Theoretical analysis on the nonlinear fractional differential equations and generalized heat equation

Abstract

\begin{abstract}
Using Schauder's fixed-point theorem, we establish sufficient conditions for the existence and uniqueness of solutions to the nonlinear fractional boundary value problem:
\begin{eqnarray}\label{X1}
\begin{cases}
\displaystyle _{C}D^\beta \zeta(x)+ f(x, \zeta(x),I^{\gamma}\zeta(x))=0, \quad x\in I=[0,1],\quad 1 <\beta \leq2, \;\; \gamma > 0, \\
\displaystyle \zeta(0)=0, \quad \zeta(1)=\phi(\zeta),
\end{cases}
\end{eqnarray}
where $ \phi $ is a functional defined on $ C(I,\mathbb{R})$. By constructing an appropriate Green’s function, we derive a Lyapunov-type inequality for a special case of the problem (\ref{X1}):
\begin{eqnarray}\label{X2}
\begin{cases}
\displaystyle _{C}D^\beta \zeta(x)+\lambda(x)I^{\gamma}\zeta(x)=\eta(x,\zeta(x)), \quad x\in I=[0,1],\quad 1 <\beta \leq2, \;\; \gamma > 0, \\
\displaystyle \zeta(0)=0, \quad \zeta(1)=\phi(\zeta).
\end{cases}
\end{eqnarray}
We further make an analysis for equation (\ref{X2}) by applying the inverse operator method and the Mittag-Leffler function with illustrative examples demonstrating applications obtained. Finally, we construct an analytic solution to the following generalized fractional heat equation with an initial condition in $n$ dimensions based on an inverse operator:
\begin{eqnarray}\label{X3}
\begin{cases}
\displaystyle _{C}D_t^\alpha u(t, x) = \triangle_{a_1(x_1), \cdots, a_n(x_n)} u(t, x) + f(t, x), \;\; (t, x) \in \mathbb R^+ \times \mathbb R^n, \;\; 0 < \alpha \leq 1, \\
\displaystyle u(0, x) = \psi(x),
\end{cases}
\end{eqnarray}
where
\[
\triangle_{a_1(x_1), \cdots, a_n(x_n)} = a_1(x_1) \frac{\partial^2}{\partial x_1^2} + \cdots + a_n(x_n) \frac{\partial^2}{\partial x_n^2}.
\]

%$$_{C}D^\beta \zeta(x)+\lambda(x)I^{\gamma}\zeta(x)=\eta(x,\zeta(x)),\hspace*{.6cm} x\in I=[0,1],\hspace*{.6cm} 1 <\beta \leq2, \ \gamma > 0$$ as the special case of main problem, under Dirichlet-type boundary conditions.
% For this problem, the functions $ \eta $ and $ \lambda $ are either bounded or their growth is controlled by certain constants.
\end{abstract}