Abstract
In this paper, we propose a novel reduced differential transform method (RDTM) to compute analytical and semianalytical approximate solutions of fractional order Airy’s ordinary differential equations and fractional order Airy’s and Airy’s type partial differential equations subjected to certain initial conditions. The performance of the proposed method was analyzed and compared with a convergent series solution form with easily computable coefficients. The behavior of approximated series solutions at different values of fractional order α and its modeling in 2-dimensional and 3-dimensional spaces are compared with exact solutions using MATLAB graphical method analysis. Moreover, the physical and geometrical interpretations of the computed graphs are given in detail within 2- and 3-dimensional spaces. Accordingly, the obtained approximate solutions of fractional order Airy’s ordinary differential equations and fractional order Airy’s and Airy’s type partial differential equations subjected to certain initial conditions exactly fit with exact solutions. Hence, the proposed method reveals reliability, effectiveness, efficiency, and strengthening of computed mathematical results in order to easily solve fractional order Airy’s type differential equations.
Highlights
In recent past years, the glorious developments have been investigated in the field of fractional calculus and fractional differential equations
Airy’s partial differential equation is one of the linear partial differential equations used in many real-world physical applications, and the Airy equation is one of the first models of water waves: a small wave traveling “wave trains” in deep water [11]. e early day of mathematical modeling of water waves was assumed that the wave height was small compared to the water depth which leads to linear dispersive equations, a representative model of which is Airy’s partial differential equation [12]
The convergence of the method in obtaining the general solutions of the three cases was illustrated. e other points we can see under these findings were the physical and geometrical applications that are obtained from the seven examples considered, which were grouped into three cases. e first two examples (Examples 1 and 2) were examples of fractional order Airy’s ordinary differential equation (FAODE), the second three examples (Examples 3–5) were examples of Fractional Order Airy’s Partial Differential Equation (FAPDE), and the last two examples (Examples 6 and 7) were examples of fractional order Airy’s type partial differential equations
Summary
The glorious developments have been investigated in the field of fractional calculus and fractional differential equations. Several real phenomena emerging in engineering and science fields can be demonstrated successfully by developing models using the fractional calculus theory. Some of these are time fractional heat equations, time fractional heat-like equations, time fractional wave equations, time fractional telegraphic equation, fractional order Airy’s ordinary differential equation, time fractional Airy’s partial differential equations, and so on. E early day of mathematical modeling of water waves was assumed that the wave height was small compared to the water depth which leads to linear dispersive equations, a representative model of which is Airy’s partial differential equation [12] Such equations are somewhat satisfying in this regard because they have solutions that resemble wave traveling along with constant speed and fixed profile along the water surface, just like one sees in nature [13]. It is an iterative procedure for obtaining Taylor series solution of differential equations [22]
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