Introduction: The modeling of many real-life physical systems leads to a set of fractional differential equations. Also fractional differential equations appear in various physical processes such as viscoelasticity and viscoplasticity, modeling of polymers and proteins, transmission of ultrasound waves, signal processing, control theory, etc. Most of fractional differential equations especially their nonlinear types do not have exact analytic solution, so numerical methods must be used. Therefore many authors have worked on the numerical solutions of this kind of equations. In recent years, many numerical methods have emerged, such as, the Adomian decomposition method, the Homotopy method, the multistep method, the extrapolation method, the spline collocation method, the product integration method and the predictor-corrector method. But most of the aforementioned methods consider the linear type of equations without a reliable theoretical justification. Then providing an efficient numerical scheme to approximate the solutions of nonlinear fractional differential equations is worthwhile and new in the literature. The main object of this paper is to develop and analyze a high order numerical method based on the collocation method when applies the orthogonal Jacobi polynomials as bases functions for the Single term nonlinear fractional differential equations. Material and methods: Due to the well-known existence and uniqueness theorems the solutions of the fractional differential equations typically suffer from singularity at the origin. Consequently direct application of the Jacobi collocation method may lead to very weak numerical results. To fix this difficulty, we introduce a smoothing transformation that removes the singularity of the exact solution and enables us to approximate the solution with a satisfactory accurate result. Convergence analysis of the proposed scheme is also presented which demonstrates that the regularization process improves the smoothness of the input data and thereby increases the order of convergence. Results and discussion: We illustrate some test problems to show the effectiveness of the proposed scheme and to confirm the obtained theoretical predictions. In overall, the reported results justify that the proposed regularization strategy works well and the obtained approximate solutions have a good accuracy. To show the applicability of our approach we solve a practical example which is developed for a micro-electrical system (MEMS) instrument that has been designed primary to measure the viscosity of fluids that are encounter during oil well exploration using the proposed scheme. Moreover, we make a comparison between our scheme and the operational Tau method to show the efficiency of our technique. The reported results approve the superiority of the proposed approach. Finally, we consider a problem that we do not have access to its exact solution. In this case, we use the “ Variational Iteration Method (VIM)” as a qualitatively correct picture of the exact solution (the source solution) to evaluate the precision of the proposed technique. The obtained results approve that our approach produces the approximate solution which is in a good agreement with source ones. Conclusion: The following conclusions were drawn from this research. A reliable numerical method based on the Jacobi collocation method to approximate the solutions of a class of nonlinear fractional differential equations was developed. To achieve an efficient approximation a regularization strategy was proposed that improves the smoothness of the input data and enables us to obtain an approximate solution with a satisfactory accuracy. Convergence analysis of the proposed method was investigated which confirmed the high order of convergence of the proposed method.

This paper is devoted to prove the existence of extremal solutions for multi-term nonlinear fractional differential equations with nonlinear boundary conditions. The fractional derivative is of Caputo type and the inhomogeneous term depends on the fractional derivatives of lower orders. By establishing a new comparison theorem and applying the monotone iterative technique, we show the existence of extremal solutions. The method is a constructive method that yields monotone sequences that converge to the extremal solutions. As an application, some examples are presented to illustrate the main results.

In this work, we extend the existing local fractional Sumudu decomposition method to solve the nonlinear local fractional partial differential equations. Then, we apply this new algorithm to resolve the nonlinear local fractional gas dynamics equation and nonlinear local fractional Klein-Gordon equation, so we get the desired non-differentiable exact solutions. The steps to solve the examples and the results obtained, showed the flexibility of applying this algorithm, and therefore, it can be applied to similar examples.

In this paper, we investigate the existence of solutions of some three{point boundary value problems for nth{order nonlinear fractional di erential equations with higher boundary conditions by using a  xed point theorem on cones.

In the current work, we present some innovative solutions for the attractivity of fractional functional q-di, erential equations involving Caputo fractional q-derivative in a k-dimensional system, by using some , xed point principle and the standard Schauder's , xed point theorem. Likewise, we look into the global attractivity of fractional q-di, erential equations involving classical Riemann-Liouville fractional q-derivative in a k-dimensional system, by employing the famous , xed point theorem of Krasnoselskii. Also, we must note that, this paper is mainly on the analysis of the model, with numerics used only to verify the analysis for checking the attractivity and global attractivity of solutions in the system. Lastly, we give two examples to illustrate our main results.