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Kesirli telegraf kısmi diferansiyel denklemin varyasyonel iterasyon metoduyla çözümü

Yıl 2022, Cilt: 24 Sayı: 1, 182 - 196, 05.01.2022

Mahmut Modanlı , Ayşe Aksoy

https://doi.org/10.25092/baunfbed.884328
Cited By: 1

Öz

Bu çalışmada, Caputo türeviyle tanımlı kesirli mertebeden telegraf kısmi diferansiyel denkleminin başlangıç-sınır değer koşullarına bağlı yaklaşık çözümü incelendi. Bu denklem için varyasyonel iterasyon metodunun çözüm prosedürü sunuldu. Bu metot için Lagrange parametresi belirlenip doğrulama fonksiyoneli oluşturuldu. Kesirli mertebeden telegraf kısmi diferansiyel denklemin örnek bir probleminin verilen başlangıç değerleri kullanılarak varyasyonel iterasyon metodu ile nümerik çözümleri elde edildi.

Anahtar Kelimeler

Kesirli telegraf kısmi diferansiyel denklemi varyasyonel iterasyon metodu Langrange parametresi yaklaşık çözüm.

Destekleyen Kurum

yok

Kaynakça

  • Podlubny, I., Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, (1999).
  • Samko, S.G., Kilbass A.A., ve Marchıev, O.I., Fractional integrals and derivatives, Theory and Applications, Gordon and Breach, Yverdon, 973s., (1993).
  • Kılbass, A., Srıvastava, H., ve Trujıllo, J., Theory and applications of fractional differential equations, North-Holland Mathematics Studies 204, North- Holland, 521 pp, (2006).
  • Caputo, M., ve Mainardi, F., Linear models of dissipation in anelastic solids, La Rivista del Nuovo Cimento, 1, 2, 161-198, (1971).
  • He, J.-H., Nonlinear oscillation with fractional derivative and its applications, in Proceedings of the International Conference on Vibrating Engineering 98, vol. 9, Dalian, China, (1998).
  • He, J.-H., Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178, 3-4, 257–262, (1999).
  • He, J.-H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering, 167, 1-2, 57–68, (1998).
  • Barone, A., Esposıto, F., Magee, C. J., ve Scott, A. C., Theory and applications of the sine-gordon equation, La Rivista del Nuovo Cimento, 1(2): 227–267, (1971).
  • Yusufoglu, E., The variational iteration method for studying the Klein- Gordon equation, Applied Mathematics Letters, 21, 7, 669–674, (2008).
  • Khan, M., Hyder Ali, S., ve Qi, H., On accelerated flows of a viscoelastic fluid with the fractional Burgers' model, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 10, 4, 2286–2296, (2009).
  • Elbeleze, A. A., Kılıçman A., ve Taib, B. M., Homotopy perturbation method for fractional black-scholes european option pricing equations using Sumudu transform, Mathematical Problems in Engineering, vol. 2013, Article ID 524852, 7 pages, (2013).
  • Wu, J., Theory and Applications of Partial Functional Differential Equations, Springer, New York, NY, USA, (1996).
  • Keller, A. A., Contribution of the delay differential equations to the complex economic macrodynamics, WSEAS Transactions on Systems, 9, 4, 358–371, (2010).
  • Sakar, M. G., Uludag, F., ve Erdogan, F., Numerical solution of time-fractional nonlinear PDEs with proportional delays by homotopy perturbation method, Applied Mathematical Modelling, 40, 13-14, 6639–6649, (2016).
  • Lıu, J., ve Hou, G., Numerical solutions of the space- and time-fractional coupled Burgers equations by generalized differential transform method, Applied Mathematics and Computation, 217, 16, 7001–7008, (2011).
  • Momanı, S., ve Odıbat, Z., Analytical solution of a time-fractional Navier–Stokes equation by Adomian decomposition method, Applied Mathematics and Computation, 177, 2, 488–494, (2006).
  • Sakar, M. G., ve Erdogan, F., The homotopy analysis method for solving the time-fractional Fornberg-Whitham equation and comparison with Adomian's decomposition method, Applied Mathematical Modelling, 37, 20-21, 8876–8885, (2013).
  • Kumar, S., ve Kumar, D., Fractional modelling for BBM-Burger equation by using new homotopy analysis transform method, Journal of the Association of Arab Universities for Basic and Applied Sciences, 16, 16–20, (2014).
  • Zubik-Kowal, B., Chebyshev pseudospectral method and waveform relaxation for differential and differential-functional parabolic equations. Applied Numerical Mathematics, 34, 2-3, 309–328, (2000).
  • Kumar, S., Kumar, D., Abbasbandy, S., ve Rashıdı, M. M., Analytical solution of fractional Navier-Stokes equation by using modified Laplace decomposition method, Ain Shams Engineering Journal, 5, 2, 569–574, (2014).
  • Kumar, S., Yıldırım, A., Khan, Y., Jafarı, H., Sayevand, K., ve Weı, L., Analytical solution of fractional Black-Scholes European option pricing equation by using Laplace transform, Journal of Fractional Calculus and Applications, 2, 8, 1-9, (2012).
  • Tanthanuch, J., Symmetry analysis of the nonhomogeneous inviscid Burgers equation with delay, Communications in Nonlinear Science and Numerical Simulation, 17, 12, 4978–4987, (2012).
  • Kurulay, M., The approximate and exact solutions of the space and time-fractional Burggres equations, International Journal of Research and Reviews in Applied Sciences, 3, 3, 257–263, (2010).
  • Momani, S., Erjaee, G. H., ve Alnasr, M. H., The modified homotopy perturbation method for solving strongly nonlinear oscillators, Computers and Mathematics with Applications, 58, 11-12, 2209–2220, (2009).
  • Sıngh, J., Kumar, D., ve Kılıçman, A., Homotopy perturbation method for fractional gas dynamics equation using Sumudu transform, Abstract and Applied Analysis, vol. 2013, Article ID 934060, 8 pages, (2013).
  • Golmankhaneh, A. K., Golmankhaneh, A. K., ve Baleanu, D., On nonlinear fractional KleinGordon equation, Signal Processing, 91, 3, 446–451, (2011).
  • Kurulay, M., Solving the fractional nonlinear Klein-Gordon equation by means of the homotopy analysis method, Advances in Difference Equations, p. 2012, (2012).
  • Yang, X. J., Baleanu, D., Khan, Y., ve Mohyud-Din, S. T., Local fractional variational iteration method for diffusion and wave equations on Cantor sets, Romanian Journal of Physics, 59, 1-2, 36-48, (2014).
  • Geng, F., Lın, Y., ve Cuı, M., A piecewise variational iteration method for Riccati differential equations, Computers and Mathematics with Applications, 58, 11-12, 2518–2522, (2009).
  • Prakash, A., Kumar, M., ve Sharma, K. K., Numerical method for solving fractional coupled Burgers equations, Applied Mathematics and Computation, 260, 314-320, (2015).
  • Sakar, M. G., ve Ergören, H., Alternative variational iteration method for solving the time-fractional Fornberg-Whitham equation, Applied Mathematical Modelling, 39, 14, 3972–3979, (2015).
  • Singh, B. K., ve Kumar, P., Fractional Variational Iteration Method for Solving Fractional Partial Differential Equations with Proportional Delay, International Journal of Differential Equations, vol. 2017, Article ID 5206380, 11 pages, (2017).
  • Elbeleze, A. A., Kiliçman, A., ve Taib, B. M., Fractional Variational Iteration Method and Its Application to Fractional Partial Differential Equation, Mathematical Problems in Engineering, vol. 2013, Article ID 543848, 10 pages, (2013).
  • Modanli, M., Faraj, B. M., ve Ahmed, F. W., Using matrix stability for variable telegraph partial differential equation, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 10(2), 237-243, (2020).
  • Modanli, M. (2018). Kesirli telegraf kısmi diferansiyel denklemlerin fark şeması metodu ile nümerik çözümü, Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 20(1), 440-449, (2018).
  • Modanli, M., On the numerical solution for third order fractional partial differential equation by difference scheme method, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 9(3), 1-5, (2019).
  • Yavuz, M., ve Abdeljawad, T., Nonlinear regularized long-wave models with a new integral transformation applied to the fractional derivative with power and Mittag-Leffler kernel, Advances in Difference Equations, 2020(1), 1-18, (2020).
  • Yavuz, M., ve Özdemir, N., Analysis of an epidemic spreading model with exponential decay law, Mathematical Sciences and Applications E-Notes, 8(1), 142-154, (2020).
  • Evirgen, F., Analyze the optimal solutions of optimization problems by means of fractional gradient based system using VIM, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 6(2), 75-83, (2016).
  • Sarp, Ü., Evirgen, F., ve İkikardeş, S., Applications of differential transformation method to solve systems of ordinary and partial differential equations, Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 20(2), 135-156, (2018).
  • Yavuz, M., ve Özdemir, N., Comparing the new fractional derivative operators involving exponential and Mittag-Leffler kernel, Discrete & Continuous Dynamical Systems-S, 13(3), 995, (2020).
  • Yokus, A., Durur, H., Ahmad, H., Thounthong, P., ve Zhang, Y. F. Construction of exact traveling wave solutions of the Bogoyavlenskii equation by (G′/G, 1/G)-expansion and (1/G′)-expansion techniques, Results in Physics, 19, 103409, (2020).
  • Yokus, A., Durur, H., ve Ahmad, H., Hyperbolic type solutions for the couple Boiti-Leon-Pempinelli system, Facta Universitatis, Series: Mathematics and Informatics, 35(2), 523-531, (2020).
  • Yokus, A., Durur, H., Kaya, D., Ahmad, H., ve Nofal, T. A., Numerical Comparison of Caputo and Conformable Derivatives of Time Fractional Burgers-Fisher Equation. Results in Physics, 104247, (2021).
  • Ahmad, H., Khan, T. A., Durur, H., Ismail, G. M., ve Yokus, A., Analytic approximate solutions of diffusion equations arising in oil pollution, Journal of Ocean Engineering and Science, 6(1), 62-69, (2021).
  • Koksal, M. E., Time and frequency responses of non-integer order RLC circuits, AIMS Mathematics, 4(1), 64-78, (2019).
  • Koksal, M. E., Senol, M., ve Unver, A. K., Numerical simulation of power transmission lines, Chinese Journal of Physics, 59, 507-524, (2019).
  • Modanli, M., Abdulazeez, S. T., ve Husien, A. M., On variational iteration method for solving pseudo-hyperbolic partial differential equations with nonlocal conditions, (under r

On the numerical solution of the fractional telegraph partial differential equation with variational iteration method

Yıl 2022, Cilt: 24 Sayı: 1, 182 - 196, 05.01.2022

Mahmut Modanlı , Ayşe Aksoy

https://doi.org/10.25092/baunfbed.884328
Cited By: 1

Öz

In this work, the fractional order telegraph partial differential equation defined by Caputo derivative depend on initial-boundry value conditions is investigated for approximate solutions. The solution procedure of the variational iteration method is presented for this fractional order telegraph partial differential equation. Lagrange parameter is determined and correction functional is formed. The numerical solutions of an example problem for the fractional order telegraph partial differential equation are obtained by variational iteration method by using the given initial values.

Anahtar Kelimeler

Fractional telegraph partial differential equations varitional iteration method, Langrange parameter approximation solution

Kaynakça

  • Podlubny, I., Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, (1999).
  • Samko, S.G., Kilbass A.A., ve Marchıev, O.I., Fractional integrals and derivatives, Theory and Applications, Gordon and Breach, Yverdon, 973s., (1993).
  • Kılbass, A., Srıvastava, H., ve Trujıllo, J., Theory and applications of fractional differential equations, North-Holland Mathematics Studies 204, North- Holland, 521 pp, (2006).
  • Caputo, M., ve Mainardi, F., Linear models of dissipation in anelastic solids, La Rivista del Nuovo Cimento, 1, 2, 161-198, (1971).
  • He, J.-H., Nonlinear oscillation with fractional derivative and its applications, in Proceedings of the International Conference on Vibrating Engineering 98, vol. 9, Dalian, China, (1998).
  • He, J.-H., Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178, 3-4, 257–262, (1999).
  • He, J.-H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering, 167, 1-2, 57–68, (1998).
  • Barone, A., Esposıto, F., Magee, C. J., ve Scott, A. C., Theory and applications of the sine-gordon equation, La Rivista del Nuovo Cimento, 1(2): 227–267, (1971).
  • Yusufoglu, E., The variational iteration method for studying the Klein- Gordon equation, Applied Mathematics Letters, 21, 7, 669–674, (2008).
  • Khan, M., Hyder Ali, S., ve Qi, H., On accelerated flows of a viscoelastic fluid with the fractional Burgers' model, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 10, 4, 2286–2296, (2009).
  • Elbeleze, A. A., Kılıçman A., ve Taib, B. M., Homotopy perturbation method for fractional black-scholes european option pricing equations using Sumudu transform, Mathematical Problems in Engineering, vol. 2013, Article ID 524852, 7 pages, (2013).
  • Wu, J., Theory and Applications of Partial Functional Differential Equations, Springer, New York, NY, USA, (1996).
  • Keller, A. A., Contribution of the delay differential equations to the complex economic macrodynamics, WSEAS Transactions on Systems, 9, 4, 358–371, (2010).
  • Sakar, M. G., Uludag, F., ve Erdogan, F., Numerical solution of time-fractional nonlinear PDEs with proportional delays by homotopy perturbation method, Applied Mathematical Modelling, 40, 13-14, 6639–6649, (2016).
  • Lıu, J., ve Hou, G., Numerical solutions of the space- and time-fractional coupled Burgers equations by generalized differential transform method, Applied Mathematics and Computation, 217, 16, 7001–7008, (2011).
  • Momanı, S., ve Odıbat, Z., Analytical solution of a time-fractional Navier–Stokes equation by Adomian decomposition method, Applied Mathematics and Computation, 177, 2, 488–494, (2006).
  • Sakar, M. G., ve Erdogan, F., The homotopy analysis method for solving the time-fractional Fornberg-Whitham equation and comparison with Adomian's decomposition method, Applied Mathematical Modelling, 37, 20-21, 8876–8885, (2013).
  • Kumar, S., ve Kumar, D., Fractional modelling for BBM-Burger equation by using new homotopy analysis transform method, Journal of the Association of Arab Universities for Basic and Applied Sciences, 16, 16–20, (2014).
  • Zubik-Kowal, B., Chebyshev pseudospectral method and waveform relaxation for differential and differential-functional parabolic equations. Applied Numerical Mathematics, 34, 2-3, 309–328, (2000).
  • Kumar, S., Kumar, D., Abbasbandy, S., ve Rashıdı, M. M., Analytical solution of fractional Navier-Stokes equation by using modified Laplace decomposition method, Ain Shams Engineering Journal, 5, 2, 569–574, (2014).
  • Kumar, S., Yıldırım, A., Khan, Y., Jafarı, H., Sayevand, K., ve Weı, L., Analytical solution of fractional Black-Scholes European option pricing equation by using Laplace transform, Journal of Fractional Calculus and Applications, 2, 8, 1-9, (2012).
  • Tanthanuch, J., Symmetry analysis of the nonhomogeneous inviscid Burgers equation with delay, Communications in Nonlinear Science and Numerical Simulation, 17, 12, 4978–4987, (2012).
  • Kurulay, M., The approximate and exact solutions of the space and time-fractional Burggres equations, International Journal of Research and Reviews in Applied Sciences, 3, 3, 257–263, (2010).
  • Momani, S., Erjaee, G. H., ve Alnasr, M. H., The modified homotopy perturbation method for solving strongly nonlinear oscillators, Computers and Mathematics with Applications, 58, 11-12, 2209–2220, (2009).
  • Sıngh, J., Kumar, D., ve Kılıçman, A., Homotopy perturbation method for fractional gas dynamics equation using Sumudu transform, Abstract and Applied Analysis, vol. 2013, Article ID 934060, 8 pages, (2013).
  • Golmankhaneh, A. K., Golmankhaneh, A. K., ve Baleanu, D., On nonlinear fractional KleinGordon equation, Signal Processing, 91, 3, 446–451, (2011).
  • Kurulay, M., Solving the fractional nonlinear Klein-Gordon equation by means of the homotopy analysis method, Advances in Difference Equations, p. 2012, (2012).
  • Yang, X. J., Baleanu, D., Khan, Y., ve Mohyud-Din, S. T., Local fractional variational iteration method for diffusion and wave equations on Cantor sets, Romanian Journal of Physics, 59, 1-2, 36-48, (2014).
  • Geng, F., Lın, Y., ve Cuı, M., A piecewise variational iteration method for Riccati differential equations, Computers and Mathematics with Applications, 58, 11-12, 2518–2522, (2009).
  • Prakash, A., Kumar, M., ve Sharma, K. K., Numerical method for solving fractional coupled Burgers equations, Applied Mathematics and Computation, 260, 314-320, (2015).
  • Sakar, M. G., ve Ergören, H., Alternative variational iteration method for solving the time-fractional Fornberg-Whitham equation, Applied Mathematical Modelling, 39, 14, 3972–3979, (2015).
  • Singh, B. K., ve Kumar, P., Fractional Variational Iteration Method for Solving Fractional Partial Differential Equations with Proportional Delay, International Journal of Differential Equations, vol. 2017, Article ID 5206380, 11 pages, (2017).
  • Elbeleze, A. A., Kiliçman, A., ve Taib, B. M., Fractional Variational Iteration Method and Its Application to Fractional Partial Differential Equation, Mathematical Problems in Engineering, vol. 2013, Article ID 543848, 10 pages, (2013).
  • Modanli, M., Faraj, B. M., ve Ahmed, F. W., Using matrix stability for variable telegraph partial differential equation, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 10(2), 237-243, (2020).
  • Modanli, M. (2018). Kesirli telegraf kısmi diferansiyel denklemlerin fark şeması metodu ile nümerik çözümü, Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 20(1), 440-449, (2018).
  • Modanli, M., On the numerical solution for third order fractional partial differential equation by difference scheme method, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 9(3), 1-5, (2019).
  • Yavuz, M., ve Abdeljawad, T., Nonlinear regularized long-wave models with a new integral transformation applied to the fractional derivative with power and Mittag-Leffler kernel, Advances in Difference Equations, 2020(1), 1-18, (2020).
  • Yavuz, M., ve Özdemir, N., Analysis of an epidemic spreading model with exponential decay law, Mathematical Sciences and Applications E-Notes, 8(1), 142-154, (2020).
  • Evirgen, F., Analyze the optimal solutions of optimization problems by means of fractional gradient based system using VIM, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 6(2), 75-83, (2016).
  • Sarp, Ü., Evirgen, F., ve İkikardeş, S., Applications of differential transformation method to solve systems of ordinary and partial differential equations, Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 20(2), 135-156, (2018).
  • Yavuz, M., ve Özdemir, N., Comparing the new fractional derivative operators involving exponential and Mittag-Leffler kernel, Discrete & Continuous Dynamical Systems-S, 13(3), 995, (2020).
  • Yokus, A., Durur, H., Ahmad, H., Thounthong, P., ve Zhang, Y. F. Construction of exact traveling wave solutions of the Bogoyavlenskii equation by (G′/G, 1/G)-expansion and (1/G′)-expansion techniques, Results in Physics, 19, 103409, (2020).
  • Yokus, A., Durur, H., ve Ahmad, H., Hyperbolic type solutions for the couple Boiti-Leon-Pempinelli system, Facta Universitatis, Series: Mathematics and Informatics, 35(2), 523-531, (2020).
  • Yokus, A., Durur, H., Kaya, D., Ahmad, H., ve Nofal, T. A., Numerical Comparison of Caputo and Conformable Derivatives of Time Fractional Burgers-Fisher Equation. Results in Physics, 104247, (2021).
  • Ahmad, H., Khan, T. A., Durur, H., Ismail, G. M., ve Yokus, A., Analytic approximate solutions of diffusion equations arising in oil pollution, Journal of Ocean Engineering and Science, 6(1), 62-69, (2021).
  • Koksal, M. E., Time and frequency responses of non-integer order RLC circuits, AIMS Mathematics, 4(1), 64-78, (2019).
  • Koksal, M. E., Senol, M., ve Unver, A. K., Numerical simulation of power transmission lines, Chinese Journal of Physics, 59, 507-524, (2019).
  • Modanli, M., Abdulazeez, S. T., ve Husien, A. M., On variational iteration method for solving pseudo-hyperbolic partial differential equations with nonlocal conditions, (under r
Toplam 48 adet kaynakça vardır.

Ayrıntılar

Birincil Dil Türkçe
Bölüm Araştırma Makalesi
Yazarlar

Mahmut Modanlı 0000-0002-7743-3512

Ayşe Aksoy Bu kişi benim 0000-0001-7179-0124

Yayımlanma Tarihi 5 Ocak 2022
Gönderilme Tarihi 9 Mart 2021
Yayımlandığı Sayı Yıl 2022 Cilt: 24 Sayı: 1

Kaynak Göster

APA Modanlı, M., & Aksoy, A. (2022). Kesirli telegraf kısmi diferansiyel denklemin varyasyonel iterasyon metoduyla çözümü. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 24(1), 182-196. https://doi.org/10.25092/baunfbed.884328
AMA Modanlı M, Aksoy A. Kesirli telegraf kısmi diferansiyel denklemin varyasyonel iterasyon metoduyla çözümü. BAUN Fen. Bil. Enst. Dergisi. Ocak 2022;24(1):182-196. doi:10.25092/baunfbed.884328
Chicago Modanlı, Mahmut, ve Ayşe Aksoy. “Kesirli Telegraf kısmi Diferansiyel Denklemin Varyasyonel Iterasyon Metoduyla çözümü”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 24, sy. 1 (Ocak 2022): 182-96. https://doi.org/10.25092/baunfbed.884328.
EndNote Modanlı M, Aksoy A (01 Ocak 2022) Kesirli telegraf kısmi diferansiyel denklemin varyasyonel iterasyon metoduyla çözümü. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 24 1 182–196.
IEEE M. Modanlı ve A. Aksoy, “Kesirli telegraf kısmi diferansiyel denklemin varyasyonel iterasyon metoduyla çözümü”, BAUN Fen. Bil. Enst. Dergisi, c. 24, sy. 1, ss. 182–196, 2022, doi: 10.25092/baunfbed.884328.
ISNAD Modanlı, Mahmut - Aksoy, Ayşe. “Kesirli Telegraf kısmi Diferansiyel Denklemin Varyasyonel Iterasyon Metoduyla çözümü”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 24/1 (Ocak 2022), 182-196. https://doi.org/10.25092/baunfbed.884328.
JAMA Modanlı M, Aksoy A. Kesirli telegraf kısmi diferansiyel denklemin varyasyonel iterasyon metoduyla çözümü. BAUN Fen. Bil. Enst. Dergisi. 2022;24:182–196.
MLA Modanlı, Mahmut ve Ayşe Aksoy. “Kesirli Telegraf kısmi Diferansiyel Denklemin Varyasyonel Iterasyon Metoduyla çözümü”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, c. 24, sy. 1, 2022, ss. 182-96, doi:10.25092/baunfbed.884328.
Vancouver Modanlı M, Aksoy A. Kesirli telegraf kısmi diferansiyel denklemin varyasyonel iterasyon metoduyla çözümü. BAUN Fen. Bil. Enst. Dergisi. 2022;24(1):182-96.

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Fractional model for blood flow under MHD influence in porous and non-porous media
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https://doi.org/10.11121/ijocta.1497