Rasyonel Üslü Cebirsel ve Üstel Eşleme Yaklaşımı ile Thomas-Fermi Denklemi için İkinci Derece Doğruluklu Sonlu Farklar Yöntemi
Öz
Doğa bilimlerine dayalı birçok problemin insanlığa hizmet etmesi için bilim insanları ve mühendisler tarafından çözülmeleri gerekir. Atomik dünyadaki iyi bilinen modellerden biri, bir denklemde yoğunlaşır ve bu denklem Thomas-Fermi denklemi olarak adlandırılır. Thomas-Fermi denklemi ağır, nötr atomların yük dağılımlarını tanımlayan ikinci dereceden bir diferansiyel denklemdir. Denklem için henüz tam bir analitik çözüm bulunamamıştır. Esasen, problemin güçlü nonlineer yapısı, tekil özellik sergilemesi ve sınırsız aralıklı tanım kümesi, yaklaşık sayısal bir çözüm elde etmede de büyük zorluklara yol açmaktadır. Bu makalede, Thomas-Fermi denklemi, sanki-doğrusallaştırma yöntemi ile birlikte ikinci dereceden doğruluklu bir sonlu farklar yöntemi kullanılarak çözülmüştür. Problemin yarı sonsuz aralığı, cebirsel ve üstel eşleme olarak adlandırılan iki farklı koordinat dönüşümü kullanılarak [0, 1) aralığına dönüştürülmüştür. Sayısal doğruluk mertebesi, sistematik ağ sıkılaştırma tekniği kullanılıp hesaplanan başlangıç eğim y'(0) değerlerinin karşılaştırılması ile kontrol edilmiştir. Başlangıç eğimi için hesaplanan sonuçların, literatürde verilen sonuçlarla iyi bir uyum içinde olduğu gösterilmiştir. Son olarak, Richardson ekstrapolasyonunun uygulanmasıyla çözümün doğruluk mertebesi arttırılmıştır.
Anahtar Kelimeler
Thomas-Fermi Denklemi nonlineer ADD Yarı sonsuz aralık Sonlu Farklar Metodu Sanki-lineerleştirme Aralık eşleme
Kaynakça
- Sommerfeld, A., 1932, Asymptotische integration der differentialgleichung des thomas fermischen atoms, Zeitschrift für Physik, 78 283–308.
- van de Vooren and A.I., Dijkstra, D., 1970, The Navier–Stokes solution for laminar flow past a semi-infinite flat plate, Journal of Engineering Mathematics, 4 9–27.
- Wazwaz, A.M., 1999 The modified decomposition method and Pade approximates for solving the Thomas–Fermi equation, Applied Mathematics and Computation, 105 11–19.
- Kumari, A. and Kukreja, V.K., 2022, Sixth order Hermite collocation method for analysis of MRLW equation, Journal of Ocean Engineering and Science xxx (xxxx) xxx, ARTICLE IN PRESS, Available online 24 June 2022, https://doi.org/10.1016/j.joes.2022.06.028
- Mekki, A. and Maâtoug M.A., 2013, Numerical simulation of Kadomtsev–Petviashvili–Benjamin– Bona–Mahony equations using finite difference method, Applied Mathematics and Computation, 219 11214–11222.
- Laurenzi, B.J., 1990, An analytic solution to the Thomas–Fermi equation, Journal of Mathematical Physics, 31 2535–2537. Bender, C.M. and Milton, K.A., Pinsky, S.S., Simmons, L.M., Jr., 1989, A new perturbative approach to nonlinear problems, Journal of Mathematical Physics, 30 1447–1455.
- Gerald, C.F., 1978, Applied Num Analysis, 2.nd Ed., Addison-wesley Publishing Publishing Company, Inc. Grosch, C.E., Orszag, S.A., 1977, Numerical solution of problems in unbounded regions: coordinate transforms, Journal of Computational Physics, 25 273–296.
- Fermi, E., 1928, Eine statistische methode zur bestimmung einiger eigenschaften des atoms und ihre anwendung auf die theorie des periodischen systems der elemente, Zeitschrift für Physik, 48 73–79.
- Hille, E., 1973, Some aspects of the Thomas–Fermi equation, Journal of Analitical Mathematics, 23(1) 147–170.
- Baker, E.B., 1930, The application of the Fermi-Thomas statistical model to the calculation of potential distribution in positive ions, Physical Reviews, 36 630–647.
- Amromin, E.L., 2015, Ships with ventilated cavitation in seaways and active flow control, Applied Ocean Research, 50 163–172.
- Amromin, E.L., 2018, Ships Bottom Cavities as Shock Absorbers in Waves, Journal of Marine Science and Application, 17 173–177.
- de Hoog, F.R., Weiss, R., 1980, An approximation theory for boundary value problems on infinite intervals, Computing, 24 227–239.
- Ahmad, F., Ullah, M.z., Jang, T.S., Alaidarous, E.S., 2017, An efficient method for the static deflection analysis of an infinite beam on a nonlinear elastic foundation of one-way spring model, Ships and Offshore Structures, 12 7, 963-970.
- Lu, F., Song, Z., Zhang, Z., 2016, A Compact Fourth-Order Fınıte Dıfference Scheme For The Improved Boussınesq Equatıon Wıth Dampıng Terms, Journal of Computational Mathematics, 34 5 462–478.
- Mason, J.C., 1964, Rational approximations to the ordinary Thomas–Fermi function and its derivative, Proceedings of Physical Societies, 84 357–359.
- Josea, J., Choib, S.J., Giljarhusc, K.E.T., 2017, Ove Tobias Gudmestada, A comparison of numerical simulations of breaking wave forces on a monopile structure using two different numerical models based on finite difference and finite volume methods, Ocean Engineering, 137 78–88.
- Parand, K., Delkhosh, M., 2017, Accurate solution of the Thomas–Fermi equation using the fractional order of rational Chebyshev functions, Journal of Computations and Applied Mathematics 317 624–642.
- Parand, K., Mazaheri, P., Yousefi, H., Delkhosh, M., 2017, Fractional order of rational Jacobi functions for solving the non-linear singular Thomas-Fermi equation, European Physical Journal Plus, 132 77.
- Richardson L.F., 1927, The deferred approach to the limit, Philosophical Transactions A, 226 299–349.
- Thomas L.H., 1927, The calculation of atomic fields, Mathematics Proceedings Cambridge, 23 542–548.
- Lentini, M., Keller, H.B., 1980, Boundary value problems on semi-infinite intervals and their numerical solutions, SIAM Journal of Numerical Analysis, 17 577–604.
- Pelka, M., Mackenberg, M., Funda, C., Hellbrück, H., 2017, Optical underwater distance estimation, Oceans - Aberdeen, 2017, pp. 1 6.
- Chawla, M.M., Katti, C.P., 1982, Finite difference methods and their convergence for a class of singular two point boundary value problems, Numerical Mathematics, 39.
- Chawla, M.M., Mckee, S., Shaw, G., 1986, Order h2 method for a singular two-point boundary value problem, BIT, 26.
- Anderson, N., Arthurs, A.M., Robinson, P.D., 1968, Variational solutions of the Thomas–Fermi equation, Nuovo Cimento 57 523.
- Setia, N., Mohanty, R.K., 2021, A third-order finite difference method on a quasi-variable mesh for nonlinear two point boundary value problems with Robin boundary conditions. Soft Computing, 25(20), 12775-12788.
- Csavinszky, P., 1968, Physical Reviews 166, 53.
- Markowich, P.A., 1982, A theory for the approximation of solution of boundary value problems on infinite intervals, SIAM Journal Of Mathematical Analysis, 13 484–513.
- Markowich, P.A., 1983, Analysis of boundary value problems on infinite intervals, SIAM Journal Of Mathematical Analysis, 14 11–37.
- Roberts, R.e., Physical Rev. (1968) 170, 8.
- Fazio, R. and Jannelli, A., 2014, Finite difference schemes on quasi-uniform grids for BVPs on infinite intervals, Journal Of Computational and Applied Mathematics 269 14–23.
- Fazio, R., 1992, The Blasius problem formulated as a free boundary value problem, Acta Mechanics, 95 1–7. Bellman, R.E. and Kalaba, R.E., 1965, Quasi-linearization and Nonlinear Boundary-Value Problems, Elsevier Publishing Company, New York.
- Pandey, R.K. and Singh, A.K., 2004, On the convergence of a finite difference method for a class of singular boundary value problems arising in physiology, Journal Of Computational Applied Mathematics, 166.
- Feynman, R.P., Metropolis, N. and Teller, E., 1949, Equations of state of elements based on the generalized Fermi-Thomas theory, Physical Reviews, 75(10) 1561–1573.
- Roul, P., Goura, V.P. and Agarwal, R., 2019, A compact finite difference method for a general class of nonlinear singular boundary value problems with Neumann and Robin boundary conditions. Applied Mathematical Computations, 350 283–304.
- Abbasbandy, S. and Bervillier, C., 2011, Analytic continuation of Taylor series and the boundary value problems of some nonlinear ordinary differential equations, Applied Mathematical Computations, 218 2178–2199.
- Kobayashi, S., Matsukuma, T., Nagi, S. and Umeda, K., 1955, Accurate value of the initial slope of the ordinary T-F function, Journal of Physical Societies Japan, 10 759–762.
- Lee, S.H., Lee, Y.G. and LeolJeong, K., 2011, Numerical simulation of three dimensional sloshing phenomena using a finite difference method with marker-density scheme, Ocean Engineering 38 206–225.
- Zhao, T., Zhang, Z. and Wang, T., 2021, A hybrid augmented compact finite volume method for the Thomas–Fermi equation. Mathematics and Computers in Simulation, 190 760-773.
- Mandelzweig, V.B. and Tabakinb, F., 2001, Quasi-linearization approach to nonlinear problems in physics with application to nonlinear ODEs, Compuations and. Physical Communications, 141 268–281.
- Ford, W., 2015, Numerical Linear Algebra with Applications, Academic Press, Pages 163-179.
- Robin, W., 2018, Another rational analytical approximation to the Thomas–Fermi equation. Journal Of Innovative Technologies Education, 5(1) 7–13.
- Zhang, X. and Boyd, J.P., 2019, Revisiting the Thomas–Fermi equation: accelerating rational Chebyshev series through coordinate transformations. Applied Numerical Mathematics, 135 186–205.
Second Order Finite Difference Method for the Thomas-Fermi Equation via Fractional Order of Algebraic and Exponential Mapping Approach
Öz
Many problems based on natural sciences need to be solved by the scientists and engineers to serve the humanity. One of the well-known model in atomic universe is condensed into an equation, and called the Thomas-Fermi equation. It is a second order differential equation, which describes charge distributions of heavy, neutral atoms. No exact analytical solution has been found for the equation yet. In fact, strong nonlinearity, singular character and unbounded interval of the problem causes great difficulty to obtain an approximate numerical solution as well. In this paper, the Thomas-Fermi equation is solved using a second order finite difference method along with application of quasi-linearization method. Semi-infinite interval of the problem is converted into [0, 1) using two different coordinate transformations, namely algebraic and exponential mapping. Numerical order of accuracy has been checked using systematic mesh refinements and comparing the calculated initial slope y'(0). Calculated results for initial slope is found in good agreement with the results available in the literature. Lastly, accuracy is improved by the application of the Richardson extrapolation.
Anahtar Kelimeler
Thomas-Fermi equation non-linear ODE semi-infinite interval finite difference method quasi-linearization interval mapping
Kaynakça
- Sommerfeld, A., 1932, Asymptotische integration der differentialgleichung des thomas fermischen atoms, Zeitschrift für Physik, 78 283–308.
- van de Vooren and A.I., Dijkstra, D., 1970, The Navier–Stokes solution for laminar flow past a semi-infinite flat plate, Journal of Engineering Mathematics, 4 9–27.
- Wazwaz, A.M., 1999 The modified decomposition method and Pade approximates for solving the Thomas–Fermi equation, Applied Mathematics and Computation, 105 11–19.
- Kumari, A. and Kukreja, V.K., 2022, Sixth order Hermite collocation method for analysis of MRLW equation, Journal of Ocean Engineering and Science xxx (xxxx) xxx, ARTICLE IN PRESS, Available online 24 June 2022, https://doi.org/10.1016/j.joes.2022.06.028
- Mekki, A. and Maâtoug M.A., 2013, Numerical simulation of Kadomtsev–Petviashvili–Benjamin– Bona–Mahony equations using finite difference method, Applied Mathematics and Computation, 219 11214–11222.
- Laurenzi, B.J., 1990, An analytic solution to the Thomas–Fermi equation, Journal of Mathematical Physics, 31 2535–2537. Bender, C.M. and Milton, K.A., Pinsky, S.S., Simmons, L.M., Jr., 1989, A new perturbative approach to nonlinear problems, Journal of Mathematical Physics, 30 1447–1455.
- Gerald, C.F., 1978, Applied Num Analysis, 2.nd Ed., Addison-wesley Publishing Publishing Company, Inc. Grosch, C.E., Orszag, S.A., 1977, Numerical solution of problems in unbounded regions: coordinate transforms, Journal of Computational Physics, 25 273–296.
- Fermi, E., 1928, Eine statistische methode zur bestimmung einiger eigenschaften des atoms und ihre anwendung auf die theorie des periodischen systems der elemente, Zeitschrift für Physik, 48 73–79.
- Hille, E., 1973, Some aspects of the Thomas–Fermi equation, Journal of Analitical Mathematics, 23(1) 147–170.
- Baker, E.B., 1930, The application of the Fermi-Thomas statistical model to the calculation of potential distribution in positive ions, Physical Reviews, 36 630–647.
- Amromin, E.L., 2015, Ships with ventilated cavitation in seaways and active flow control, Applied Ocean Research, 50 163–172.
- Amromin, E.L., 2018, Ships Bottom Cavities as Shock Absorbers in Waves, Journal of Marine Science and Application, 17 173–177.
- de Hoog, F.R., Weiss, R., 1980, An approximation theory for boundary value problems on infinite intervals, Computing, 24 227–239.
- Ahmad, F., Ullah, M.z., Jang, T.S., Alaidarous, E.S., 2017, An efficient method for the static deflection analysis of an infinite beam on a nonlinear elastic foundation of one-way spring model, Ships and Offshore Structures, 12 7, 963-970.
- Lu, F., Song, Z., Zhang, Z., 2016, A Compact Fourth-Order Fınıte Dıfference Scheme For The Improved Boussınesq Equatıon Wıth Dampıng Terms, Journal of Computational Mathematics, 34 5 462–478.
- Mason, J.C., 1964, Rational approximations to the ordinary Thomas–Fermi function and its derivative, Proceedings of Physical Societies, 84 357–359.
- Josea, J., Choib, S.J., Giljarhusc, K.E.T., 2017, Ove Tobias Gudmestada, A comparison of numerical simulations of breaking wave forces on a monopile structure using two different numerical models based on finite difference and finite volume methods, Ocean Engineering, 137 78–88.
- Parand, K., Delkhosh, M., 2017, Accurate solution of the Thomas–Fermi equation using the fractional order of rational Chebyshev functions, Journal of Computations and Applied Mathematics 317 624–642.
- Parand, K., Mazaheri, P., Yousefi, H., Delkhosh, M., 2017, Fractional order of rational Jacobi functions for solving the non-linear singular Thomas-Fermi equation, European Physical Journal Plus, 132 77.
- Richardson L.F., 1927, The deferred approach to the limit, Philosophical Transactions A, 226 299–349.
- Thomas L.H., 1927, The calculation of atomic fields, Mathematics Proceedings Cambridge, 23 542–548.
- Lentini, M., Keller, H.B., 1980, Boundary value problems on semi-infinite intervals and their numerical solutions, SIAM Journal of Numerical Analysis, 17 577–604.
- Pelka, M., Mackenberg, M., Funda, C., Hellbrück, H., 2017, Optical underwater distance estimation, Oceans - Aberdeen, 2017, pp. 1 6.
- Chawla, M.M., Katti, C.P., 1982, Finite difference methods and their convergence for a class of singular two point boundary value problems, Numerical Mathematics, 39.
- Chawla, M.M., Mckee, S., Shaw, G., 1986, Order h2 method for a singular two-point boundary value problem, BIT, 26.
- Anderson, N., Arthurs, A.M., Robinson, P.D., 1968, Variational solutions of the Thomas–Fermi equation, Nuovo Cimento 57 523.
- Setia, N., Mohanty, R.K., 2021, A third-order finite difference method on a quasi-variable mesh for nonlinear two point boundary value problems with Robin boundary conditions. Soft Computing, 25(20), 12775-12788.
- Csavinszky, P., 1968, Physical Reviews 166, 53.
- Markowich, P.A., 1982, A theory for the approximation of solution of boundary value problems on infinite intervals, SIAM Journal Of Mathematical Analysis, 13 484–513.
- Markowich, P.A., 1983, Analysis of boundary value problems on infinite intervals, SIAM Journal Of Mathematical Analysis, 14 11–37.
- Roberts, R.e., Physical Rev. (1968) 170, 8.
- Fazio, R. and Jannelli, A., 2014, Finite difference schemes on quasi-uniform grids for BVPs on infinite intervals, Journal Of Computational and Applied Mathematics 269 14–23.
- Fazio, R., 1992, The Blasius problem formulated as a free boundary value problem, Acta Mechanics, 95 1–7. Bellman, R.E. and Kalaba, R.E., 1965, Quasi-linearization and Nonlinear Boundary-Value Problems, Elsevier Publishing Company, New York.
- Pandey, R.K. and Singh, A.K., 2004, On the convergence of a finite difference method for a class of singular boundary value problems arising in physiology, Journal Of Computational Applied Mathematics, 166.
- Feynman, R.P., Metropolis, N. and Teller, E., 1949, Equations of state of elements based on the generalized Fermi-Thomas theory, Physical Reviews, 75(10) 1561–1573.
- Roul, P., Goura, V.P. and Agarwal, R., 2019, A compact finite difference method for a general class of nonlinear singular boundary value problems with Neumann and Robin boundary conditions. Applied Mathematical Computations, 350 283–304.
- Abbasbandy, S. and Bervillier, C., 2011, Analytic continuation of Taylor series and the boundary value problems of some nonlinear ordinary differential equations, Applied Mathematical Computations, 218 2178–2199.
- Kobayashi, S., Matsukuma, T., Nagi, S. and Umeda, K., 1955, Accurate value of the initial slope of the ordinary T-F function, Journal of Physical Societies Japan, 10 759–762.
- Lee, S.H., Lee, Y.G. and LeolJeong, K., 2011, Numerical simulation of three dimensional sloshing phenomena using a finite difference method with marker-density scheme, Ocean Engineering 38 206–225.
- Zhao, T., Zhang, Z. and Wang, T., 2021, A hybrid augmented compact finite volume method for the Thomas–Fermi equation. Mathematics and Computers in Simulation, 190 760-773.
- Mandelzweig, V.B. and Tabakinb, F., 2001, Quasi-linearization approach to nonlinear problems in physics with application to nonlinear ODEs, Compuations and. Physical Communications, 141 268–281.
- Ford, W., 2015, Numerical Linear Algebra with Applications, Academic Press, Pages 163-179.
- Robin, W., 2018, Another rational analytical approximation to the Thomas–Fermi equation. Journal Of Innovative Technologies Education, 5(1) 7–13.
- Zhang, X. and Boyd, J.P., 2019, Revisiting the Thomas–Fermi equation: accelerating rational Chebyshev series through coordinate transformations. Applied Numerical Mathematics, 135 186–205.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Matematiksel Fizik, Uygulamalı Matematik, Mühendislik |
| Bölüm | Makaleler |
| Yazarlar | |
| Erken Görünüm Tarihi | 22 Haziran 2023 |
| Yayımlanma Tarihi | 28 Haziran 2023 |
| Gönderilme Tarihi | 29 Temmuz 2022 |
| Yayımlandığı Sayı | Yıl 2023 Cilt: 23 Sayı: 3 |
