Exact solutions of partial differential equations of Caputo fractional order

In this paper, three types of fractional order partial differential equations, including the fractional Cauchy–Riemann equation, fractional acoustic wave equation, and two-dimensional space partial differential equation with time-fractional-order, are considered, and these models are obtained from the standard equations by replacing an integer-order derivative with a fractional-order derivative in Caputo sense. Firstly, we discuss the fractional integral and differential properties of several functions which are derived from the Mittag-Leffler function. Secondly, by using the homotopy analysis method, the exact solutions for fractional order models mentioned above with suitable initial boundary conditions are obtained. Finally, we draw the computer graphics of the exact solutions, the approximate solutions (truncation of finite terms), and absolute errors in the limited area, which show that the effectiveness of the homotopy analysis method for solving fractional order partial differential equations.

Download Full-text

Fractional Order Airy’s Type Differential Equations of Its Models Using RDTM

Mathematical Problems in Engineering ◽

10.1155/2021/3719206 ◽

2021 ◽

Vol 2021 ◽

pp. 1-21

Author(s):

Daba Meshesha Gusu ◽

Dechasa Wegi ◽

Girma Gemechu ◽

Diriba Gemechu

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Exact Solutions ◽

Ordinary Differential Equations ◽

Fractional Order ◽

Initial Conditions ◽

Approximate Solutions ◽

Partial Differential ◽

Transform Method ◽

3 Dimensional

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.

Download Full-text

Invariant solutions of fractional-order spatio-temporal partial differential equations

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2019-0239 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Nkosingiphile Mnguni ◽

Sameerah Jamal

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Fractional Order ◽

Growth Models ◽

Spatial Patterning ◽

Partial Differential ◽

Symmetry Approach ◽

Time Component ◽

Spatio Temporal ◽

Lie Symmetry Approach

Abstract This paper considers two categories of fractional-order population growth models, where a time component is defined by Riemann–Liouville derivatives. These models are studied under the Lie symmetry approach, and we reduce the fractional partial differential equations to nonlinear ordinary differential equations. Subsequently, solutions of the latter are determined numerically or with the aid of Laplace transforms. Graphical representations for integral and trigonometric solutions are presented. A key feature of these models is the connection between spatial patterning of organisms versus competitive coexistence.

Download Full-text