Related papers: Generalized Bernoulli Differential Equation
This chapter presents some numerical methods to solve problems in the fractional calculus of variations and fractional optimal control. Although there are plenty of methods available in the literature, we concentrate mainly on approximating…
A new method for approximating fractional derivatives of the Gaussian function and Dawson's integral are presented. Unlike previous approaches, which are dominantly based on some discretization of Riemann-Liouville integral using polynomial…
The $q$-fractional differential equation usually describe the physics process imposed on the time scale set $T_q$. In this paper, we first propose a difference formula for discretizing the fractional $q$-derivative $^cD_q^\alpha x(t)$ on…
The main purpose of this paper is to study the existence of solutions for the following hybrid nonlinear fractional pantograph equation $$ \left\{\begin{aligned} &D_{0+}^\alpha…
Based on the continuous time random walk, we derive the Fokker-Planck equations with Caputo-Fabrizio fractional derivative, which can effectively model a variety of physical phenomena, especially, the material heterogeneities and structures…
We propose the symmetry reduction method of partial differential equations to the system of differential equations with fewer number of independent variables. We also obtain generalized sufficient conditions for the solution found by…
In this paper, by using a characterization of functions having fractional derivative, we propose a rigorous fractional Lyapunov function candidate method to analyze stability of fractional-order nonlinear systems. First, we prove an…
We discuss alternative iteration methods for differential equations. We provide a convergence proof for exactly solvable examples and show more convenient formulas for nontrivial problems.
A classical formula of Allwright on the general solution of a scalar differential equation is generalized to a system of differential equations by means of the Kronecker product.The Allwright formula is connected with the Riccati equation,…
We consider some possible approaches to the fractional-order generalization of definition of variation (functional) derivative. Some problems of formulation of a fractional-order variational derivative are discussed. To give a consistent…
We consider fractional differential equations of order $\alpha \in (0,1)$ for functions of one independent variable $t\in (0,\infty)$ with the Riemann-Liouville and Caputo-Dzhrbashyan fractional derivatives. A precise estimate for the order…
The goal of this paper is to investigate Gevrey properties of formal solutions of certain generalized linear partial differential equations with variable coefficients. In particular, we extend the notion of moment partial differential…
We give stability and consistency results for higher order Gr\"unwald-type formulae used in the approximation of solutions to fractional-in-space partial differential equations. We use a new Carlson-type inequality for periodic Fourier…
A method for finding the general solution to the partial differential equations: \ $F(u_x,u_y)=0$; \ $F(f(x)\:u_x,u_y)=0$ \ (or \ $F(u_x,h(y)\:u_y)=0$) \ is presented, founded on a Legendre like transformation and a theorem for Pfaffian…
We provide a novel representation of the total n-th derivative of the multivariate composite function $f \circ g$, i.e. a generalized Fa\`a di Bruno's formula. To this end, we make use of properties of the Kronecker product and the n-th…
We apply an integral inequality to obtain a rigorous apriori estimate of the accuracy of the partial sum to the power series solution of the celebrated Riccati-Bernoulli differential equation
We consider the fractional generalization of nonholonomic constraints defined by equations with fractional derivatives and provide some examples. The corresponding equations of motion are derived using variational principle.
One of the well-studied equations in the theory of ODEs is the Mathieu differential equation. A common approach for obtaining solutions is to seek solutions via Fourier series by converting the equation into an infinite system of linear…
Generalized summability results are obtained regarding formal solutions of certain families of linear moment integro-differential equations with time variable coefficients. The main result leans on the knowledge of the behavior of the…
In this paper we deal with Mellin convolution of generalized Gamma densities which leads to integrals of modified Bessel functions of the second kind. Such convolutions allow us to explicitly write the solutions of the time-fractional…