On Long-time Behavior of Solutions for Some Nonlinear Fractional Integro-differential Equations

The study of the asymptotic/long-time behavior (limiting behavior for large values of time) of solutions for differential equations (DEs) is of great importance in theory and applications. It is particularly capital for the DEs where solutions cannot be found explicitly. This task is more challenging when the DEs are nonlinear and of non-integer order. Due to the difficulties caused by the nonlocality nature of the fractional derivatives and the singularity of the kernels, the investigation requires new tools and special techniques In this project, we propose to investigate the asymptotic behavior of non-trivial global solutions for some initial value problems of fractional differential equations (FDEs). The left-hand side of the equations involves fractional derivatives of Hadamard or Caputo types of sub-first or sub-second orders. The (nonlinear) source function in the right-hand side depends on a lower order fractional derivative of the solution and an integral of a kernel involving the solution or a lower order fractional derivative of the solution. In applications, these derivatives may account for fractional damping. Our objective is to investigate the behavior of solutions at infinity and determine, if possible, the type of that behavior. In particular, we intend to characterize some reasonable conditions under which solutions are eventually bounded or behave like lines, power or logarithmic functions. For this purpose, we plan to use some appropriate modified estimation techniques, generalized inequalities, and asymptotic theories. Some examples will be provided to illustrate the results. Our findings could be used to predict the behavior of many physical systems modeled by FDEs for which no explicit analytical solutions are available. In the area of mathematical and numerical analysis, our results will help in developing some numerical treatments for approximate solutions. In addition, they will extend and enrich the limited available results for fractional order problems.