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We examine the transition region between two solids which state differs from the state of contacting media. Small thickness of the intermediate region allows us to reduce a three-dimensional problem to a two-dimensional one for a median surface endowed with equivalent physical properties. In the present paper, we consider the generalized boundary conditions of nonperfect thermal contact for the time-fractional heat conduction equation with the Caputo derivative and solve the problem for a composite medium consisting of two semi-infinite regions. Numerical results are illustrated graphically. [ABSTRACT FROM AUTHOR]

In this article, the numerical method based on operational matrices of fractional order derivatives and integrals in the Caputo and Riemann-Liouville senses of two-parametric orthogonal shifted Jacobi polynomials is proposed for studying the approximate solutions for a generalized class of fractional order partial differential equations. The technique is extended herein to generalized classes of fractional order coupled systems having mixed partial derivatives terms. One salient aspect of this article is the development of a new operational matrix for mixed partial derivatives in the sense of Caputo. Validity of the method is established by comparing our simulated results with literature solutions obtained otherwise, yielding negligible errors. Furthermore, as a result of the comparative study, some results presented in the literature are extended and improved in the investigation herein. [ABSTRACT FROM AUTHOR]

We use the Laplace transform method to solve certain families of fractional order differential equations. Fractional derivatives that appear in these equations are defined in the sense of Caputo fractional derivative or the Riemann-Liouville fractional derivative. We first state and prove our main results regarding the solutions of some families of fractional order differential equations, and then give examples to illustrate these results. In particular, we give the exact solutions for the vibration equation with fractional damping and the Bagley-Torvik equation. [ABSTRACT FROM AUTHOR]

The time-fractional heat conduction equation with the Caputo derivative is solved for an infinite plane with an external half-infinite crack which surfaces are exposed to the heat flux loading. The solution is obtained using the integral transform technique and is expressed in terms of the Mittag-Leffler function. The stress intensity factor is calculated for different values of the order of fractional derivative. Numerical results are illustrated graphically. [ABSTRACT FROM AUTHOR]

This paper is dedicated to several original (weighted) Hölder continuity results for Riemann-Liouville fractional integrals of weighted integrable functions. As an application, we prove a new weighted continuity result for solutions to nonlinear Riemann-Liouville fractional Cauchy problems with Carathéodory dynamics. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we construct a discontinuous Galerkin finite element scheme for the multi-order fractional ordinary differential equation. The analysis of the stability shows the scheme is L 2 stable. The existence and uniqueness of the numerical solution are discussed in detail. The convergence study gives the approximation orders under L 2 norm and L ∞ norm. Numerical examples demonstrate the effectiveness of the theoretical results. The oscillation phenomena are also found during numerical tracing a non-linear multi-order fractional initial problem. [ABSTRACT FROM AUTHOR]

In study, we show the existence and integral representation of solution for energy- dependent fractional Sturm-Liouville impulsive problem of order with a (1,2] impulsive and boundary conditions. An existence theorem is proved for energy-dependent fractional Sturm-Liouville impulsive problem by using Schaefer fixed point theorem. Furthermore, in the last part of the article, an application is given for the problem and visual results are shown by figures. [ABSTRACT FROM AUTHOR]

Abstract This work deals with the generalization of the Lorentz model. It is made using the Caputo fractional derivative of order 0 < γ ≤ 1 in the frequency domain. The fractional complex dielectric function, index of refraction, reflectance, electric susceptibility were obtained. It is shown that all of them depend not only on the frequency of applied field, but also on the fractional order of the differential equation 0 < γ ≤ 1. This opens the possibility to describe more accurately the experimental results in complex systems. The ordinary results are obtained when γ = 1. [ABSTRACT FROM AUTHOR]

Purpose The purpose of this paper was to verify the possibility of applying differential calculus of incomplete order to describe relaxation of the material obtained using selective laser sintering (SLS) technology.Design/methodology/approach The samples were made using the incremental technology for three print directions. Relaxation tests were conducted. The theoretical curves, which are the solution of the equation describing the five-parameter Maxwell-Wiechert model for derivatives in relation to the total time of complete order and fractional order, were adjusted to the obtained experimental curves.Findings The SLS technology creates new possibilities regarding modelling polymeric elements which might be applied as functional models (products). Therefore, it is necessary to conduct an in-depth study of their properties, including relaxation properties, which is associated with the necessity to use proper mathematical tools to describe those properties. The differential calculus of incomplete order was applied herein to describe the anisotropy of relaxation properties because of the print direction in relation to the relaxation curves adjusted with the five-parameter Maxwell-Wiechert model.Research limitations/implications As a result of the conducted considerations, the authors obtained the dependencies describing the anisotropy of relaxation properties with the use of coefficients alpha and beta, which stand for the derivative order of the differential equation, whereas coefficient kappa stands for the translation coefficient which is an innovative application of this type of mathematical apparatus.Practical implications The developed method might be applied to describe the anisotropy of a broader group of materials manufactured with the use of incremental technologies.Originality/value The application of the differential calculus of incomplete order to describe the anisotropy of the materials manufactured from polyamide powder using the SLS technology is a distinctive feature of this paper. A crucial cognitive element of the conducted research is the fact which confirms that the dynamic viscosity coefficients have the greatest impact on the anisotropy of material properties depending on the print directions. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a new fractional derivative of the Caputo type is proposed and some basic properties are studied. The form of the definition shows that the new derivative is the natural extension of the Caputo one, and that it yields the Caputo derivative with designated memory length. By adaptively changing the memory length, the new definition is capable of capturing local memory effect in a distinct way, which is critical in modelling complex systems where the short memory properties has to be considered. Another attractive property of the new derivative is that it is naturally associated with the Riemann–Liouville definition and as a result, the well established Grünwald–Letnikov approach for numerically solving the fractional differential equation can be readily embedded to approximate the solution of differential equation that involves the new derivatives. Numerical simulations demonstrate the changeable memory effect of the new definition. Highlights • The paper presents a new fractional derivative of the Caputo type. • Some important properties and integral transform of the new derivative is derived. • The new derivative offers particular value for capturing the local memory effect. [ABSTRACT FROM AUTHOR]

Abstract In this report we establish Green's formula for an integro-differential operator, and apply it to describe a class of self-adjoint fractional order differential operators. A found symmetric fractional order Caputo–Riemann–Liouville type operator can be considered as a fractional analogue of the classical Sturm–Liouville operator in some sense. [ABSTRACT FROM AUTHOR]

In this remark, an anomalous diffusion phenomena for a fractional kinetic equation is studied. Here, we find well‐posedness conditions for the anomalous diffusion equation for the fractional kinetic process in homogeneous and non‐homogeneous cases, given a discussion about an application to biology. Finally, we derive some numerical experiments. [ABSTRACT FROM AUTHOR]

For two‐dimensional autonomous linear incommensurate fractional‐order dynamical systems with Caputo derivatives of different orders, necessary and sufficient conditions are obtained for the asymptotic stability and instability of the null solution. These conditions are expressed in terms of the elements of the system's matrix, as well as of the fractional orders of the Caputo derivatives, leading to a generalization of the well known Routh‐Hurwitz conditions. These theoretical results are then used to investigate the stability properties of a two‐dimensional fractional‐order FitzHugh‐Nagumo neuronal model. The occurrence of Hopf bifurcations is also discussed. Numerical simulations are provided with the aim of exemplifying the theoretical results, revealing rich spiking behavior, in comparison with the classical integer‐order FitzHugh‐Nagumo model. [ABSTRACT FROM AUTHOR]

This paper is devoted to the application of the method of lines to solve one-dimensional diffusion equation where the classical (integer) second derivative is replaced by a fractional derivative of the Caputo type of order α less than 2 as the space derivative. A system of initial value problems approximates the solution of the fractional diffusion equation with spline approximation of the Caputo derivative. The result is a numerical approach of order O ( Δ x 2 + Δ t m ) , where Δx and Δt denote spatial and temporal step-sizes, and 1 ≤  m  ≤ 5 is an integer which is set by an ODE integrator that we used. The convergence and numerical stability of the method are considered, and numerical tests to investigate the efficiency and feasibility of the scheme are provided. [ABSTRACT FROM AUTHOR]

In this paper, the approximate analytic solution of any order space-time fractional differential equations is constructed by means of semi-analytical method, named as residual power series method (RPSM). The first step is to reduce space-time fractional differential equation to either a space fractional differential equations or a time fractional differential equations before applying RSPM. The main step is to obtain fractional power series solutions by RSPM. At the final step, it is shown that RPSM is very efficacious, plain and powerful for obtaining the solution of any-order space-time fractional differential equations in the form of fractional power series by illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, a discrete form is proposed for solving time fractional convection–diffusion equation. Firstly, we obtain a time discrete scheme based on finite difference method. Secondly, we prove that the time discrete scheme is unconditionally stable, and the numerical solution converges to the exact one with order O ( τ 2 − α ) , where τ is the time step size. Finally, two numerical examples are proposed respectively, to verify the order of convergence. [ABSTRACT FROM AUTHOR]

Symmetric stress distribution in an infinite isotropic plane with a line crack, in which surfaces are exposed to the heat flux loading is considered in the framework of fractional thermoelasticity. The heat conduction is described by the time-fractional heat conduction equations with the Caputo derivative of fractional order α. The solution is obtained using the integral transform technique and is expressed in terms of the Mittag-Leffler function. The stress intensity factor is calculated for different values of the order of fractional derivative. The numerical results are illustrated graphically. [ABSTRACT FROM AUTHOR]

This paper investigates robust finite-time stability problem of fractional-order systems (FOSs) with time-varying delay and nonlinear perturbation. Based on a generalized Gronwall inequality, a delay-dependent sufficient condition for robust finite-time stability of such systems is established in terms of the Mittag-Leffler function. Compared with the finite-time stability conditions in the literature, our condition is simple and convenient for testing and application. The obtained results are applied to finite-time stability of linear uncertain FOSs and linear non-autonomous FOSs with time-varying delays. A numerical example is given to show the effectiveness of the obtained result. [ABSTRACT FROM AUTHOR]

The main aim of the current paper is to propose an efficient numerical technique for solving two-dimensional space-multi-time fractional Bloch–Torrey equations. The current research work is a generalization of [6] . The temporal direction is based on the Caputo fractional derivative with multi-order fractional derivative and the spatial directions are based on the Riemann–Liouville fractional derivative. Thus, to achieve a numerical technique, the time variable is discretized using a finite difference scheme with convergence order O ( τ 2 − α ) . Also, the space variable is discretized using the finite element method. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. Finally, four test problems have been illustrated to verify the efficiency and simplicity of the proposed technique on irregular computational domains. [ABSTRACT FROM AUTHOR]

The study of oscillation theory for fractional differential equations has been initiated by Grace et al. [5]. In this paper we establish some new criteria for the oscillation of fractional differential equations with the Caputo derivative of the form D a r C ⁢ x ⁢ (t) = e ⁢ (t) + f ⁢ (t , x ⁢ (t)) , t > 0 , a > 1 {{}^{C}D_{a}^{r}x(t)=e(t)+f(t,x(t)),t>0,a>1} , where r = α + n - 1 , α ∈ (0 , 1) {r=\alpha+n-1,\alpha\in(0,1)} , and n ≥ 1 {n\geq 1} is a natural number. We also present the conditions under which all solutions of this equation are asymptotic to t n - 1 {t^{n-1}} as t → ∞ {t\to\infty}. [ABSTRACT FROM AUTHOR]

In this paper, the reduced differential transform method is modified and successfully used to solve the fractional heat transfer equations. The numerical examples show that the new method is efficient, simple, and accurate. [ABSTRACT FROM AUTHOR]

Bees are the main contributors of pollination and it is predicted that disruption of pollination causes some serious problems in economics, agriculture and ecology. It has been reported that honeybee colony collapse has increased dramatically for some time in different parts of the world. In order to investigate the causes of colony collapse, we establish a fractional multi-order honeybee colony population model. Two different models with Caputo and Caputo–Fabrizio differentiation operators have been compared. We justify the existence and uniqueness of the solution. Stability analysis has been presented. We find the numerical values of the fractional order ( p , q ) so that the numerical results are close to the experimental findings more than the integer-order case. [ABSTRACT FROM AUTHOR]

In this work, we are concerned with how to find an explicit approximate solution (AS) for the telegraph equation of space-fractional order (TESFO) using Sumudu transform method (STM). In this method, the space-fractional order derivatives are defined in the Caputo idea. The Sumudu method (SM) is established to be reliable and accurate. Three examples are discussed to check the applicability and the simplicity of this method. Finally, the Numerical results are tabulated and displayed graphically whenever possible to make comparisons between the AS and exact solution (ES). [ABSTRACT FROM AUTHOR]

In this paper, the high-dimensional Caputo-type parabolic equation with fractional Laplacian is studied by using the finite difference method. The convergence and error estimate of the established finite difference scheme are shown. And the illustrative examples are also displayed which support the theoretical analysis. [ABSTRACT FROM AUTHOR]

We have applied the new approach of homotopy perturbation method (NHPM) for Schrödinger equation featuring time-fractional derivative. A combination of NHPM and multiple fractional power series form has been used the first time to present analytical solution. In order to illustrate the simplicity and ability of the suggested approach, some specific and clear examples have been given. All numerical calculations in this manuscript have been carried out with Mathematica. [ABSTRACT FROM AUTHOR]

We compare the model of heat transfer proposed by Cattaneo, Maxwell, and Vernotte with another one, formulated in terms of fractional differential equations, in one and two dimensions. These are only some of the numerous models that have been proposed in the literature over many decades to model heat transport and possibly heat waves, in place of the classical heat equation due to Fourier. These models are characterized by sound as well as by critical properties. In particular, we found that the Cattaneo model does not exhibit necessarily oscillations or negative values of the (absolute) temperature when the relaxation parameter, $\tau$, drops below some value. On the other hand, the fractional derivative model may be affected by oscillations, depending on the specific initial profile. We also estimate the error made when the Cattaneo equation is adopted in place of the heat equation, and show that the approximation error is of order $\tau$. Moreover, the solution of the Cattaneo equation converges uniformly to that of the heat equation as $\tau \to 0+$ in the full closed time interval $[0,T]$ (for any given $T > 0$), while this does not occur for the time derivative, and the higher-order time derivatives blow up. [ABSTRACT FROM AUTHOR]

We propose a generalized definition of Caputo derivatives from t = 0 of order γ∈in (0, 1) using a convolution group, and we build a convenient framework for studying initial value problems of general nonlinear time fractional differential equations. Our strategy is to define a modified Riemann-Liouville fractional calculus which agrees with the traditional Riemann-Liouville definition for t > 0 but includes some singularities at t = 0 so that the group property holds. Then, making use of this fractional calculus, we introduce the generalized definition of Caputo derivatives. The new definition is consistent with various definitions in the literature while revealing the underlying group structure. The underlying group property makes many properties of Caputo derivatives natural. In particular, it allows us to deconvolve the fractional differential equations to integral equations with completely monotone kernels, which then enables us to prove the general comparison principle with the most general conditions. This then allows for a priori energy estimates of fractional PDEs. Since the new definition is valid for locally integrable functions that can blow up in finite time, it provides a framework for solutions to fractional ODEs and fractional PDEs. Many fundamental results for fractional ODEs are revisited within this framework under very weak conditions. [ABSTRACT FROM AUTHOR]

The time-fractional diffusion equation with mass absorption is studied in a half-line domain under the Dirichlet boundary condition varying harmonically in time. The Caputo derivative is employed. The solution is obtained using the Laplace transform with respect to time and the sin-Fourier transform with respect to the spatial coordinate. The results of numerical calculations are illustrated graphically. [ABSTRACT FROM AUTHOR]

In this paper, using the piecewise linear and quadratic Lagrange interpolation functions, we propose a novel numerical approximate method for the Caputo fractional derivative. For the obtained explicit recursion formula, the truncation error is investigated, which shows the involved convergence order isO(τ3−β) withβ∈(0,1). As an application, we use this proposed numerical approximation to solve the time fractional diffusion equations by the barycentric rational interpolations in space. The resultant systems of algebraic equations, truncation error, convergence, and stability are analyzed. Theoretical analysis and numerical examples show this constructed method enjoys accuracy of, wheredis the degree of the rational polynomial. [ABSTRACT FROM AUTHOR]

In this paper, we present a hybridizable discontinuous Galerkin (HDG) method for solving a class of fractional boundary value problems involving Caputo derivatives. The HDG methods have the computational advantage of eliminating all internal degrees of freedom and the only globally coupled unknowns are those at the element interfaces. Furthermore, the global stiffness matrix is tridiagonal, symmetric, and positive definite. Internal degrees of freedom are recovered at an element-by-element postprocessing step. We carry out a series of numerical experiments to ascertain the performance of the proposed method. [ABSTRACT FROM AUTHOR]

This paper studies the existence and uniqueness of polynomial solutions for a class of fractional differential equations and systems involving Caputo derivatives. Several illustrative examples with explicit solutions are also presented. [ABSTRACT FROM AUTHOR]

This paper deals with the study of heat transfer and thermal damage in triple layer skin tissue using fractional bioheat model. Here, we consider three types of heating viz. sinusoidal heat flux, constant temperature and constant heat flux heating on skin surface. An implicit finite difference scheme is obtained by approximating fractional time derivative by quadrature formula and space derivative by central difference formula. The temperature profiles and thermal damage in the skin tissue are obtained to study the effect of fractional parameter on diffusion process for constant temperature and heat flux boundary heating on skin surface. A parametric study for sinusoidal heat flux at skin surface has also been made. [ABSTRACT FROM AUTHOR]

In this work, we apply fractional calculus to analyze a class of economic growth modelling (EGM) of the Spanish economy. More precisely, the Grünwald-Letnnikov and Caputo derivatives are used to simulate GDP by replacing the previous integer order derivatives with the help of Matlab, SPSS and R software. As a result, we find that the data raised from the Caputo derivative are better than the data raised from the Grünwald-Letnnikov derivative. We improve the previous result in [12]. [ABSTRACT FROM AUTHOR]

This paper presents a new computational technique for solving fractional pantograph differential equations. The fractional derivative is described in the Caputo sense. The main idea is to use Müntz-Legendre wavelet and its operational matrix of fractional-order integration. First, the Müntz-Legendre wavelet is presented. Then a family of piecewise functions is proposed, based on which the fractional order integration of the Müntz-Legendre wavelets are easy to calculate. The proposed approach is used this operational matrix with the collocation points to reduce the under study problem to a system of algebraic equations. An estimation of the error is given in the sense of Sobolev norms. The efficiency and accuracy of the proposed method are illustrated by several numerical examples. [ABSTRACT FROM AUTHOR]

In this work, the variational iteration method (VIM) is used to solve a class of fractional optimal control problems (FOCPs). New Lagrange multipliers are determined and some new iterative formulas are presented. The fractional derivative (FD) in these problems is in the Caputo sense. The necessary optimality conditions are achieved for FOCPs in terms of associated Euler–Lagrange equations and then the VIM is used to solve the resulting fractional differential equations. This technique rapidly provides the convergent successive approximations of the exact solution and the solutions approach the classical solutions of the problem as the order of the FDs approaches 1. To achieve the solution of the FOCPs using VIM, four illustrative examples are included to demonstrate the validity and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a spatial fractional-order thermal transport equation with the Caputo derivative is proposed to describe convective heat transfer of nanofluids within disordered porous media in boundary layer flow. This equation arises naturally when the effect of anomalous migration of nanoparticles on heat transfer is considered. The numerical results show that local Nusselt numbers of four different kinds of nanofluids are all inversely proportional to the fractional derivative exponent β . Based on this finding, it is concluded that the anomalous diffusion of nanoparticles improves the convective heat transfer of nanofluids and the space fractional thermal transport equation may serve as a candidate model for studying nanofluids. Additionally, the effects of other involved physical parameters on temperature distribution and Nusselt number are presented and analyzed. [ABSTRACT FROM AUTHOR]

This work studies the approximate controllability of a class of impulsive neutral stochastic differential equations with infinite delay and Poisson jumps involving generalized Caputo fractional derivative under the condition that the corresponding linear system is approximately controllable. Utilizing the fixed point theory and sectorial operator theory, the existence of the mild solution of the impulsive neutral stochastic equation is established imposing weaker regularity on nonlinear terms. A set of sufficient conditions establishing controllability results is derived with the help of stochastic analysis and fractional calculus. Finally, an example is provided to illustrate the obtained abstract result. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a numerical solution of Lienard's equation. The residual power series (RPS) method is implemented to find an approximate solution to this problem. The proposed method is a combination of the fractional Taylor series and the residual functions. Numerical and theoretical results are presented. [ABSTRACT FROM AUTHOR]

This paper describes an algorithm for reconstruction the boundary condition and order of derivative for the heat conduction equation of fractional order. This fractional order derivative was applied to time variable and was defined as the Caputo derivative. The heat transfer coefficient, occurring in the boundary condition of the third kind, was reconstructed. Additional information for the considered inverse problem is given by the temperature measurements at selected points of the domain. The direct problem was solved by using the implicit finite difference method. To minimize functional defining the error of approximate solution an Artificial Bee Colony (ABC) algorithm and Nelder-Mead method were used. In order to stabilize the procedure the Tikhonov regularization was applied. The paper presents examples to illustrate the accuracy and stability of the presented algorithm. [ABSTRACT FROM AUTHOR]

A simple and inexpensive preprocessing of an initial–boundary value problem with a Caputo time derivative is shown theoretically and numerically to yield an enhanced convergence rate for the L1 scheme. The same preprocessing can also be used with other methods for time-dependent problems. [ABSTRACT FROM AUTHOR]

In this paper, a variable-order fractional derivative nonlinear cable equation is considered. It is commonly accepted that fractional differential equations play an important role in the explanation of many physical phenomena. For this reason we need a reliable and efficient technique for the solution of fractional differential equations. This paper deals with the numerical solution of class of fractional partial differential equation with variable coefficient of fractional differential equation in various continues functions of spatial and time orders. Our main aim is to generalize the Chebyshev cardinal operational matrix to the fractional calculus. Finally, illustrative examples are included to demonstrate the validity and applicability of the presented technique. [ABSTRACT FROM AUTHOR]

In this paper, the semi-analytic numerical technique for the solution of time-space fractional telegraph equation is applied. This numerical technique is based on coupling of the homotopy analysis method and sumudu transform. It shows the clear advantage with mess methods like finite difference method and also with polynomial methods similar to perturbation and Adomian decomposition methods. It is easily transform the complex fractional order derivatives in simple time domain and interpret the results in same meaning. [ABSTRACT FROM AUTHOR]

In his first paper on the generalization of the tautochrone problem, that was published in 1823, Niels Henrik Abel presented a complete framework for fractional-order calculus, and used the clear and appropriate notation for fractional-order integration and differentiation. [ABSTRACT FROM AUTHOR]

For a class of tempered fractional terminal value problems of the Caputo type, we study the existence and uniqueness of the solution, analyze the continuous dependence on the given data, and using a shooting method we present and discuss three numerical schemes for the numerical approximation of such problems. Some numerical examples are considered in order to illustrate the theoretical results and evidence the efficiency of the numerical methods. [ABSTRACT FROM AUTHOR]

In this paper, we proposed a new numerical scheme to solve the time fractional nonlinear Klein–Gordon equation. The fractional derivative is described in the Caputo sense. The method consists of expanding the required approximate solution as the elements of Sinc functions along the space direction and shifted Chebyshev polynomials of the second kind for the time variable. The proposed scheme reduces the solution of the main problem to the solution of a system of nonlinear algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces accurate results. [ABSTRACT FROM AUTHOR]

The present work studies the effect of an impulsive perturbation to classical and fractional classes of initial value problems. Scattering properties of the time sequences corresponding to the impulses are studied in the both integer and non-integer order cases. It is shown that, the sign-changing property of the perturbation function determines the mentioned scattering property of the kick times. [ABSTRACT FROM AUTHOR]

In this paper, we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operatorwhereand the fractional derivatives,,are in the Caputo sense. Applying integral transform methods, we describe a complete family of eigenfunctions and fundamental solutions of the operatorin classes of functions admitting a summable fractional derivative. The solutions are expressed using the Mittag–Leffler function. From the family of fundamental solutions obtained, we deduce a family of fundamental solutions of the corresponding fractional Dirac operator, which factorizes the fractional Laplace operator introduced in this paper. [ABSTRACT FROM AUTHOR]

The generalized fractional integral operator and the Caputo type generalized fractional derivative operator are defined which contain the Riemann-Liouville integral operator, the Hadamard fractional integral operator, the Caputo fractional derivative operator and the Caputo type Hadamard fractional derivative operator as special cases. General solutions (the explicit solutions) of the impulsive Caputo type generalized fractional differential equations are given. Applying our results, existence results of solutions of boundary value problems for an impulsive fractional differential equations involved with the Caputo type generalized fractional derivatives are established. Examples and some remarks on recent published papers are presented to illustrate the main theorems. [ABSTRACT FROM AUTHOR]