A new Mittag-Leffler function undetermined coefficient method and its applications to fractional homogeneous partial differential equations

Abstract

In this paper, we develop a new application of the Mittag-Leffler function that will extend the application to fractional homogeneous differential equations, and propose a Mittag-Leffler function undetermined coefficient method. A new solution is constructed in power series. When a very simple ordinary differential equation is satisfied, no matter the original equation is linear or nonlinear, the method is valid, then combine the alike terms, compare the coefficient with identical powers, and the undetermined coefficient will be obtained. The fractional derivatives are described in the Caputo sense. To illustrate the reliability of the method, some examples are provided, and the solutions are in the form of generalized Mittag-Leffler function. The results reveal that the approach introduced here are very effective and convenient for solving homogeneous differential equations with fractional order.

An Efficient Approach for Solving Fractional Dynamics of a Predator-Prey System

“…It should be noted that the results obtained in this research are completely different from the results published in Liu et al (2017) where we are using a different approach. As the method used in the mentioned paper solve only one partial fractional differential equation and the nonlinear part in this equation is always equal to zero, and this is not the case in our method presented in this article.…”

“…So, the method of the GMLFM proved its efficiency and its lightness in solving the ordinary fractional differential equations Arafa et al (2013a,b). Moreover, the Mittag-Leffler function undetermined coefficient method is contributed to the solution of fractional homogeneous partial differential equations Liu et al (2017).…”

Research and Application of PID Controller with Feedforward Filtering Function

“…Numerical differentiation [13][14][15][16] approximates the function value of the unknown objective function at certain points based on the information of the known finite discrete sampling points. According to the literature, numerical differentiation mainly includes: finite difference type [17], polynomial interpolation type [13,15], regularization method [18,19], and undetermined coefficients [20]. Among them, the commonly used method is finite difference, but its effect is not very ideal for the high-frequency noise existing in the measurement process [21].…”

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“…(1.1) it has been proved that it possesses wildly application fields in Hilbert spaces [18,43], uniformly convex [12] and uniformly smooth Banach spaces [34,36]. At present, there exist many effective algorithms working in it, such as the traditional Newton method [4,21,31,32,48], the wave method [45,46], the BFGS method [16,19], the Levenberg-Marquardt method [3,42], the trust region method [7,8,44], the conjugate gradient algorithm [9,27], the limited BFGS method [22], etc.…”