Novel Methods for Solving the Conformable Wave Equation

Mohammed Kaabar

EN

Novel Methods for Solving the Conformable Wave Equation

Abstract

In this paper, a two-dimensional conformable fractional wave equation describing a circular membrane undergoing axisymmetric vibrations is formulated. It was found that the analytical solutions of the fractional wave equation using the conformable fractional formulation can be easily and efficiently obtained using separation of variables and double Laplace transform methods. These solutions are compared with the approximate solution obtained using the differential transform method for certain cases

Keywords

conformable derivative, wave equation, double Laplace transform, dierential transform method

References

R. Almeida, D. Tavares, and D. F. Torres, The Variable-Order Fractional Calculus of Variations, arXiv preprint:1805.00720l, Springer International Publishing, 2018.
O. P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational prob- lems,Journal of Mathematical Analysis and Applications 272(1) (2002) 368-379 .
M. Klimek, Fractional sequential mechanics-models with symmetric fractional deriva- give, Czechoslovak Journal of Physics 51(12) (2001) 1348-1354.
A. B. Malinowska and D. F. Torres, Fractional variational calculus in terms of a combined Caputo derivative,arXiv preprint:1007.0743, 2010.
S. G. Samko and B. Ross, Integration and differentiation to a variable fractional order, Integral Transforms and Special Functions 1(4) (1993) 277-300.
Z. Yi, Fractional differential equations of motion in terms of combined Riemann-Liouville deriva- tives,Chinese Physics B. 21(8) (2012) 084502.
M. Abu Hammad and R. Khalil, Conformable fractional heat differential equation, Int. J. Pure Appl. Math. 94(2) (2014) 215-221.
R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics 264 (2014) 65-70.
M. J. Lazo and D. F. Torres, Variational calculus with conformable fractional derivatives, IEEE/CAA Journal of Automatica Sinica 4(2) (2017) 340-352.
Y. Cenesiz and A. Kurt, The new solution of time fractional wave equation with conformable fractional derivative denition, Journal of New Theory 7 (2015) 79-85.

O. Tasbozan, Y. Cenesiz, A. Kurt, and D. Baleanu, New analytical solutions for conformable fractional PDEs arising in mathematical physics by exp-function method, Open Physics 15(1) (2017) 647-651.
R. Khalil and H. Abu-Shaab, Solution of some conformable fractional dierential equations, Int. J. Pure Appl. Math. 103(4) (2015) 667-673.
M. Yavuz, 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) (2017) 1-7.
M. Yavuz and B. Yaskiran, Conformable Derivative Operator in Modelling Neuronal Dynamics, Applications & Applied Mathematics 13(2) (2018) 803-817.
M. Yavuz and B. Yaskiran, Approximate-analytical solutions of cable equation using conformable fractional operator, New Trends in Mathematical Sciences 5(4) (2017) 209-219.
M. Yavuz and N. Ozdemir, A different approach to the European option pricing model with new fractional operator, Mathematical Modelling of Natural Phenomena 13(1) (2018) 12.
D. Zhao and M. Luo, General conformable fractional derivative and its physical interpretation, Calcolo 54(3) (2017) 903-917.
O. Acan, D. Baleanu, M. M. A. Qurashi, and M. G. Sakar, Analytical approximate solutions of (n+1)-dimensional fractal heat-like and wave-like equations, Entropy 19(7) (2017) 296.
J. He, Fractal calculus and its geometrical explanation, Results in Physics 10 (2018) 272-276.
J. He, Z. Li, and Q. Wang, A new fractional derivative and its application to explanation of polar bear hairs, Journal of King Saud University-Science 28(2) (2016) 190-192.
J. He, A New Fractal Derivation, Thermal Science 15(1) (2011) 145-147.
Y. Hu and J. He, On fractal space-time and fractional calculus, Thermal Science 20(3) (2016) 773-777.
H. Liu, Z. Li, and Y. Yao, A fractional nonlinear system for release oscillation of silver ions from hollow bers, Journal of Low Frequency Noise, Vibration and Active Control 38(1) (2019) 88-92.
A. Atangana, D. Baleanu, and A. Alsaedi, New properties of conformable derivative, Open Mathematics 13(1) (2015) 889-898.
J. E. N. Napoles Valdes, J. A. C. Medina, P. M. Guzman, and L. M. Lugo, A New Local Fractional Derivative of q-uniform Type, Discontinuity, Nonlinearity, and Complexity 8(1) (2019) 101-109.
J.Weberszpil and W. Chen, Generalized Maxwell relations in thermodynamics with metric deriva- tives, Entropy 19(8) (2017) 407.
J. He, A tutorial review on fractal spacetime and fractional calculus, International Journal of Theoretical Physics 53(11) (2014) 3698-3718.
Y. Liang, W. Chen, and W. Cai, Hausdorff Calculus: Applications to Fractal Systems, Walter de Gruyter GmbH & Co KG, 6, 2019.
W. Chen, F. Wang, B. Zheng, and W. Cai, Non-Euclidean distance fundamental solution of Hausdorff derivative partial differential equations, Engineering Analysis with Boundary Elements 84 (2017) 213-219.
S. Meng and Y. Cui, The Extremal Solution To Conformable Fractional Differential Equations Involving Integral Boundary Condition, Mathematics 7(2) (2019) 186.
F. Silva, D. M. Moreira, and M. A. Moret, Conformable Laplace Transform of Fractional Differ- ential Equations, Axioms 7(3) (2018) 55.
T. Abdeljawad and M. Grossman, On geometric fractional calculus, Journal of Semigroup Theory and Applications 2 (2016) 2051-2937.
P. M. Guzman, G. Langton, L. M. Lugo, J. Medina, and J. E. N. Napoles Valdes, A new definition of a fractional derivative of local type, Journal of Mathematical Analysis 9(2) (2018) 88-98.
J. E. N. Napoles Valdes, P. M. Guzman, and L. M. Lugo, Some New Results on Nonconformable Fractional Calculus, Advances in Dynamical Systems and Applications 13(2) (2018) 167-175.
A. F. Imbert, J. Mendez-Bermudez, J. E. N. Napoles Valdes, and J. M. Sigarreta Almira, On fractional Lienard-type systems, Revista Mexicana de Fisic 65(6) (2019) 618-625.
P. M. Guzman and J. E. N. Napoles Valdes, A Note on the Oscillatory Character of Some Non Conformable Generalized Linebard System, Advanced Mathematical Models & Applications 4(2) (2019) 127-133.
F. Martines, P. O. Mohammed, and J. E. N. Napoles Valdes, Non-Conformable Fractional Laplace Transform, Kragujevac Journal of Mathematics 46(3) (2022) 341-354.
J. E. N. Napoles Valdes, J. M. Rodriguez, and J. M. Sigarreta, New Hermite{Hadamard Type Inequalities Involving Non-Conformable Integral Operators, Symmetry 11(9) (2019) 1108.
M. Y. Jarin, Dynamic ltration with rotating disks, and rotating and vibrating membranes: an update, Current Opinion in Chemical Engineering 1(2) (2012) 171-177.
A. Kola, Y. Ye, P. Le-Clech, and V. Chen, Transverse vibration as novel membrane fouling mitigation strategy in anaerobic membrane bioreactor applications, Journal of membrane science 455 (2014) 320-329.
A. Kola, Y. Ye, A. Ho, P. Le-Clech, and V. Chen, Application of low frequency transverse vibration on fouling limitation in submerged hollow bre membranes, Journal of membrane science 409 (2012) 54-65.
H. Singh, H. M. Srivastava, and D. Kumar, A reliable numerical algorithm for the fractional vibration equation, Chaos, Solitons & Fractals 103 (2017) 131-138.
N. H. Asmar, Partial differential equations with Fourier series and boundary value problems, Courier Dover Publications, 2016.
D. J. Wollkind and B. J. Dichone, Comprehensive Applied Mathematical Modeling in the Natural and Engineering Sciences, Springer International Publishing, 2017.
T. Abdeljawad, On conformable fractional calculus, Journal of Computational and Applied Mathematics 279 (2015) 57-66.
T. A. Estrin and T. J. Higgins, The solution of boundary value problems by multiple Laplace transformations, Journal of the Franklin Institute 252(2) (1951) 153-167.
L. Debnath, The double Laplace transforms and their properties with applications to functional, integral and partial differential equations, International Journal of Applied and Computational Mathematics 2(2) (2016) 223-241.
A. M. O. Anwar, F. Jarad, D. Baleanu, and F. Ayaz, Fractional Caputo heat equation within the double Laplace transform, Rom. Journ. Phys. 58 (2013) 15-22.
H. Eltayeb and A. Kılıçman, On double Sumudu transform and double Laplace transform, Malaysian Journal of Mathematical Sciences 4(1) (2010) 17-30.
H. Eltayeb and A. Kılıçman, A note on solutions of wave, Laplace's and heat equations with convolution terms by using a double Laplace transform, Applied Mathematics Letters 21(12) (2008) 1324-1329.
A. Kılıçman and H. E. Gadain, On the applications of Laplace and Sumudu transforms, Journal of the Franklin Institute 347(5) (2010) 848-862.
R. R. Dhunde and G. L. Waghmare, Double Laplace transform method for solving space and time fractional telegraph equations, International Journal of Mathematics and Mathematical Sciences, 2016 (2016) 1414595.
M. Omran and A. Kılıçman, Fractional double Laplace transform and its properties, AIP Conference Proceedings 1795(1) (2017) 020021.
O. Korfal and I. B. Parlak, Vibrations of a Circular Membrane Subjected To a Pulse, Vibration Problems ICOVP 2005, Springer International Publishing, 2006.
E. Unal and A. Gökdoğan, Solution of conformable fractional ordinary differential equations via differential transform method, Optik-International Journal for Light and Electron Optics. 128 (2017) 264-273.
A. Arikoglu and I. Ozkol, Solution of fractional differential equations by using differential trans- form method, Chaos, Solitons & Fractals 34(5) (2007) 1473-1481.
J. K. Zhou, Differential transformation and its application for electrical circuits, Huazhong University Press, Wuhan, China, 1986.
O. Acan, O. Firat, Y. Keskin, and G. Oturanc, Solution of conformable fractional partial differen- tial equations by reduced differential transform method, Selcuk Journal of Applied Mathematics, 2016.
O. Acan and D. Baleanu, A New Numerical Technique for Solving Fractional Partial Differential Equations, Miskolc Mathematical Notes 19(1) (2018) 3-18.
D. Avcı, B. Eroglu, and N. Ozdemir, Conformable heat problem in a cylinder, Proceedings, International Conference on Fractional Differentiation and its Applications (2016) 572-581.
T. Sandev and Z. Tomovski, The general time fractional wave equation for a vibrating string, Journal of Physics A: Mathematical and Theoretical 43(5) (2010) 055204.
E. Demirci and N. Ozalp, A method for solving differential equations of fractional order, Journal of Computational and Applied Mathematics 236(11) (2012) 2754-2762.
T. Blaszczyk and M. Ciesielski, Numerical solution of fractional Sturm-Liouville equation in integral form, Fractional Calculus and Applied Analysis 17(2) (2014) 307-320.
C. Celik and M. Duman, Crank{Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, Journal of computational physics, 231(4) (2012) 1743-1750.
[65] S. Momani and Z. Odibat, Numerical comparison of methods for solving linear differential equa- tions of fractional order, Chaos, Solitons & Fractals, 31(5) (2007) 1248-1255.
[66] N. A. Pirim and F. Ayaz, Hermite collocation method for fractional order differential equations, An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 8(2) (2018) 228-236.

Details

Primary Language

English

Subjects

Mathematical Sciences, Applied Mathematics

Journal Section

Research Article

Authors

Mohammed Kaabar ^*
United Arab Emirates

Publication Date

June 30, 2020

Submission Date

September 12, 2019

Acceptance Date

June 20, 2020

Published in Issue

Year 2020 Number: 31

IZ

https://izlik.org/JA84PK96NF

APA

Kaabar, M. (2020). Novel Methods for Solving the Conformable Wave Equation. Journal of New Theory, 31, 56-85. https://izlik.org/JA84PK96NF

AMA

1.Kaabar M. Novel Methods for Solving the Conformable Wave Equation. JNT. 2020;(31):56-85. https://izlik.org/JA84PK96NF

Chicago

Kaabar, Mohammed. 2020. “Novel Methods for Solving the Conformable Wave Equation”. Journal of New Theory, nos. 31: 56-85. https://izlik.org/JA84PK96NF.

EndNote

Kaabar M (June 1, 2020) Novel Methods for Solving the Conformable Wave Equation. Journal of New Theory 31 56–85.

IEEE

[1]M. Kaabar, “Novel Methods for Solving the Conformable Wave Equation”, JNT, no. 31, pp. 56–85, June 2020, [Online]. Available: https://izlik.org/JA84PK96NF

ISNAD

Kaabar, Mohammed. “Novel Methods for Solving the Conformable Wave Equation”. Journal of New Theory. 31 (June 1, 2020): 56-85. https://izlik.org/JA84PK96NF.

JAMA

1.Kaabar M. Novel Methods for Solving the Conformable Wave Equation. JNT. 2020;:56–85.

MLA

Kaabar, Mohammed. “Novel Methods for Solving the Conformable Wave Equation”. Journal of New Theory, no. 31, June 2020, pp. 56-85, https://izlik.org/JA84PK96NF.

Vancouver

1.Mohammed Kaabar. Novel Methods for Solving the Conformable Wave Equation. JNT [Internet]. 2020 Jun. 1;(31):56-85. Available from: https://izlik.org/JA84PK96NF