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Year 2017, Volume: 5 Issue: 4, 209 - 219, 01.10.2017

Abstract

References

  • Alkahtania, B. S. T., et al., New Numerical Analysis of Riemann-Liouville Time-Fractional Schrodinger with Power, Exponential Decay, and Mittag-Leffler Laws, Journal of Nonlinear Sciences and Applications, 10(8), 4231-4243, (2017).
  • Evirgen, F. and N. Özdemir, A Fractional Order Dynamical Trajectory Approach for Optimization Problem with Hpm, in: Fractional Dynamics and Control, (Ed. D. Baleanu, Machado, J.A.T., Luo, A.C.J.), Springer, 145-155, (2012).
  • Momani, S. and Z. Odibat, Analytical Approach to Linear Fractional Partial Differential Equations Arising in Fluid Mechanics, Physics Letters A, 355(4), 271-279, (2006).
  • Özdemir, N. and M. Yavuz, Numerical Solution of Fractional Black-Scholes Equation by Using the Multivariate Padé Approximation, Acta Physica Polonica A, 132(3), 1050-1053, (2017).
  • Turut, V. and N. Güzel, Multivariate Pade Approximation for Solving Nonlinear Partial Differential Equations of Fractional Order, Abstract and Applied Analysis, 2013, (2013).
  • Yavuz, M., et al., Generalized Differential Transform Method for Fractional Partial Differential Equation from Finance, Proceedings, International Conference on Fractional Differentiation and its Applications, Novi Sad, Serbia, 778-785, (2016).
  • Duan, J.-S., et al., A Review of the Adomian Decomposition Method and Its Applications to Fractional Differential Equations, Communications in Fractional Calculus, 3(2), 73-99, (2012).
  • Wazwaz, A.-M., A New Algorithm for Calculating Adomian Polynomials for Nonlinear Operators, Applied Mathematics and Computation, 111(1), 33-51, (2000).
  • Adomian, G., A Review of the Decomposition Method and Some Recent Results for Nonlinear Equations, Mathematical and Computer Modelling, 13(7), 17-43, (1990).
  • Bildik, N. and H. Bayramoglu, The Solution of Two Dimensional Nonlinear Differential Equation by the Adomian Decomposition Method, Applied mathematics and computation, 163(2), 519-524, (2005).
  • Bildik, N., et al., Solution of Different Type of the Partial Differential Equation by Differential Transform Method and Adomian’s Decomposition Method, Applied Mathematics and Computation, 172(1), 551-567, (2006).
  • Daftardar-Gejji, V. and H. Jafari, Adomian Decomposition: A Tool for Solving a System of Fractional Differential Equations, Journal of Mathematical Analysis and Applications, 301(2), 508-518, (2005).
  • El-Sayed, A. and M. Gaber, The Adomian Decomposition Method for Solving Partial Differential Equations of Fractal Order in Finite Domains, Physics Letters A, 359(3), 175-182, (2006).
  • El-Wakil, S., et al., Adomian Decomposition Method for Solving the Diffusion–Convection–Reaction Equations, Applied Mathematics and Computation, 177(2), 729-736, (2006).
  • Evirgen, F. and N. Özdemir, Multistage Adomian Decomposition Method for Solving Nlp Problems over a Nonlinear Fractional Dynamical System, Journal of Computational and Nonlinear Dynamics, 6(2), 021003, (2011).
  • 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, 6(2), 75, (2016).
  • He, J. H., Variational Iteration Method: Some Recent Results and New Interpretations, Journal of Computational and Applied Mathematics, 207(1), 3-17, (2007).
  • Turut, V. and N. Güzel, On Solving Partial Differential Equations of Fractional Order by Using the Variational Iteration Method and Multivariate Padé Approximations, European Journal of Pure and Applied Mathematics, 6(2), 147-171, (2013).
  • İbiş, B. and M. Bayram, Approximate Solution of Time-Fractional Advection-Dispersion Equation Via Fractional Variational Iteration Method, The Scientific World Journal, 2014, (2014).
  • İbiş, B. and M. Bayram, Analytical Approximate Solution of Time-Fractional Fornberg–Whitham Equation by the Fractional Variational Iteration Method, Alexandria Engineering Journal, 53(4), 911-915, (2014).
  • Zhang, Y., A Finite Difference Method for Fractional Partial Differential Equation, Applied Mathematics and Computation, 215(2), 524-529, (2009).
  • Meerschaert, M. M. and C. Tadjeran, Finite Difference Approximations for Two-Sided Space-Fractional Partial Differential Equations, Applied numerical mathematics, 56(1), 80-90, (2006).
  • Shawagfeh, N. T., Analytical Approximate Solutions for Nonlinear Fractional Differential Equations, Applied Mathematics and Computation, 131(2), 517-529, (2002).
  • Khalil, R., et al., A New Definition of Fractional Derivative, Journal of Computational and Applied Mathematics, 264, 65-70, (2014).
  • Anderson, D. and D. Ulness, Newly Defined Conformable Derivatives, Adv. Dyn. Syst. Appl, 10(2), 109-137, (2015).
  • Atangana, A., et al., New Properties of Conformable Derivative, Open Mathematics, 13(1), (2015).
  • Abdeljawad, T., On Conformable Fractional Calculus, Journal of computational and Applied Mathematics, 279, 57-66, (2015).
  • Avcı, D., et al., Conformable Fractional Wave-Like Equation on a Radial Symmetric Plate, in: Theory and Applications of Non-Integer Order Systems, (Ed. A. Babiarz, Czornik, A., Klamka, J., Niezabitowski, M.), Springer, 137-146, (2017).
  • Avcı, D., et al., Conformable Heat Problem in a Cylinder, Proceedings, International Conference on Fractional Differentiation and its Applications, 572-581, (2016).
  • Avci, D., et al., Conformable Heat Equation on a Radial Symmetric Plate, Thermal Science, 21(2), 819-826, (2017).
  • Eroğlu, B. B. İ., et al., Optimal Control Problem for a Conformable Fractional Heat Conduction Equation, Acta Physica Polonica A, 132(3), 658-662, (2017).
  • Evirgen, F., Conformable Fractional Gradient Based Dynamic System for Constrained Optimization Problem, Acta Physica Polonica A, 132(3), 1066-1069, (2017).
  • Yavuz, M., Novel Solution Methods for Initial Boundary Value Problems of Fractional Order with Conformable Differentiation, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 8(1), 1-7, (2018).
  • Liu, F., et al., Stability and Convergence of Two New Implicit Numerical Methods for Fractional Cable Equation, Proceedings, Proceeding of the ASME 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, IDETC/CIE, San Diego, California, USA, (2009).
  • Hu, X. and L. Zhang, Implicit Compact Difference Schemes for the Fractional Cable Equation, Applied Mathematical Modelling, 36(9), 4027-4043, (2012).
  • Quintana-Murillo, J. and S. Yuste, An Explicit Numerical Method for the Fractional Cable Equation, International Journal of Differential Equations, 2011, (2011).
  • Acan, O. and D. Baleanu, A New Numerical Technique for Solving Fractional Partial Differential Equations, arXiv preprint arXiv:1704.02575, (2017).
  • Acan, O., et al., Conformable Variational Iteration Method, New Trends in Mathematical Sciences, 5(5), 172-178, (2017).

Approximate-analytical solutions of cable equation using conformable fractional operator

Year 2017, Volume: 5 Issue: 4, 209 - 219, 01.10.2017

Abstract

In the present work, we have introduced a new formulation for the approximate-analytical solution of the fractional one-dimensional cable differential equation (FCE) by using the conformable fractional derivative. First of all, we have redefined Adomian decomposition method (CADM) and variational iteration method (CVIM) in the conformable sense. Then, we have solved by using the mentioned methods, which can analytically solve the fractional partial differential equations (FPDEs). In order to represent the efficiencies of these proposed methods, we have compared the numerical and exact solutions of the (FCE). Also, we have found out that the proposed models defined with the conformable derivative operator are very efficient and powerful techniques in finding approximate- analytical solutions for the cable equation of fractional order. In addition, the classical derivative and integral properties are recovered partially when the fractional term (alpha) is equal to one.

References

  • Alkahtania, B. S. T., et al., New Numerical Analysis of Riemann-Liouville Time-Fractional Schrodinger with Power, Exponential Decay, and Mittag-Leffler Laws, Journal of Nonlinear Sciences and Applications, 10(8), 4231-4243, (2017).
  • Evirgen, F. and N. Özdemir, A Fractional Order Dynamical Trajectory Approach for Optimization Problem with Hpm, in: Fractional Dynamics and Control, (Ed. D. Baleanu, Machado, J.A.T., Luo, A.C.J.), Springer, 145-155, (2012).
  • Momani, S. and Z. Odibat, Analytical Approach to Linear Fractional Partial Differential Equations Arising in Fluid Mechanics, Physics Letters A, 355(4), 271-279, (2006).
  • Özdemir, N. and M. Yavuz, Numerical Solution of Fractional Black-Scholes Equation by Using the Multivariate Padé Approximation, Acta Physica Polonica A, 132(3), 1050-1053, (2017).
  • Turut, V. and N. Güzel, Multivariate Pade Approximation for Solving Nonlinear Partial Differential Equations of Fractional Order, Abstract and Applied Analysis, 2013, (2013).
  • Yavuz, M., et al., Generalized Differential Transform Method for Fractional Partial Differential Equation from Finance, Proceedings, International Conference on Fractional Differentiation and its Applications, Novi Sad, Serbia, 778-785, (2016).
  • Duan, J.-S., et al., A Review of the Adomian Decomposition Method and Its Applications to Fractional Differential Equations, Communications in Fractional Calculus, 3(2), 73-99, (2012).
  • Wazwaz, A.-M., A New Algorithm for Calculating Adomian Polynomials for Nonlinear Operators, Applied Mathematics and Computation, 111(1), 33-51, (2000).
  • Adomian, G., A Review of the Decomposition Method and Some Recent Results for Nonlinear Equations, Mathematical and Computer Modelling, 13(7), 17-43, (1990).
  • Bildik, N. and H. Bayramoglu, The Solution of Two Dimensional Nonlinear Differential Equation by the Adomian Decomposition Method, Applied mathematics and computation, 163(2), 519-524, (2005).
  • Bildik, N., et al., Solution of Different Type of the Partial Differential Equation by Differential Transform Method and Adomian’s Decomposition Method, Applied Mathematics and Computation, 172(1), 551-567, (2006).
  • Daftardar-Gejji, V. and H. Jafari, Adomian Decomposition: A Tool for Solving a System of Fractional Differential Equations, Journal of Mathematical Analysis and Applications, 301(2), 508-518, (2005).
  • El-Sayed, A. and M. Gaber, The Adomian Decomposition Method for Solving Partial Differential Equations of Fractal Order in Finite Domains, Physics Letters A, 359(3), 175-182, (2006).
  • El-Wakil, S., et al., Adomian Decomposition Method for Solving the Diffusion–Convection–Reaction Equations, Applied Mathematics and Computation, 177(2), 729-736, (2006).
  • Evirgen, F. and N. Özdemir, Multistage Adomian Decomposition Method for Solving Nlp Problems over a Nonlinear Fractional Dynamical System, Journal of Computational and Nonlinear Dynamics, 6(2), 021003, (2011).
  • 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, 6(2), 75, (2016).
  • He, J. H., Variational Iteration Method: Some Recent Results and New Interpretations, Journal of Computational and Applied Mathematics, 207(1), 3-17, (2007).
  • Turut, V. and N. Güzel, On Solving Partial Differential Equations of Fractional Order by Using the Variational Iteration Method and Multivariate Padé Approximations, European Journal of Pure and Applied Mathematics, 6(2), 147-171, (2013).
  • İbiş, B. and M. Bayram, Approximate Solution of Time-Fractional Advection-Dispersion Equation Via Fractional Variational Iteration Method, The Scientific World Journal, 2014, (2014).
  • İbiş, B. and M. Bayram, Analytical Approximate Solution of Time-Fractional Fornberg–Whitham Equation by the Fractional Variational Iteration Method, Alexandria Engineering Journal, 53(4), 911-915, (2014).
  • Zhang, Y., A Finite Difference Method for Fractional Partial Differential Equation, Applied Mathematics and Computation, 215(2), 524-529, (2009).
  • Meerschaert, M. M. and C. Tadjeran, Finite Difference Approximations for Two-Sided Space-Fractional Partial Differential Equations, Applied numerical mathematics, 56(1), 80-90, (2006).
  • Shawagfeh, N. T., Analytical Approximate Solutions for Nonlinear Fractional Differential Equations, Applied Mathematics and Computation, 131(2), 517-529, (2002).
  • Khalil, R., et al., A New Definition of Fractional Derivative, Journal of Computational and Applied Mathematics, 264, 65-70, (2014).
  • Anderson, D. and D. Ulness, Newly Defined Conformable Derivatives, Adv. Dyn. Syst. Appl, 10(2), 109-137, (2015).
  • Atangana, A., et al., New Properties of Conformable Derivative, Open Mathematics, 13(1), (2015).
  • Abdeljawad, T., On Conformable Fractional Calculus, Journal of computational and Applied Mathematics, 279, 57-66, (2015).
  • Avcı, D., et al., Conformable Fractional Wave-Like Equation on a Radial Symmetric Plate, in: Theory and Applications of Non-Integer Order Systems, (Ed. A. Babiarz, Czornik, A., Klamka, J., Niezabitowski, M.), Springer, 137-146, (2017).
  • Avcı, D., et al., Conformable Heat Problem in a Cylinder, Proceedings, International Conference on Fractional Differentiation and its Applications, 572-581, (2016).
  • Avci, D., et al., Conformable Heat Equation on a Radial Symmetric Plate, Thermal Science, 21(2), 819-826, (2017).
  • Eroğlu, B. B. İ., et al., Optimal Control Problem for a Conformable Fractional Heat Conduction Equation, Acta Physica Polonica A, 132(3), 658-662, (2017).
  • Evirgen, F., Conformable Fractional Gradient Based Dynamic System for Constrained Optimization Problem, Acta Physica Polonica A, 132(3), 1066-1069, (2017).
  • Yavuz, M., Novel Solution Methods for Initial Boundary Value Problems of Fractional Order with Conformable Differentiation, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 8(1), 1-7, (2018).
  • Liu, F., et al., Stability and Convergence of Two New Implicit Numerical Methods for Fractional Cable Equation, Proceedings, Proceeding of the ASME 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, IDETC/CIE, San Diego, California, USA, (2009).
  • Hu, X. and L. Zhang, Implicit Compact Difference Schemes for the Fractional Cable Equation, Applied Mathematical Modelling, 36(9), 4027-4043, (2012).
  • Quintana-Murillo, J. and S. Yuste, An Explicit Numerical Method for the Fractional Cable Equation, International Journal of Differential Equations, 2011, (2011).
  • Acan, O. and D. Baleanu, A New Numerical Technique for Solving Fractional Partial Differential Equations, arXiv preprint arXiv:1704.02575, (2017).
  • Acan, O., et al., Conformable Variational Iteration Method, New Trends in Mathematical Sciences, 5(5), 172-178, (2017).
There are 38 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Mehmet Yavuz

Burcu Yaşkıran This is me

Publication Date October 1, 2017
Published in Issue Year 2017 Volume: 5 Issue: 4

Cite

APA Yavuz, M., & Yaşkıran, B. (2017). Approximate-analytical solutions of cable equation using conformable fractional operator. New Trends in Mathematical Sciences, 5(4), 209-219.
AMA Yavuz M, Yaşkıran B. Approximate-analytical solutions of cable equation using conformable fractional operator. New Trends in Mathematical Sciences. October 2017;5(4):209-219.
Chicago Yavuz, Mehmet, and Burcu Yaşkıran. “Approximate-Analytical Solutions of Cable Equation Using Conformable Fractional Operator”. New Trends in Mathematical Sciences 5, no. 4 (October 2017): 209-19.
EndNote Yavuz M, Yaşkıran B (October 1, 2017) Approximate-analytical solutions of cable equation using conformable fractional operator. New Trends in Mathematical Sciences 5 4 209–219.
IEEE M. Yavuz and B. Yaşkıran, “Approximate-analytical solutions of cable equation using conformable fractional operator”, New Trends in Mathematical Sciences, vol. 5, no. 4, pp. 209–219, 2017.
ISNAD Yavuz, Mehmet - Yaşkıran, Burcu. “Approximate-Analytical Solutions of Cable Equation Using Conformable Fractional Operator”. New Trends in Mathematical Sciences 5/4 (October 2017), 209-219.
JAMA Yavuz M, Yaşkıran B. Approximate-analytical solutions of cable equation using conformable fractional operator. New Trends in Mathematical Sciences. 2017;5:209–219.
MLA Yavuz, Mehmet and Burcu Yaşkıran. “Approximate-Analytical Solutions of Cable Equation Using Conformable Fractional Operator”. New Trends in Mathematical Sciences, vol. 5, no. 4, 2017, pp. 209-1.
Vancouver Yavuz M, Yaşkıran B. Approximate-analytical solutions of cable equation using conformable fractional operator. New Trends in Mathematical Sciences. 2017;5(4):209-1.