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Abstract: The aim of this manuscript is to analyze the existence of a periodic mild solution to the problem of the following nonlinear fractional differential equation where denotes the Riemann–Liouville fractional derivative. We obtained the expressions of the general solution for the linear fractional differential equation by making use of the Laplace and inverse Laplace transforms. By making use of the Banach contraction mapping principle and the Schaefer fixed point theorem, the existence results of one or at least one mild solution for a nonlinear fractional differential equation were given. [Copyright &y& Elsevier]

Abstract: In this work, the controllability result of a class of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems in a Banach space has been established by using the theory of fractional calculus, fixed point technique and also we introduced a new concept called -resolvent family. As an application that illustrates the abstract results, an example is given. [Copyright &y& Elsevier]

Abstract: In this work we report the convergence of the Chebyshev polynomials combined with the method for the steady state transport equation using the fractional derivative. The procedure is based on the expansion of the angular flux in a truncated series of orthogonal polynomials that results in the transformation of the multidimensional problem into a system of fractional differential equations. The convergence of this approach is studied in the context of the multidimensional discrete-ordinates equations. [Copyright &y& Elsevier]

Abstract: Fractional calculus is a powerful tool that has been applied successfully for the analysis of the complex systems. One interesting example of a complex mixture is given by the multicomponent pharmaceutical samples having constant matrix content. The main aim of this study is to develop a new approach based on the combined use of the fractional wavelet transform (FWT) and the continuous wavelet transform (CWT) in order to quantify atorvastatin (ATO) and amlodipine (AML) in their mixtures without requiring a chemical pretreatment. In the first step, the absorption spectra of the compounds and their samples were processed by the FWT method. In the next step, the CWT approach was applied to the fractional wavelet spectra obtained in the above step. The aim of the application of FWT is data reduction corresponding to the spectra of compounds and their commercial samples. In the following step, the CWT was used for the quantitative resolution of the composite signals of the analyzed compounds. After method validation, the proposed signal processing methods based on the combined use of the FWT and the CWT were successfully applied to the resolution of the composite spectra for the quantitation of atorvastatin (ATO) and amlodipine (AML) in tablets. [Copyright &y& Elsevier]

Enormous application of fractional order partial differential equations (FPDEs) subjected to some constrains in the form of nonlocal boundary conditions motivated the interest of many scientists around the world. The prime objective of this article is to find approximate solution of a general FPDEs subject to nonlocal integral type boundary conditions on both ends of the domain. The proposed method is based on spectral method. We construct some new operational matrices which have the ability to handle integral type non-local boundary constrains. These operational matrices can be effectively applied to convert the FPDEs together with its integral types boundary conditions to easily solvable matrix equation. The accuracy and efficiency of proposed method are demonstrated by solving some bench mark problems. The proposed method has the ability to solve non-local FPDEs with high accuracy and low computational cost. Different aspects of presented approach are compared with two other recently developed methods, Haar wavelets collocation method and a family of collocation methods which are based on Radial base functions. It is observed that the proposed method is highly accurate, robust, efficient and stable as compared to these methods. [ABSTRACT FROM AUTHOR]

Abstract: We consider a class of fractional-order chaotic systems which undergoes unknown perturbations. We revisit the problem of sliding mode controller design for robust stabilization of chaotic systems using one control input. In the recent works, it was assumed that one of the system equations are perturbed by uncertainties. For this case we show that the sliding mode dynamics are globally stable which is not addressed so far. Next, we allow that all the system’s equations depend on uncertain terms and provide a theoretical justification for applicability of the existing design. We also determine the least amount of precise information about the chaotic system that is needed to design the controller. [Copyright &y& Elsevier]

Abstract: The Bloch equation provides the fundamental description of nuclear magnetic resonance (NMR) and relaxation ( and ). This equation is the basis for both NMR spectroscopy and magnetic resonance imaging (MRI). The fractional-order Bloch equation is a generalization of the integer-order equation that interrelates the precession of the , and components of magnetization with time- and space-dependent relaxation. In this paper we examine transient chaos in a non-linear version of the Bloch equation that includes both fractional derivatives and a model of radiation damping. Recent studies of spin turbulence in the integer-order Bloch equation suggest that perturbations of the magnetization may involve a fading power law form of system memory, which is concisely embedded in the order of the fractional derivative. Numerical analysis of this system shows different patterns in the stability behavior for near 1.00. In general, when is near 1.00, the system is chaotic, while for 0.98 0.94, the system shows transient chaos. As the value of decreases further, the duration of the transient chaos diminishes and periodic sinusoidal oscillations emerge. These results are consistent with studies of the stability of both the integer and the fractional-order Bloch equation. They provide a more complete model of the dynamic behavior of the NMR system when non-linear feedback of magnetization via radiation damping is present. [Copyright &y& Elsevier]

Abstract: In this paper we investigate a fractional generalization of the Bloch equation that includes both fractional derivatives and time delays. The appearance of the fractional derivative on the left side of the Bloch equation encodes a degree of system memory in the dynamic model for magnetization. The introduction of a time delay on the right side of the equation balances the equation by also adding a degree of system memory on the right side of the equation. The analysis of this system shows different stability behavior for the and the relaxation processes. The decay is stable for the range of delays tested (1–100 μs), while the relaxation in this model exhibited a critical delay (typically 6 μs) above which the system was unstable. Delays are expected to appear in NMR systems, in both the system model and in the signal excitation and detection processes. Therefore, by including both the fractional derivative and finite time delays in the Bloch equation, we believe that we have established a more complete and more realistic model for NMR resonance and relaxation. [Copyright &y& Elsevier]

Abstract: By considering the initial and boundary conditions corresponding to parallel subsurface drains, the linear form of a one-dimensional fractional Boussinesq equation was solved and an analytical mathematical model was developed to predict the water table profile between two parallel subsurface drains. The developed model is a generalization of the Glover–Dumm’s mathematical model. As a result, the new model is applicable for both homogeneous and heterogeneous soils. It considers the degree of heterogeneity of soil as a determinable parameter. This parameter was called the heterogeneity index. The laboratory and field tests were conducted to study the performance of the proposed mathematical model in a homogenous soil and in an agricultural soil. The optimal values of parameters of the fractional model developed in this study and Glover–Dumm’s model were estimated using the inverse problem method. In the proposed inverse model, a bees algorithm (BA) was used. The results showed that in the homogenous soil, the heterogeneity index was nearly equal to 2 and therefore, the developed mathematical model reduced to the Glover–Dumm’s mathematical model. The heterogeneity index of the experimental field soil considered was equal to 1.04; hence, this soil was classified as a very heterogeneous soil. In the experimental field soil, the proposed mathematical model better represented the water table profile between two parallel subsurface drains than the Glover–Dumm’s mathematical model. Therefore, it appears that the proposed fractional model presented is a highly general and effective method to estimate the water table profile between two parallel subsurface drains, and the scale effects are robustly reflected by the introduced heterogeneity index. [Copyright &y& Elsevier]