The residual power series method for solving fractional Klein-Gordon equation

In this article, the residual power series method (RPSM) for solving fractional Klein-Gordon equations is introduced. Residual power series algorithm gets Maclaurin expansion of the solution. The solutions of our equation are computed in the form of rapidly convergent series with easily calculable components by using mathematica software package. Reliability of the method is given with graphical consequences and series solutions. The found consequences show that the method is a power and efficient method in determination of solution the time fractional Klein-Gordon equations.

Kesirli Klein-Gordon denklemi için residual power seri metodu

Bu makalede kesirli Klein-Gordon denklemlerinin çözümleri için Residual Power Seri metodu (RPSM) uygulanmıştır. Residual Power Seri algoritması çözümün Maclaurin açılımını verir. Bu denklemlerin çözümleri, Mathematica programı kullanılarak kolayca hesaplanan bileşenler ile hızlı yakınsak seriler formunda hesaplanmıştır. Metodun güvenilirliği, seri çözümler ve grafik sonuçlar yardımıyla verilmiştir. Bulunan sonuçlar, kullandığımız metodun kesirli Klein-Gordon denklemlerinin seri çözümlerinin belirlenmesinde güçlü ve etkili bir metot olduğunu göstermektedir.

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A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, (2006).

I.Podlubny, Fractional Differential Equation, Academic Press, San Diego, (1999).

J.Sabatier, O.P. Agrawal, J.A.T. Machado (Eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, (2007).

S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Switzerland, (1993).

D. Baleanu, K. Diethelm, E. Scalas, J. Truj llo, Fractional Calculus Models and Numerical Methods, Complexity,Nonlinearity and Chaos, World Scientific, Boston, Mass, USA, (2012).

J.S. Duan, R. Rach, D. Baleanu, A.M. Wazwaz, ‘’A review of the Adomian decomposition method and its applications to fractional differential equations’’, Commun. Frac. Calc.,vol.3, pp.7-99, 2012.

R. Mag n, X. Feng, D. Baleanu, ‘’Solving the Fractional Order Bloch Equation, Concepts in Magnetic Resonance’’, vol.34 no.A (1), pp.16-23,2009.

A. Kadem, D. Baleanu, ‘’Homotopy perturbation method for the coupled fractional Lotka-Volterra equations’’, Rom. J. Phys.,no.56, pp.332-338, 2011

D. Baleanu, ‘’New Applications of Fractional Variational Principles’’, Reports on Mathematical Physics, vol.61, no.2, 199-206, 2008.

R.L. Magin, O. Abdullah, D. Baleanu, X.J. Zhou, ‘’Anomalous diffusion expressed through fractional order differential operators in the Bloch--Torrey equation’’, Journal of Magnetic Resonance, vol.190, pp. 255-270,2008.

O. Abu Arqub, ‘’Series solution of fuzzy differential equations under strongly generalized differentiability’’, Journal of Advanced Research in Applied Mathematics, no.5, pp. 31-52, 2013.

O. Abu Arqub, A El-Ajou, A. Bataineh, I. Hashim, ‘’A representation of the exact solution of generalized Lane Emden equations using a new analytical method’’, Abstract and Applied Analysis, pp. 1-10, 2013.

A. El-Ajou, O. Abu Arqub, Z. Al Zhour, S. Momani, ‘’New results on fractional power series: theories and applications’’, Entropy, vol.15, pp. 5305- 5323, 2013.

O. Abu Arqub, A. El-Ajou, Z. Al Zhour, S. Momani, ‘’Multiple solutions of nonlinear boundary value problems of fractional order: a new analytic iterative technique’’, Entropy, vol.16, pp. 471-493.2014.

O. Abu Arqub, A. El-Ajou, S. Momani, ‘’Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations’’, Journal of Computational Physics, vol.293, pp.385-399, 2015.

A. El-Ajou, O. Abu Arqub, S. Momani, ‘’Approximate analytical solution of the nonlinear fractional KdV-Burgers equation: a new iterative algorithm’’, Journal of Computational Physics, vol.293, pp.81-95,2015.

M. Alquran, K. Al-Khaled, J. Chattopadhyay, ‘’Analytical Solutions of Fractional Population Diffusion Model: Residual Power Series’’, Nonlinear Studies, vol.22, no.1, pp.31-39,2015.

A. El-Ajou, O. Abu Arqub, S. Momani, D. Baleanu, A. Alsaedi, ‘’A novel expansion iterative method for solving linear partial differential equations of fractional order’’, Applied Mathematics and Computation, vol.257, vol.119-133, 2015.

A.K. Golmankhaneh, D. Baleanu, ‘’On nonlinear fractional Klein--Gordon equation’’, Sigal Process. Vol. 91,pp.446--451.2011.

B. Lu, ‘’The first integral method for some time fractional differential equations’’, Journal of Mathematical Analysis and Applications, vol. 395, pp.684-693,2012.