Cauchy problem for non-autonomous fractional evolution equations with nonlocal conditions of order (1, 2)

The Cauchy problem for non-autonomous fractional evolution equations with nonlocal conditions of order (1, 2) typically involves a partial differential equation (PDE) with a fractional derivative, initial conditions, and nonlocal conditions. The problem can be stated as follows:

Consider the non-autonomous fractional evolution equation with a Caputo fractional derivative of order α:

${�}_{�}^{�}�(�)=�(�)�(�)+�(�,�(�)),�>0,$

where ${�}_{�}^{�}$ is the Caputo fractional derivative of order α, $�(�)$ is a time-dependent operator, $�(�,�(�))$ is a nonlinear function, and $�(�)$ is the unknown function to be determined.

The initial condition is given by:

$�(0)={�}_0,$

where ${�}_0$ is a given initial function.

Additionally, the nonlocal conditions of order (1, 2) can be expressed as:

${�}^{\mathrm{\prime }}(0)={�}_1(�),$ ${\int }_0^1�(�)��={�}_2(�)\mathrm{.}$

Here, ${�}_1(�)$ and ${�}_2(�)$ are given functions of $�(�)$ representing the first-order and second-order nonlocal conditions, respectively.

The solution to this problem involves finding the function $�(�)$ that satisfies the fractional evolution equation, the initial condition, and the nonlocal conditions.

Solving this type of problem may involve various mathematical techniques, such as the method of lines, Laplace transform, fractional calculus, and functional analysis, depending on the specific form of $�(�)$, $�(�,�(�))$, ${�}_1(�)$, and ${�}_2(�)$.

It's important to note that the specific details of the problem will depend on the concrete form of the equation, the operator $�(�)$, and the functions $�(�,�(�))$, ${�}_1(�)$, and ${�}_2(�)$. The solution may require specialized methods and tools based on the particular characteristics of the problem at hand.

Download