A non-linear diffusion problem with power-law diffusivity: An approximate solution experimenting with a modified sinc function

Jordan Hristov

doi:10.53391/mmnsa.1545438

Araştırma Makalesi

Yıl 2024, Cilt: 4 Sayı: 5-Special Issue: ICAME'24, 6 - 44, 31.12.2024

Jordan Hristov

https://doi.org/10.53391/mmnsa.1545438

Cited By: 3

Öz

Kaynakça

[1] Goodman, T.R. The heat-balance integral and its application to problems involving a change of phase. Transactions of the American Society of Mechanical Engineers, 80(2), 335-342, (1958).
[2] Wood, A.S. and Kutluay, S. A heat balance integral model of the thermistor. International Journal of Heat and Mass Transfer, 38(10), 1831-1840, (1995).
[3] Moghtaderi, B., Novozhilov, V., Fletcher, D. and Kent, J.H. An integral model for the transient pyrolysis of solid materials. Fire and Materials, 21(1), 7-16, (1997).
[4] Staggs, J.E.J. Approximate solutions for the pyrolysis of char forming and filled polymers under thermally thick conditions. Fire and Materials, 24(6), 305-308, (2000).
[5] Hristov, J. The heat-balance integral method by a parabolic profile with unspecified exponent: analysis and benchmark exercises. Thermal Science, 13(2), 27-48, (2009).
[6] Hristov J. An approximate analytical (integral-balance) solution to a nonlinear heat diffusion equation. Thermal Science, 19(2), 723-733, (2015).
[7] Bollati, J. and Tarzia, D.A. Approximate solutions to one-phase Stefan-like problems with space-dependent latent heat. European Journal of Applied Mathematics, 32(2), 337-369, (2021).
[8] Canzian, E.P, Santiago, F., Brito Lopes, A.V. and Barañano, A.G. Spherical solidification: An application of the integral methods. International Journal of Thermal Sciences, 177, 107575, (2022).
[9] Canzian, E.P., Santiago, F., Brito Lopes, A.V., Barbosa, M.R. and Barañano, A.G. On the application of the double integral method with quadratic temperature profile for spherical solidification of lead and tin metals. Applied Thermal Engineering, 219, 119528, (2023).
[10] Hristov, J. Multiple integral-balance method: Basic idea and an example with Mullin’s model of thermal grooving. Thermal Science, 21(3), 1555-1560, (2017).
[11] Hristov, J. The heat radiation diffusion equation: Explicit analytical solutions by improved integral-balance method. Thermal Science, 22(2), 777-788, (2018).
[12] Bollati, J., Natale, M.F., Semitiel, J.A., Tarzia, D.A. Integral balance methods applied to non-classical Stefan problem. ArXiv Prints, ArXiv:1810.06370, (2018).
[13] Ceretani, A.N., Salva, N.N. and Tarzia, D.A. An exact solution to a Stefan problem with variable thermal conductivity and a Robin boundary condition. Nonlinear Analysis: Real World Applications, 40, 243-259, (2018).
[14] Bollati, J., Natale, M.F., Semitiel, J.A. and Tarzia, D.A. A two-phase Stefan problem with power-type temperature-dependent thermal conductivity. Existence of a solution by two fixed points and numerical results. AIMS Mathematics, 9(8), 21189-21211, (2024).
[15] Hristov J. Integral solutions to transient nonlinear heat (mass) diffusion with a power-law diffusivity: a semi-infinite medium with fixed boundary conditions. Heat and Mass Transfer, 52, 635-655, (2016).
[16] Hill, J.M. Similarity solutions for nonlinear diffusion-a new integration procedure. Journal of Engineering Mathematics, 23, 141-155, (1989).
[17] Smyth, N.F. and Hill, J.M. High-order nonlinear diffusion. IMA Journal of Applied Mathematics, 40(2), 73-86, (1988).
[18] Buckmaster, J. Viscous sheets advancing over dry beds. Journal of Fluid Mechanics, 81(4), 735-756, (1977).
[19] Marino, B.M., Thomas, L.P., Gratton, R., Diez, J.A., Betelú, S. and Gratton, J. Waiting-time solutions of a nonlinear diffusion equation: Experimental study of a creeping flow near a waiting front. Physical Review E, 54, 2628, (1996).
[20] Prasad, S.N. and Salomon, J.B. A new method for analytical solution of a degenerate diffusion equation. Advances in Water Resources, 28(10), 1091-1101, (2005).
[21] Aronson, D.G. The porous medium equation. In: Nonlinear Diffusion Problems (pp. 1-46). Springer-Verlag: Berlin, (1986).
[22] Bradley, D.M. Note on Dirichlet’s sinc integral. The American Mathematical Monthly, 128(3), 273-274, (2021).
[23] Stenger, F. Numerical methods based on Whittaker cardinal, or sinc functions. Siam Review, 23(2), 165-224, (1981).
[24] Bercu, G. Sharp bounds on the sinc function via the Fourier series method. Journal of Mathematical Inequalities, 13(2), 495-504, (2019).
[25] Hu, Y. and Mortici, C. A lower bound of the sinc function and its applications. The Scientific World Journal, 2014, 571218, (2014).
[26] Yunlong, W. Sinc sum function and its application on FIR filter design. Acta Applicandae Mathematicae, 110, 1037-1056, (2010).
[27] Gomopoulos, N., Cheng, W., Efremov, M., Nealey, P.F. and Fytas,G. Out-of-plane longitudinal elastic modulus of supported polymer thin films. Macromolecules, 42(18), 7164-7167, (2009).
[28] Abrarov, S.M. and Quine, B.M. A rational approximation of the sinc function based on sampling and the Fourier transforms. Applied Numerical Mathematics, 150, 65-75, (2020).
[29] Stenger, F. Numerical Methods Based on Sinc and Analytic Functions (Vol. 20). Springer-Verlag: New York, (1993).
[30] Morlet, A.C. Convergence of the Sinc method for a fourth-order ordinary differential equation with an application. SIAM Journal on Numerical Analysis, 32(5), 1475-1503, (1995).
[31] Mescia, L., Bia, P. and Caratelli, D. Fractional-calculus-based electromagnetic tool to study pulse propagation in arbitrary dispersive dielectrics. Physica Status Solidi A, 216(3), 1800557, (2018).
[32] Stenger, F. Fourier series for zeta function via Sinc. Linear Algebra and its Applications, 429(10), 2636-2639, (2008).
[33] Cattani, C. Sinc-fractional operator on Shannon wavelet space. Frontiers in Physics, 6, 118, (2018).
[34] Slavnov, N.A. Integral operators with the generalized sine kernel on the real axis. Theoretical and Mathematical Physics, 165, 1262-1274, (2010).
[35] Yang, X.J., Gao, F., Tenreiro Machado, J.A. and Baleanu, D. A new fractional derivative involving the normalized sinc function without singular kernel. The European Physical Journal Special Topics, 226, 3567-3575, (2017).
[36] Yavuz, M. Characterizations of two different fractional operators without singular kernel. Mathematical Modelling of Natural Phenomena, 14(3), 302, (2019).
[37] Odibat, Z. and Baleanu, D. A new fractional derivative operator with generalized cardinal sine kernel: Numerical simulation. Mathematics and Computers in Simulation, 212, 224-233, (2023).
[38] Odibat, Z., Al-Refai, M. and Baleanu, D. On some properties of generalized cardinal sine kernel fractional operators: Advantages and applications of the extended operators. Chinese Journal of Physics, 91, 349-360, (2024).
[39] Şelariu, M.E., Smarandache, F. and Nitu, M. Cardinal functions and integral functions. International Journal of Geometry, 1(1), 27-40,(2012).
[40] Heaslet, M.A. and Alksne, A. Diffusion from a fixed surface with a concentration-dependent coefficient. Journal of the Society for Industrial and Applied Mathematics, 9(4), 584-596, (1961).
[41] Hristov, J. The heat-balance integral: 2. Parabolic profile with a variable exponent: The concept, analysis and numerical experiments. Comptes Rendus Mécanique, 340(7), 493-500, (2012).
[42] Crank, J. The Mathematics of Diffusion. Oxford University Press: New York, (1956).
[43] King, J.R. Approximate solutions to a nonlinear diffusion equation. Journal of Engineering Mathematics, 22, 53-72, (1988).
[44] Philip, J. Numerical solution of equations of the diffusion type with diffusivity concentrationdependent. Transactions of the Faraday Society, 51, 885-892, (1955).
[45] Weisberg, L.R. and Blanc, J. Diffusion with interstitial-substitutional equilibrium. Zinc in GaAs. Physical Review, 131, 1548, (1963).
[46] Brutsaert, W. The adaptability of an exact solution to horizontal infiltration. Water Resources Research, 4(4), 785-789, (1968).
[47] Brutsaert, W. A solution for vertical infiltration into a dry porous medium. Water Resources Research, 4(5), 1031-1038, (1968).
[48] Brutsaert, W. and Weisman, R.N. Comparison of solutions of a nonlinear diffusion equation. Water Resources Research, 6(2), 642-644, (1970).
[49] Tuck, B. Some explicit solutions to the non-linear diffusion equation. Journal of Physics D: Applied Physics, 9(11), 1559, (1976).
[50] Parlange, M.B., Prasad, S.N, Parlange, J.-Y. and Römkens, M.J.M. Extension of the HeasletAlksne technique to arbitrary soil water diffusivities. Water Resources Research, 28(10), 2793- 2797, (1992).
[51] Lockington, D., Parlange, J.Y. and Dux, P. Sorptivity and the estimation of water penetration into unsaturated concrete. Materials and Structures, 32, 342-347, (1999).
[52] Lockington, D. Estimating the sorptivity for a wide range of diffusivity dependence on water content. Transport in Porous Media, 10, 95-101, (1993).
[53] Ioannou, I., Andreou, A., Tsikouras, B., Hatzipanagiotou, K. Application of the sharp front model to capillary absorption in a vuggy limestone. Engineering Geology, 105(1-2), 20-23, (2009).
[54] Küntz, M. and Lavallée, P. Experimental evidence and theoretical analysis of anomalous diffusion during water infiltration in porous building materials. Journal of Physics D: Applied Physics, 34(16), 2547, (2001).
[55] Hristov,J., El-Ganaoui,M. Modelling capillary absorption in building materials with emphasis on the fourth root time law: Time-fractional models, solutions and analysis. In Fractional Differential Equations: Theoretical Aspects and Applications (pp. 153-165). Academic Press: New York, (2024).
[56] Hristov, J. A new closed-form approximate solution to diffusion with quadratic Fujita’s nonlinearity: the case of diffusion controlled sorption kinetics relevant to rectangular adsorption isotherms. Heat and Mass Transfer, 55, 261-279, (2019).
[57] Hristov, J. Magnetic field diffusion in ferromagnetic materials: fractional calculus approaches. An International Journal of Optimization and Control: Theories & Applications, 11(3), 1-15, (2021).
[58] Xiao, B., Wang, G., Zhao, L. Feng, C. and Shu, S. An exact solution for the magnetic diffusion problem with a step-function resistivity model. The European Physical Journal Plus, 139, 305, (2024).

A non-linear diffusion problem with power-law diffusivity: An approximate solution experimenting with a modified sinc function

Yıl 2024, Cilt: 4 Sayı: 5-Special Issue: ICAME'24, 6 - 44, 31.12.2024

Jordan Hristov

https://doi.org/10.53391/mmnsa.1545438

Cited By: 3

Öz

Employing a modified version of the cardinal $Sinc_{\pi} \left(\pi x^{n} \right)$ function as the assumed profile, the work presents approximate solutions of a non-linear (degenerate) diffusion equation with a power-law-type concentration-dependent diffusivity in a semi-infinite domain by the integral-balance method (double integration technique). The behavior and basic features of a modified function $ Sinc_{\pi}\left(x^{n} \right)$ are addressed, highlighting how it is used in the generated approximate solutions. It has been successful in implementing the concept of the modified $sinc (x)$ function's variable (argument-dependent) exponent. To demonstrate the suitability of the suggested technique, comparative examinations concerning well-known approximate analytical and numerical problem solutions have been developed.

Anahtar Kelimeler

Transient diffusion, approximate solutions, integral-balance method, sinc function

Kaynakça

[1] Goodman, T.R. The heat-balance integral and its application to problems involving a change of phase. Transactions of the American Society of Mechanical Engineers, 80(2), 335-342, (1958).
[2] Wood, A.S. and Kutluay, S. A heat balance integral model of the thermistor. International Journal of Heat and Mass Transfer, 38(10), 1831-1840, (1995).
[3] Moghtaderi, B., Novozhilov, V., Fletcher, D. and Kent, J.H. An integral model for the transient pyrolysis of solid materials. Fire and Materials, 21(1), 7-16, (1997).
[4] Staggs, J.E.J. Approximate solutions for the pyrolysis of char forming and filled polymers under thermally thick conditions. Fire and Materials, 24(6), 305-308, (2000).
[5] Hristov, J. The heat-balance integral method by a parabolic profile with unspecified exponent: analysis and benchmark exercises. Thermal Science, 13(2), 27-48, (2009).
[6] Hristov J. An approximate analytical (integral-balance) solution to a nonlinear heat diffusion equation. Thermal Science, 19(2), 723-733, (2015).
[7] Bollati, J. and Tarzia, D.A. Approximate solutions to one-phase Stefan-like problems with space-dependent latent heat. European Journal of Applied Mathematics, 32(2), 337-369, (2021).
[8] Canzian, E.P, Santiago, F., Brito Lopes, A.V. and Barañano, A.G. Spherical solidification: An application of the integral methods. International Journal of Thermal Sciences, 177, 107575, (2022).
[9] Canzian, E.P., Santiago, F., Brito Lopes, A.V., Barbosa, M.R. and Barañano, A.G. On the application of the double integral method with quadratic temperature profile for spherical solidification of lead and tin metals. Applied Thermal Engineering, 219, 119528, (2023).
[10] Hristov, J. Multiple integral-balance method: Basic idea and an example with Mullin’s model of thermal grooving. Thermal Science, 21(3), 1555-1560, (2017).
[11] Hristov, J. The heat radiation diffusion equation: Explicit analytical solutions by improved integral-balance method. Thermal Science, 22(2), 777-788, (2018).
[12] Bollati, J., Natale, M.F., Semitiel, J.A., Tarzia, D.A. Integral balance methods applied to non-classical Stefan problem. ArXiv Prints, ArXiv:1810.06370, (2018).
[13] Ceretani, A.N., Salva, N.N. and Tarzia, D.A. An exact solution to a Stefan problem with variable thermal conductivity and a Robin boundary condition. Nonlinear Analysis: Real World Applications, 40, 243-259, (2018).
[14] Bollati, J., Natale, M.F., Semitiel, J.A. and Tarzia, D.A. A two-phase Stefan problem with power-type temperature-dependent thermal conductivity. Existence of a solution by two fixed points and numerical results. AIMS Mathematics, 9(8), 21189-21211, (2024).
[15] Hristov J. Integral solutions to transient nonlinear heat (mass) diffusion with a power-law diffusivity: a semi-infinite medium with fixed boundary conditions. Heat and Mass Transfer, 52, 635-655, (2016).
[16] Hill, J.M. Similarity solutions for nonlinear diffusion-a new integration procedure. Journal of Engineering Mathematics, 23, 141-155, (1989).
[17] Smyth, N.F. and Hill, J.M. High-order nonlinear diffusion. IMA Journal of Applied Mathematics, 40(2), 73-86, (1988).
[18] Buckmaster, J. Viscous sheets advancing over dry beds. Journal of Fluid Mechanics, 81(4), 735-756, (1977).
[19] Marino, B.M., Thomas, L.P., Gratton, R., Diez, J.A., Betelú, S. and Gratton, J. Waiting-time solutions of a nonlinear diffusion equation: Experimental study of a creeping flow near a waiting front. Physical Review E, 54, 2628, (1996).
[20] Prasad, S.N. and Salomon, J.B. A new method for analytical solution of a degenerate diffusion equation. Advances in Water Resources, 28(10), 1091-1101, (2005).
[21] Aronson, D.G. The porous medium equation. In: Nonlinear Diffusion Problems (pp. 1-46). Springer-Verlag: Berlin, (1986).
[22] Bradley, D.M. Note on Dirichlet’s sinc integral. The American Mathematical Monthly, 128(3), 273-274, (2021).
[23] Stenger, F. Numerical methods based on Whittaker cardinal, or sinc functions. Siam Review, 23(2), 165-224, (1981).
[24] Bercu, G. Sharp bounds on the sinc function via the Fourier series method. Journal of Mathematical Inequalities, 13(2), 495-504, (2019).
[25] Hu, Y. and Mortici, C. A lower bound of the sinc function and its applications. The Scientific World Journal, 2014, 571218, (2014).
[26] Yunlong, W. Sinc sum function and its application on FIR filter design. Acta Applicandae Mathematicae, 110, 1037-1056, (2010).
[27] Gomopoulos, N., Cheng, W., Efremov, M., Nealey, P.F. and Fytas,G. Out-of-plane longitudinal elastic modulus of supported polymer thin films. Macromolecules, 42(18), 7164-7167, (2009).
[28] Abrarov, S.M. and Quine, B.M. A rational approximation of the sinc function based on sampling and the Fourier transforms. Applied Numerical Mathematics, 150, 65-75, (2020).
[29] Stenger, F. Numerical Methods Based on Sinc and Analytic Functions (Vol. 20). Springer-Verlag: New York, (1993).
[30] Morlet, A.C. Convergence of the Sinc method for a fourth-order ordinary differential equation with an application. SIAM Journal on Numerical Analysis, 32(5), 1475-1503, (1995).
[31] Mescia, L., Bia, P. and Caratelli, D. Fractional-calculus-based electromagnetic tool to study pulse propagation in arbitrary dispersive dielectrics. Physica Status Solidi A, 216(3), 1800557, (2018).
[32] Stenger, F. Fourier series for zeta function via Sinc. Linear Algebra and its Applications, 429(10), 2636-2639, (2008).
[33] Cattani, C. Sinc-fractional operator on Shannon wavelet space. Frontiers in Physics, 6, 118, (2018).
[34] Slavnov, N.A. Integral operators with the generalized sine kernel on the real axis. Theoretical and Mathematical Physics, 165, 1262-1274, (2010).
[35] Yang, X.J., Gao, F., Tenreiro Machado, J.A. and Baleanu, D. A new fractional derivative involving the normalized sinc function without singular kernel. The European Physical Journal Special Topics, 226, 3567-3575, (2017).
[36] Yavuz, M. Characterizations of two different fractional operators without singular kernel. Mathematical Modelling of Natural Phenomena, 14(3), 302, (2019).
[37] Odibat, Z. and Baleanu, D. A new fractional derivative operator with generalized cardinal sine kernel: Numerical simulation. Mathematics and Computers in Simulation, 212, 224-233, (2023).
[38] Odibat, Z., Al-Refai, M. and Baleanu, D. On some properties of generalized cardinal sine kernel fractional operators: Advantages and applications of the extended operators. Chinese Journal of Physics, 91, 349-360, (2024).
[39] Şelariu, M.E., Smarandache, F. and Nitu, M. Cardinal functions and integral functions. International Journal of Geometry, 1(1), 27-40,(2012).
[40] Heaslet, M.A. and Alksne, A. Diffusion from a fixed surface with a concentration-dependent coefficient. Journal of the Society for Industrial and Applied Mathematics, 9(4), 584-596, (1961).
[41] Hristov, J. The heat-balance integral: 2. Parabolic profile with a variable exponent: The concept, analysis and numerical experiments. Comptes Rendus Mécanique, 340(7), 493-500, (2012).
[42] Crank, J. The Mathematics of Diffusion. Oxford University Press: New York, (1956).
[43] King, J.R. Approximate solutions to a nonlinear diffusion equation. Journal of Engineering Mathematics, 22, 53-72, (1988).
[44] Philip, J. Numerical solution of equations of the diffusion type with diffusivity concentrationdependent. Transactions of the Faraday Society, 51, 885-892, (1955).
[45] Weisberg, L.R. and Blanc, J. Diffusion with interstitial-substitutional equilibrium. Zinc in GaAs. Physical Review, 131, 1548, (1963).
[46] Brutsaert, W. The adaptability of an exact solution to horizontal infiltration. Water Resources Research, 4(4), 785-789, (1968).
[47] Brutsaert, W. A solution for vertical infiltration into a dry porous medium. Water Resources Research, 4(5), 1031-1038, (1968).
[48] Brutsaert, W. and Weisman, R.N. Comparison of solutions of a nonlinear diffusion equation. Water Resources Research, 6(2), 642-644, (1970).
[49] Tuck, B. Some explicit solutions to the non-linear diffusion equation. Journal of Physics D: Applied Physics, 9(11), 1559, (1976).
[50] Parlange, M.B., Prasad, S.N, Parlange, J.-Y. and Römkens, M.J.M. Extension of the HeasletAlksne technique to arbitrary soil water diffusivities. Water Resources Research, 28(10), 2793- 2797, (1992).
[51] Lockington, D., Parlange, J.Y. and Dux, P. Sorptivity and the estimation of water penetration into unsaturated concrete. Materials and Structures, 32, 342-347, (1999).
[52] Lockington, D. Estimating the sorptivity for a wide range of diffusivity dependence on water content. Transport in Porous Media, 10, 95-101, (1993).
[53] Ioannou, I., Andreou, A., Tsikouras, B., Hatzipanagiotou, K. Application of the sharp front model to capillary absorption in a vuggy limestone. Engineering Geology, 105(1-2), 20-23, (2009).
[54] Küntz, M. and Lavallée, P. Experimental evidence and theoretical analysis of anomalous diffusion during water infiltration in porous building materials. Journal of Physics D: Applied Physics, 34(16), 2547, (2001).
[55] Hristov,J., El-Ganaoui,M. Modelling capillary absorption in building materials with emphasis on the fourth root time law: Time-fractional models, solutions and analysis. In Fractional Differential Equations: Theoretical Aspects and Applications (pp. 153-165). Academic Press: New York, (2024).
[56] Hristov, J. A new closed-form approximate solution to diffusion with quadratic Fujita’s nonlinearity: the case of diffusion controlled sorption kinetics relevant to rectangular adsorption isotherms. Heat and Mass Transfer, 55, 261-279, (2019).
[57] Hristov, J. Magnetic field diffusion in ferromagnetic materials: fractional calculus approaches. An International Journal of Optimization and Control: Theories & Applications, 11(3), 1-15, (2021).
[58] Xiao, B., Wang, G., Zhao, L. Feng, C. and Shu, S. An exact solution for the magnetic diffusion problem with a step-function resistivity model. The European Physical Journal Plus, 139, 305, (2024).

Toplam 58 adet kaynakça vardır.

Ayrıntılar

Birincil Dil	İngilizce
Konular	Teorik ve Uygulamalı Mekanik Matematiği
Bölüm	Araştırma Makalesi
Yazarlar	Jordan Hristov 0000-0002-7957-8192
Erken Görünüm Tarihi	9 Ocak 2025
Yayımlanma Tarihi	31 Aralık 2024
Gönderilme Tarihi	8 Eylül 2024
Kabul Tarihi	25 Kasım 2024
Yayımlandığı Sayı	Yıl 2024 Cilt: 4 Sayı: 5-Special Issue: ICAME'24

Kaynak Göster

APA	Hristov, J. (2024). A non-linear diffusion problem with power-law diffusivity: An approximate solution experimenting with a modified sinc function. Mathematical Modelling and Numerical Simulation With Applications, 4(5-Special Issue: ICAME’24), 6-44. https://doi.org/10.53391/mmnsa.1545438