory and Implementations, Cambridge University Press, 2003. DOI: 10.1017/CBO9780511543241.
  41. Fornberg, B. and Flyer, N. Solving PDEs with radial basis functions. Acta Numerica, 2015, 24, 215–258. DOI: 10.1017/S0962492914000130.

А. Л. Казаков, Л. Ф. Спевак

АНАЛИТИЧЕСКОЕ И ЧИСЛЕННОЕ РЕШЕНИЕ НЕЛИНЕЙНОГО УРАВНЕНИЯ ТЕПЛОПРОВОДНОСТИ С ДРОБНОЙ ПРОИЗВОДНОЙ ПО ВРЕМЕНИ

Работа посвящена аналитическому и численному исследованию вырождающегося нелинейного уравнения теплопроводности, содержащего конформную дробную производную по времени. Приведен краткий обзор применения дробного исчисления в математических моделях естественнонаучных процессов и явлений, а также методов решения задач, содержащих производные дробного порядка. Для рассматриваемого уравнения доказана теорема существования и единственности аналитического решения типа тепловой волны при заданном нулевом фронте. Аналитическое решение строится в виде степенного ряда. Предложен пошаговый алгоритм построения численного решения с разностной аппроксимацией по времени. На каждом шаге применяется итерационный подход, основанный на методе коллокаций и аппроксимации радиальными базисными функциями. Для верификации численного алгоритма проведены тестовые расчеты, в которых численные решения сравнивались с отрезками рядов, в виде которых представлены аналитические решения.

Благодарность: Работа выполнена при финансовой поддержке Минобрнауки России в рамках проекта «Исследование и моделирование явлений реологии и тепломассопереноса в средах с внутренней структурой при переменных начальных и граничных условиях» (№ 124020600042-9).

Ключевые слова: параболическое уравнение, нелинейное уравнение теплопроводности, дробная производная, конформная производная, теорема существования и единственности, аналитическое решение, численное решение, метод коллокаций, радиальные базисные функции

Библиография:

  1. Tarasov V. E. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. – Berlin; Heidelberg : Springer, 2010. – 505 p. – DOI: 10.1007/978-3-642-14003-7.
  2. Fractional heat conduction with heat absorption in a solid with a spherical cavity under time-harmonic heat flux / Y. Povstenko, T. Kyrylych, B. Woźna-Szcześniak, A. Yatsko // Applied Sciences. – 2024. – Vol. 14. – 1627. – DOI: 10.3390/app14041627.
  3. Weia S., Chena W., Zhang J. Time-fractional derivative model for chloride ions sub-diffusion in reinforced concrete // European Journal of Environmental and Civil Engineering. – 2017. – Vol. 21 (3). – P. 319–331. – DOI: 10.1080/19648189.2015.1116467.
  4. A time fractional convection-diffusion equation to model gas transport through heterogeneous soil and gas reservoirs / A. Chang, H. G. Sun, C. Zheng, B. Lu, C. Lu, R. Mae, Y. Zhang // Physica A. – 2018. – Vol. 502. – P. 356–369. – DOI: 10.1016/j.physa.2018.02.080.
  5. Bolster D., Benson D. A., Singha K. Upscaling chemical reactions in multicontinuum systems: when might time fractional equations work? // Chaos, Solitons and Fractals. – 2017. – Vol. 102. – P. 414–425. – DOI: 10.1016/j.chaos.2017.04.028.
  6. Fractional description of time-dependent mechanical property evolution in materials with strain softening behavior / R. Meng, D. Yin, C. Zhou, H. Wu // Applied Mathematical Modelling. – 2016. – Vol. 40. – P. 398–406. – DOI: 10.1016/j.apm.2015.04.055.
  7. A new collection of real-world applications of fractional calculus in science and engineering / H. G. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Q. Chen // Communications in Nonlinear Science and Numerical Simulation. – 2018. – Vol. 64. – P. 213–231. – DOI: 10.1016/j.cnsns.2018.04.019.
  8. A new definition of fractional derivative / R. Khalil, M. Al Horani, A. Yousef, M. Sababheh // Journal of Computational and Applied Mathematics. – 2014. – Vol. 264. – P. 65–70. – DOI: 10.1016/j.cam.2014.01.002.
  9. Abdeljawad T. On conformable fractional calculus // Journal of Computational and Applied Mathematics. – 2015. – Vol. 279. – P. 57–66. – DOI: 10.1016/j.cam.2014.10.016.
  10. Abu Hammad M., Khalil R. Conformable fractional heat differential equation // International Journal of Pure and Applied Mathematics. – 2014. – Vol. 94 (2). – P. 215–221. – DOI: 10.12732/ijpam.v94i2.8.
  11. Zhou H. W., Yang S., Zhang S. Q. Conformable derivative approach to anomalous diffusion // Physica A. – 2018. – Vol. 491. – P. 1001–1013. – DOI: 10.1016/j.physa.2017.09.101.
  12. Kazakov A. L., Spevak L. F. On the construction of a heat wave generated by a boundary condition on a moving border // Diagnostics, Resource and Mechanics of materials and structures. – 2021. – Iss. 6. – P. 54–67. – DOI: 10.17804/2410-9908.2021.6.054-067. – URL: http://dream-journal.org/issues/2021-6/2021-6_350.html
  13. Kazakov A. L., Spevak L. F. Analytical and numerical radially symmetric solutions to a heat equation with arbitrary nonlinearity // Diagnostics, Resource and Mechanics of materials and structures. – 2023. – Iss. 2. – P. 49–64. – DOI: 10.17804/2410-9908.2023.2.049-064. – URL: http://dream-journal.org/issues/2021-6/2021-6_350.html
  14. Kazakov A. L., Spevak L. F. Self-similar solutions to a multidimensional singular heat equation with power nonlinearity // Diagnostics, Resource and Mechanics of materials and structures. – 2024. – Iss. 2. – P. 6–19. – DOI: 10.17804/2410-9908.2024.2.006-019. – URL: http://dream-journal.org/issues/2023-2/2023-2_400.html
  15. Kazakov A. L., Spevak L. F. Constructing exact and approximate diffusion wave solutions for a quasilinear parabolic equation with power nonlinearities // Mathematics. – 2022. – Vol. 10 (9). – 1559. – DOI: 10.3390/math10091559.
  16. Kazakov A. L., Spevak L. F. Exact and approximate solutions of a problem with a singularity for a convection-diffusion equation // Journal of Applied Mechanics and Technical Physics. – 2021. – Vol. 62 (1). – P. 18–26. – DOI: 10.1134/S002189442101003X.
  17. Казаков А. Л., Орлов С. С. О некоторых точных решениях нелинейного уравнения теплопроводности // Труды Института математики и механики УрО РАН. – 2016. – Т. 22 (1). – С. 112–123.
  18. On the analytical and numerical study of a two-dimensional nonlinear heat equation with a source term / A. L. Kazakov, L. F. Spevak, O. A. Nefedova, A. A. Lempert // Symmetry. – 2020. – Vol. 12 (6). – 921. – DOI: 10.3390/sym12060921.
  19. Сидоров А. Ф. Избранные труды. Математика, механика. – М. : Физматлит, 2001. – 576 c.
  20. Сорокин П. Н. Некоторые обобщения формулы Фаа Ди Бруно // Чебышевский сборник. – 2023. – Т. 24 (5). – С. 180–193. – DOI: 10.22405/2226-8383-2023-24-5-180-193.
  21. Zeid S. S. Approximation methods for solving fractional equations // Chaos, Solitons and Fractals. – 2019. – Vol. 125. – P. 171–193. – DOI: 10.1016/j.chaos.2019.05.008.
  22. Zhang Y. A finite difference method for fractional partial differential equation // Applied Mathematics and Computation. – 2009. – Vol. 215 (2). – P. 524–529. – DOI: 10.1016/j.amc.2009.05.018.
  23. Du R., Cao W. R., Sun Z. Z. A compact difference scheme for the fractional diffusion-wave equation // Applied Mathematical Modelling. – 2010. – Vol. 34 (10). – P. 2998–3007. – DOI: 10.1016/j.apm.2010.01.008.
  24. Chen M., Deng W. Fourth order accurate scheme for the space fractional diffusion equations // SIAM Journal on Numerical Analysis. – 2014. – Vol. 52 (3). – P. 1418–1438. – DOI: 10.1137/130933447.
  25. Monotone iterative technique for solving finite difference systems of time fractional parabolic equations with initial/periodic conditions / A. A. Hamou, Z. Hammouch, E. Azroul, P. Agarwal // Applied Numerical Mathematics. – 2022. – Vol. 181. – P. 561–593. – DOI: 10.1016/j.apnum.2022.04.022.
  26. Jiang Y., Ma J. High-order finite element methods for time-fractional partial differential equations // Journal of Computational and Applied Mathematics. – 2011. – Vol. 235 (11). – P. 3285–3290. – DOI: 10.1016/j.cam.2011.01.011.
  27. A two-grid mixed finite element method for a nonlinear fourth-order reaction-diffusion problem with time-fractional derivative / Y. Liu, Y. Du, H. Li, J. Li, S. He // Computers & Mathematics with Applications. – 2015. – Vol. 70 (10). – P. 2474–2492. – DOI: 10.1016/j.camwa.2015.09.012.
  28. Finite element method for space-time fractional diffusion equation / L. B. Feng, P. Zhuang, F. Liu, I. Turner, Y. T. Gu // Numerical Algorithms. – 2016. – Vol. 72. – P. 749–767. – DOI: 10.1007/s11075-015-0065-8.
  29. Senol M., Tasbozan O., Kurt A. Numerical solutions of fractional Burgers’ type equations with conformable derivative // Chinese Journal of Physics. – 2019. – Vol. 58. – P. 75–84. – DOI: 10.1016/j.cjph.2019.01.001.
  30. Bhrawy A. H., Baleanu D. A spectral Legendre–Gauss–Lobatto collocation method for a space-fractional advection diffusion equations with variable coefficients // Reports on Mathematical Physics. – 2013. – Vol. 72 (2). – P. 219–233. – DOI: 10.1016/S0034-4877(14)60015-X.
  31. Bhrawy A. H., Zaky M. A., Baleanu D. New numerical approximations for space–time fractional burgers equations via a Legendre spectral-collocation method // Romanian Reports in Physics. – 2015. – Vol. 67 (2). –P. 340–349.
  32. Numerical solutions of the fractional fisher’s type equations with Atangana–Baleanu fractional derivative by using spectral collocation methods / K. M. Saad, M. M Khader., J. F. Gomez-Aguilar, D. Baleanu // Chaos: An Interdisciplinary Journal of Nonlinear Science. – 2019. – Vol. 29 (2). – 023116. – DOI: 10.1063/1.5086771.
  33. Solving fractional differential equations of variable-order involving operators with Mittag-Leffler kernel using artificial neural networks / C. J. Zúñiga-Aguilar, H. M. Romero-Ugalde, J. F. Gómez-Aguilar, R. F. Escobar-Jiménez, M. Valtierra-Rodríguez // Chaos, Solitons & Fractals. – 2017. – Vol. 103. – P. 382–403. – DOI: 10.1016/j.chaos.2017.06.030.
  34. Deep neural network methods for solving forward and inverse problems of time fractional diffusion equations with conformable derivative / Y. Ye, H. Fan, Y. Li, X. Liu, H. Zhang // Neurocomputing. – 2022. – Vol. 509. – P. 177–192. – DOI: 10.1016/j.neucom.2022.08.030.
  35. Jafari H., Malidareh B. F., Hosseini V. R. Collocation discrete least squares meshless method for solving nonlinear multi-term time fractional differential equations // Engineering Analysis with Boundary Elements. – 2024. – Vol. 158. – P. 107–120. – DOI: 10.1016/j.enganabound.2023.10.014.
  36. Time-dependent fractional advection–diffusion equations by an implicit MLS meshless method / P. Zhuang, Y. Gu, F. Liu, I. Turner, P. K. D. V. Yarlagadda // International Journal for Numerical Methods in Engineering. – 2011. – Vol. 88 (13). – P. 1346–1362. – DOI: 10.1002/nme.3223.
  37. Hosseini V. R., Chen W., Avazzadeh Z. Numerical solution of fractional telegraph equation by using radial basis functions // Engineering Analysis with Boundary Elements. – 2014. – Vol. 38. – P. 31–39. – DOI: 10.1016/j.enganabound.2013.10.009.
  38. Dehghan M., Abbaszadeh M., Mohebbi A. An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein–Gordon equations // Engineering Analysis with Boundary Elements. – 2015. – Vol. 50. – P. 412–434. – DOI: 10.1016/j.enganabound.2014.09.008.
  39. Chen W., Fu Z.-J, Chen C. S. Recent Advances in Radial Basis Function Collocation Methods. – Berlin; Heidelberg : Springer, 2013. – 90 p. – DOI:10.1007/978-3-642-39572-7.
  40. Buhmann M. D. Radial Basis Functions. – Cambridge : Cambridge University Press, 2003. – 259 p. – DOI: 10.1017/CBO9780511543241.
  41. Fornberg B., Flyer N. Solving PDEs with radial basis functions // Acta Numerica. – 2015. – Vol. 24. – P. 215–258. – DOI: 10.1017/S0962492914000130.


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Библиографическая ссылка на статью

Spevak A. L. Kazakov and L. F. Analytical and Numerical Solutions to a Nonlinear Heat Equation with a Fractional Time Derivative // Diagnostics, Resource and Mechanics of materials and structures. - 2025. - Iss. 3. - P. 16-30. -
DOI: 10.17804/2410-9908.2025.3.016-030. -
URL: http://dream-journal.org/issues/2025-3/2025-3_512.html
(accessed: 18.04.2026).

 

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