The Yang-Laplace Transform for Solving the IVPs with Local Fractional Derivative

Abstract

The IVPs with local fractional derivative are considered in this paper. Analytical solutions for the homogeneous and nonhomogeneous local fractional differential equations are discussed by using the Yang-Laplace transform.

On the Existence and Uniqueness of Solutions for Local Fractional Differential Equations

“…Indeed, fractional differential equations have been subjected to many studies due to their frequent occurrence in different applications in physics, fluid mechanics, physiology, engineering, electrochemistry, and signals [1][2][3][4][5][6][7][8]. Therefore, numerical and analytical techniques have been developed to deal with fractional differential equations [9][10][11].…”

Characteristic equation method for fractal heat-transfer problem via local fractional calculus

“…The differential equations involving the local fractional calculus [1] were utilized to investigate the non-differentiable problems, e. g., fractal diffusions [2][3][4][5][6][7], fractal oscillator [8], fractal wave [9], fractal Laplace [10,11], fractal heat-conduction [12,13], fractal Fokker-Planck [14], fractal Helmholtz [15] equations and others [16,17]. Let us recall the local fractional derivative (LFD) of the function ( ) ζ Π of order (0 1) θ θ < < at 0 , ζ ζ = defined by [10][11][12][13][14][15][16][17][18]: where κ is a heat-diffusive coefficient and ω -a constant related to the density and specific heat of fractal materials.…”

“…There are a lot of numerical and analytical methods for the local fractional partial differential equations, such as the decomposition method [2,4,15], differential transform [3], variational iteration method [5,12,14], homotopy perturbation method [6], similarity variable method [7], Laplace variational iteration method [9], series expansion method [10], function decomposition method [11], Fourier transform [13], exp-function method [16], Fourier transform [17], and characteristic equation method (CEM) [19]. The main aim of this paper is to present the CEM to solve the heat-transfer equation in fractal media.…”

Results for Mild solution of fractional coupled hybrid boundary value problems

“…For the exact solutions of different classes of FDEs, we refer to valuable efforts of Y. J. Yang et al regarding Laplace equation [11]. In [12], C. G. Zhao et al produced exact solutions of many initial value problems of local FDEs. Besides the exact solutions, we also have valuable efforts of scientist for numerical approximations of FDEs.…”