Abstract
This analysis proposes an analytical-numerical approach for providing solutions of a class of nonlinear fractional Klein-Gordon equation subjected to appropriate initial conditions in Caputo sense by using the Fractional Reduced Differential Transform Method (FRDTM). This technique provides the solutions very accurately and efficiently in convergent series formula with easily computable coefficients. The behavior of the approximate series solution for different values of fractional-order "a" is shown graphically. A comparative study is presented between the FRDTM and Implicit Runge-Kutta approach to illustrate the efficiency and reliability of the proposed technique. Our numerical investigations indicate that the FRDTM is simple, powerful mathematical tool and fully compatible with the complexity of such problems.
Highlights
Fractional Partial Differential Equations (FPDEs) are widely used in interpretation and modeling of many of realism matters appear in applied mathematics and physics including fluid mechanics, electrical circuits, diffusion, damping laws, relaxation processes, mathematical biology (Klimek, 2005; Kilbas et al, 2010; Baleanu et al, 2009; Jumarie, 2009; Ortigueira, 2010; Mainardi, 2010)
∂ ∂ t u where, ܽ and ܾ are real constants, f(x,t), g0(x) and g1(x) are known analytical functions, G (u) is a nonlinear function, u is an unknown function of x and t to be determined. This model is derived from well-known Klein-Gordon Equations (KGEs) by replacing the time order derivative with fractional derivative of order ߙ
T =t0 t kα if α = 1, the FRDT of Equation 7 reduces to the classical RDT method
Summary
Fractional Partial Differential Equations (FPDEs) are widely used in interpretation and modeling of many of realism matters appear in applied mathematics and physics including fluid mechanics, electrical circuits, diffusion, damping laws, relaxation processes, mathematical biology (Klimek, 2005; Kilbas et al, 2010; Baleanu et al, 2009; Jumarie, 2009; Ortigueira, 2010; Mainardi, 2010). Consider the following nonlinear Klein-Gordon equations of one-dimensional time fractional model: This model is derived from well-known Klein-Gordon Equations (KGEs) by replacing the time order derivative with fractional derivative of order ߙ.
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