02- Progress in Fractional Differentiation and Applications An International Journal

Forthcoming

 On the Stability of Commensurate Fractional-Order Lorenz System Abstract : In this paper, two special cases related to the stability of the equilibria for the non-linear autonomous Fractional-order Lorenz System (FoLS) are discussed and verified numerically based on Adomian Decomposition Method (ADM). These two cases can be very useful for discriminating between several cases in the stability of this system. All numerical plots, in this study, have been performed using MATLAB (R2011a).

 Fractional Frobenius Series Solutions of Confluent α-Hypergeometric Differential Equation Abstract : In this work, the so-called conformable fractional derivative definition is employed to obtain the fractional Frobenius series solutions around the regular α-singular point x = 0 for the confluent α-hypergeometric differential equation. Of course, such solutions for this equation in its classical case are just confluent hypergeometric functions. The proposed method is straightforward to be applied as an algorithm.

 FRACTIONAL HAMILTONIAN SYSTEMS WITH VANISHING POTENTIALS Abstract : Using minimax techniques involving Cerami sequences, we prove the existence of at least one pair of positive solutions for an fractional Hamiltonian system in R with potentials vanishing at infinity and subcritical nonlinearities which are super-linear at the origin and at infinity. We establish new estimates in order to prove the boundedness of a Cerami sequence. Our result is new even in the integer case.

 Numerical investigation on the solution of a space-fractional via preconditioned SOR iterative method Abstract : This paper considers a numerical investigation on the solution of a one-dimensional linear space-fractional partial differential equation with the application of an unconditionally implicit finite difference method and the Caputos space-fractional derivative. We formulate the Caputosimplicit finite difference approximation equation to form a corresponding linear system in which its coefficient matrix is large-sized and has a great sparsity. We construct a preconditioned linear system intending to speed up the convergence rate in computing the solutions of the linear system using the SOR iterative method. We present two examples of the one-dimensional linear space-fractional partial differential equation problem to illustrate the effectiveness and efficiency of our proposed PSOR iterative method. Through the investigation, the numerical results show that the proposed PSOR iterative method is superior to the Preconditioned Gauss-Seidel and Gauss-Seidel iterative methods in terms of efficiency.

 A Game Theory-Based Fractional Order Model for The simulation of Human Responses in an Emerging Epidemic Abstract : A non-integer order model stemmed from an evolutionary game theory is presented in this work to study the impact of impact of changes in human behavior in an emerging epidemic. Introducing fractional order derivatives in Caputo sense to the integer order model may give a better understanding of the numerous effects of memory and learning process of individuals on the prevalence of the infectious diseases when epidemics occur. Numerical simulations are presented to figure out the features of using the non-integer order (fractional order) models in epidemiology.

 Fractional-Order Delayed Salmonella Transmission Model: A Numerical Simulation Abstract : In this paper, a fractional dynamical model of Salmonella with time delay is studied numerically. The proposed model is administered by a system of fractional delay differential equations, where the fractional derivative is defined in the sense of the Caputo definition. The parameters are modified regarding to the order of the fractional derivative. The stability of the disease free equilibrium point and endemic equilibrium point is investigated for any time delay. Weighted difference numerical technique is introduced to simulate the proposed model. This scheme was unconditionally stable when the weight factor is less than 1. Numerical simulations with some comparison are introduced to show the applicability and effectivity of the proposed method to solve such stiff systems of fractional delay differential equations and to confirm the theoretical studies.

 Hadamard Inequality for (k − r) Riemann-Liouville Fractional Integral Operator via Convexity Abstract : Recently,manyresearchershavepublishedworkontheHermite-Hadamardinequalities,duetotheirimmenseimportancein the fields of numerical analysis, statistics, optimization and convexity theory. In this paper, certain new Hermite-Hadamard type integral inequalities have been established using the (k − r) Riemann-Liouville fractional integral operator. We present various inequalities based on different types of the convex functions such as quasi-convex, l-convex, η-convex in the second sense and (β,l)-convex functions. Also, we derive Hermite-Hadamard type inequalities for the product of two l-convex functions and two (β,l)-convex functions using (k − r) Riemann-Liouville fractional integral operator. The results obtained in our work will be helpful in the further study of the convex functions and in the evaluation of the certain mathematical problems.

 Iterative Methods for Solving Seventh-Order Nonlinear Time Fractional Equations Abstract : The fundamental aim of the present work is to investigate the numerical solutions of the seventh order Caputo fractional time Kaup-Kupershmidt, Sawada-Kotera and Lax’s Korteweg-de Vries equations using two reliable techniques, namely, the fractional reduced differential transform method and q-homotopy analysis transform method. These equations are the mathematical formulation of physical phenomena that arise in chemistry, engineering and physics. For instance, in the motions of long waves in shallow water under gravity, nonlinear optics, quantum mechanics, plasma physics, fluid mechanics and so on. With these two methods, we construct series solution to these problems in the recurrence relation form. We present error estimates to further investigate the accuracy and reliability of the proposed techniques. The outcome of the study reveals that the two techniques used are computationally accurate, reliable and easy to implement when solving fractional nonlinear complex phenomena that arise in physics, biology, chemistry and mathematics.

 Continuity with respect to fractional order of the time a linear fractional pseudo-parabolic equation Abstract : In this paper, we consider a pseudo-parabolic equation with the Caputo fractional derivative. We study the existence and uniqueness of a class of mild solutions of these equations. For a nonlinear problem, we first investigate the global solution under the initial data $u_0 \in L^2$. In the case of initial data $u_0 \in L^q,~q \neq 2$, we obtain the local existence result. Our main tool here is using fundamental tools, namely Banach fixed point theorem and Sobolev embeddings.

 Analysis and Modelling of Fractional Order HIV/AIDS Model with Parameter Estimation Abstract : In this paper, nonlinear fractional order HIV/AIDS mathematical model is discussed for the complex transmission of the disease epidemic problems. It is accepted that susceptible wind up contaminated by means of sexual contacts with infective and all infective eventually create AIDS. The point of this task was to amend transmission models recently created to represent HIV transmission and AIDS related mortality. The Caputo-Fabrizio fractional derivative operator of order $\beta \in (0, 1]$ is used to obtain fractional differential equations structure. The stability fractional order model was developed and the unique non-negative solution was tested. The numerical simulations are performed using an iterative technique. Some new results are being viewed with the help of Sumudo transform. Nonetheless, according to Banach, the related findings are given nonlinear functional analysis and fixed point theory. However, mathematical simulations are also acknowledged to evaluate the impact of the models parameter by decreasing the fractional values and showing the effect of the $\beta$ fractional parameter on our solutions obtained. The impact of various parameters is represented graphically.

 Analysis and Modeling of HIV Dynamical Transmission Abstract : In this paper, we developed the fractional order HIV transmission model for treatment and control to reduce its effect on a population which play an important role for public health. Qualitative analysis has been made to verify the the steady state and uniqueness of the system is also developed for reliable results. Caputo fractional derivative operator $\beta\in(0, 1]$ works to achieve the fractional differential equations. Laplace with Adomian Decomposition Method successfully solved the fractional differential equations. Ultimately, numerical simulations are also developed to evaluate the effects of the device parameter on spread of disease and effect of fractional parameter $\beta$ on obtained solution which are also assessed by tabulated results.

 The Drug Administration via Fractional-order PI^\lambda D^\delta -Controller Abstract : Amiodarone is one of the most commonly used antiarrhythmic drugs. Some of its features, like its anomalous and its non-exponential pharmacokinetics make it so important in clinical implications. Recently, more accurate fractional-order models were proposed to describe its pharmacokinetics in a consistent manner. Hence, fractional-order PID controllers will be used and tuned to regulate the dynamic behavior of the drug administration and to provide good closed-loop response. This controller is designed based on the Particle Swarm Optimization (PSO) algorithm by approximating s^alpha; 0 < alpha <= 1. The PSO algorithm is used to minimize the ITAE, IAE, ISE and ITSE error functions. A comparison between Oustaloup’s and El-Khazali’s approximations are presented to show the effectiveness and the cost of the controller design. All results are verified via numerical simulations.

 A fractional-order model of dengue fever with awareness effect model: Numerical solutions and asymptotic stability analysis Abstract : In this paper, we investigate Adams-type predictor-corrector method for solving fractional dengue fever model. Asymptotic stability Analysis of the solutions is provided. Our results are presented graphically. It is found that this method is more convenient and accurate for solving nonlinear fractional differential equations.

 Newly Proposed Solutions using Caputo, Caputo–Fabrizio and Atangana–Baleanu Fractional Derivatives: A Comparison Abstract : The aim of this paper is to demonstrate the extent to which the new iterative Sumudu transform method (NISTM) helps in solving fractional KdV–Burgers equation (KdVB). In fact, new explanatory solutions are being obtained by using Caputo sense, which represents kernels power law type, Caputo–Fabrizio (CF) standing for exponentially with decaying type kernel and the Atangana–Baleanu (AB) representing the Mittag-Leffler type kernel. The NISTM is a powerful algorithm that has been found effective and certainly adds to the weight of the current study.

 Impulsive fractional differential equations under uncertainty: Application in fluid mechanics Abstract : In this research, we study impulsive fractional differential equations (IFDEs) under interval uncertainty using Laplace transforms. For this purpose, the solution of IFDEs is obtained under Riemann-Liouville differentiability. Also, the Bagley-Torvik equation involving additive delta function on the interval right-hand side is solved to validate the theoretical results. the Bagley-Torvik equation arises in fluid mechanics.

 Controllability of Hilfer fractional non-autonomous evolution equations with nonlocal initial conditions Abstract : This paper is concerned with the controllability of Hilfer fractional non-autonomous evolution equations with nonlocal initial conditions in Banach spaces. The main keys in this investigation are the Krasnoselskii’s fixed point theorem and the properties of evolution operators. An example is given to illustrate the main results.

 Existence and Stability of Nonlinear Implicit Caputo-Exponential Fractional Differential Equations Abstract : In this paper, We discuss the existence and stability of solution for the following fractional problem with Caputo--Exponential fractional derivative \begin{equation*} _{c}^{e}D_{0^{+}}^{\mu}y(t)= f(t,y(t),\ _{c}^{e}D_{0^{+}}^{\mu}y(t)), \ \mbox{for each}, \ t\in J:=[0, b], \ b>0, \ 0<\mu\leq 1, \end{equation*} \begin{equation*} y(0)= y_{0}. \end{equation*} The arguments are based upon the Banach contraction principle, Schauder fixed point theorem and the nonlinear alternative of Leray--Schauder type. As applications, two examples are included to show the applicability of our results.

 Trigonometric Commutative Caputo fractional Korovkin theory for Stochastic processes Abstract : Here we consider and study from the trigonometric point of view ex- pectation commutative stochastic positive linear operators acting on L1- continuous stochastic processes which are Caputo fractional differentiable. Under some mild, general and natural assumptions on the stochastic processes we produce related trigonometric Caputo fractional stochastic Shisha-Mond type inequalities pointwise and uniform. All convergences are produced with rates and are given by the trigonometric fractional stochastic inequalities involving the first modulus of continuity of the ex- pectation of the α-th right and left fractional derivatives of the engaged stochastic process, α > 0, α ∈/ N. The amazing fact here is that the basic non-stochastic real Korovkin test functions assumptions impose the conclusions of our trigonometric Caputo fractional stochastic Korovkin theory. We include also a detailed trigonometric application to stochastic Bernstein operators.

 Reduced Order Hybrid Dislocated Synchronization Of Complex Fractional Order Chaotic Systems Abstract : In this manuscript,we have analyzed the complex Lorenz and complex Duffing systems of fractional order of different dimension.Here,we have blended the ideas of reduced order synchronization with dislocated synchronization schemes.Using the stability theory of Lyapunov,sufficient conditions have been derived for accomplishing reduced order hybrid dislocated synchronization.Numerical simulations have been performed in MATLAB to validate the efficacy of the method proposed. The obtained results show the usefulness and suitability of the method used to achieve the synchronization.

 Numerical Solution of Fractional Bratu’s Initial Value Problem using Compact Finite Difference Scheme Abstract : In this paper, we consider initial value problem of Bratu-type equations of fractional order 1 < α ≤ 2. Compact finite difference schemes corresponding to α = 2 and 1 < α < 2 are proposed and convergence analysis of the methods are discussed separately. Some examples are also given to show the efficiency of the methods

 A novel approach for solving time fractional nonlinear Fisher’s equation by using Chebyshev Spectral Collocation method Abstract : In this paper, the Chebyshev spectral method is applied to solve the nonlinear Fisher fractional equation with initial boundary conditions. Many researchers have perused time and space fractional Fisher’s equation including , differential transform method , variational iteration method , etc. Here, the fractional derivative is considered in Caputo type, Then, by using the Chebyshev spectral collocation method, the problem is transformed into an algebraic system. The results show that this method is an acceptable method for numerical solution of the Fisher equation.

 Numerical Treatments for a Complex Order Fractional HIV Infection Model with Drug Resistance During Therapy Abstract : In this article, we develop the numerical technique for solving HIV mathematical model of complex order with drug resistance during the therapy treatment, where the derivative is defined in Caputo sense. Two numerical methods are presented to study numerically the complex order fractional HIV model. The proposed numerical methods are the non-standard finite difference method and the generalized Euler method. Comparative studies and numerical simulations are given to validate the theoretical results.

 Existence Results For a New Fractional Boundary Value Problem By Variational Methods Abstract : In this paper, we consider a new fractional boundary value problem with the Riemann-Liouville fractional derivatives of higher order n-1<\alpha

 Dynamics and Sensitivity Analysis of Fractional-Order Delay Differential Model For Coronavirus Infection Abstract : Recently, an outbreak of a new coronavirus (CoV) that began in the Chinese city of Wuhan has already killed at least 106 people in China, according to data released early Tuesday 21 January 2020. In this article, we provide a mathematical model to give us a better understanding of what drives the intensity of symptoms, infectivity of the virus and the host, and duration of the disease. The model is governed by a system fractional-order differential equations with time-delay. We incorporate a fractional-order and time-delay in the model to naturally represent both long-run memory and non-locality effects in the dynamics of the model. We investigate the qualitative behaviour of the model and deduce some interesting sufficient conditions which ensure the asymptotic stability of the steady states. Sensitivity analyses are conducted to provide insight into how uncertainty in the parameters affects the model outputs and which parameters tend to drive these variations. We investigate the sensitivity to variations in the rate of interferon, in the rate of innate immunity cells, rate of adaptive immunity cells, and in pathogen virulence. Some numerical simulations are provided to show the effectiveness of the derived theoretical results. This study may help in delineate the interaction between respiratory viruses and the host immune system, and determine the most effective parameters for treatment.

 Construction of Caputo-Fabrizio fractional differential mask for image enhancement. Abstract : The aim of this paper is to introduce an algorithm based on the Caputo-Fabrizio (CF) fractional differential mask for image contrast enhancement. Experiments show that the method can control the degree of contrast enhancement by varying the fractional differential order. The contrast performance is measured by using Peak Signal to Noise Ratio (PSNR).The final numerical procedure is given for contrast enhancement, and the experimental results verify the effectiveness of the algorithm (higher PSNR values) when compared with other proposed fractional differential mask.

 A New Nonsingular Fractional Model for a Biological Snap Oscillator with Chaotic Attractors Abstract : The research aims to propose a new non-integer order model to analyse the chaotic behaviour of a biological snap oscillator. The suggested model consists of a newly developed fractional derivative with Mittag-Leffler kernel. To investigate the model, the time-domain response and the phase portrait are taken into account. In addition, a powerful numerical method is employed in order to implement the model in an appropriate, precise manner. The existence of chaotic attractors are shown by some simulations and experiments. Finally, to control the chaos, a stabilizing controller is designed and its effectiveness is illustrated and verified.

 On the entropic order quantity model based on the conformable calculus Abstract : The economic order quantity (EOQ) pattern resolves the value that minimizes the collection aggregate of cost expense functions. One of the resent active models is that based on entropy, which is called entropy order quantity (EnOQ). It’s recommended that it might be potential to develop production systems presentation by employing the theory of information (Entropy). In this effort, we suggest a model for EOQ, by using the conformable calculus calling conformable entropy order quantity (C-EnOQ). In this case, we deliver the cost functions with respect to time in a returning period. In this model, we investigate the connected optimization problem and develop an undisturbed technique for computing a bounded interval covering the optimal sequence distance, utilizing the Tsallis fractional entropy. In addition, for an extraordinary class of transport functions, we investigate these cost functions to calculate the optimal size.

 New Results on the Generator of Conformable Fractional Semigroups Abstract : The theory of semigroups of bounded linear operators is a useful tool for solving differential equations. This approach is used for solving a􀀀abstract Cauchy problem. This paper is concerned with certain aspects of fractional semigroups. Mainly, we try to give an answer to the following question “when can a linear operator A generate a fractional semigroup?” To do this in the sequel, we introduce and prove new properties of fractional semigroups of operators similar to that of strongly continuous semigroups of operators.

 Analytic Solutions of a 3-D Propagated Wave Dynamical Equation Formulated by Conformable Calculus Abstract : Researchers show that there is a fundamental association between the symmetric and traveling wave solutions. They have shown that all symmetric waves are traveling waves. In this paper, we establish new analytic solution collections of nonlinear conformable time-fractional wave dynamical equation, equations of Khokhlov-Zabolotskaya (KZ) type in a complex domain. For this purpose, we build a new definition of a symmetric conformable differential operator (SCDO). The operator has a symmetric illustration in the open unit disk. By using SCDO, we propagate a class of special wave dynamical equation type KZ equation. The consequences show that the obtainable methods are powerful, dependable and formulate to apply to all classes of complex differential equations.

 Ritz Method and Genocchi Polynomials for Solution of Fractional Partial Differential Equations Abstract : In this work, Ritz approximation based on Genocchi polynomials have been used to ob- tain numerical solutions large category of fractional partial differential equations (FPDEs) that fractional derivatives are of Capotou type. We transforme fractional diffrential equa- tions into optimization problem by Genocchi polynomial and get the system of nonlinear algebraic equation. Solving the system of nonlinear algebraic equation, we obtain the coefficients of polynomial expansion. In the following, Some numerical examples are presented which illustrate the performance of the method.

 Sumudu Variational Iteration Method for Solving Burger’s and Coupled Burger’s Equations of Fractional Order Abstract : In this paper, the Sumudu variational iteration method (SVIM) is employed to handle the fractional Burger’s and coupled fractional Burger’s equations. The fractional derivative is described in the Caputo sense. The approximate solutions are obtained by using SVIM, which is the coupling method of fractional variational iteration method and Sumudu transform. The method in general is easy to implement and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the proposed technique.

 Linear Differential Equations of Fractional Order with Recurrence Relationship Abstract : This paper introduce the basic general theory for Linear Sequential Fractional Differential Equations which includes a recurrence relationship, involving the Riemann-Liouville fractional operator. The presented equation is not a generalization of the known Sequential Linear Fractional Differential Equations, but both are closely related.

 NEW INEQUALITIES OF HERMITE-HADAMARD TYPE FOR h-CONVEX FUNCTIONS VIA GENERALIZED FRACTIONAL INTEGRALS Abstract : In this paper, we establish new inequalities of Hermite-Hadamard type for h-convex functions by using generalized fractional integral. Our established results are extension of previous research.

 Fractional Integral Inequalities using Marichev-Saigo-Maeda Fractional Integral Operator Abstract : Here, we obtain several new fractional integral inequalities using Marichev Saigo Maeda fractional integral operator for synchronous functions which are concern with extended Chebyshev functional.

 Analytic solutions of a 3-D propagated wave dynamical equation formulated by conformable calculus Abstract : Researchers show that there is a fundamental association between the symmetric and traveling wave solutions. They have shown that all symmetric waves are traveling waves. In this paper, we establish new analytic solution collections of nonlinear conformable time-fractional wave dynamical equation, equations of Khokhlov-Zabolotskaya (KZ) type in a complex domain. For this purpose, we build a new definition of a symmetric conformable differential operator (SCDO). The operator has a symmetric illustration in the open unit disk. By using SCDO, we propagate a class of special wave dynamical equation type KZ equation. The consequences show that the obtainable methods are powerful, dependable and formulate to apply to all classes of complex differential equations.

 A Convenience Approximate Method for Solving an Inverse Heat Conduction Problem Abstract : In this research, fractional type one-dimensional inverse heat conduction problem (FIHCP) have been studied. This problem is devoted to calculating the temperature distribution in its range and the thermal flux on the bound while the temperature is clear at some of the domain points. The new approach of homotopic perturbation method (NHPM) is employed to recovering unknown functions and obtaining a solution for the problem. At the end, some appropriate examples are given for introducing and implementing the proposed approach in solving FIHCP.

 Study of memory effect in an Economic Order Quantity model for completely backlogged demand during shortage Abstract : The most commonly developed inventory models are the classical economic order quantity model, is governed by the integer order differential equations. We want to come out from the traditional thought i.e. classical order inventory model where the memory phenomena are absent. Here, we want to incorporate the memory effect that is based on the fact economic agents remember the history of changes of exogenous and endogenous variables. In this paper, we have proposed and solved a fractional order economic order quantity model with constant demand rate where the demand is fully backlogged during shortage time. Finally, a numerical example has been illustrated for this model to show the memory dependency of the system. The numerical example clears that for the considered system the profit is maximum in long memory affected system compared to the low memory affected or memory less system.

 Fractional Derivative Inventory Model for Deteriorating Items with no Shortages Abstract : In this paper, a nonlinear inventory model with demand as a derivative of time is developed for deteriorating items with no shortages are allowed. The model is solved initially by using Differential Transform Method (DTM) and further, the model is modified to Caputo Derivative Fractional Order (CDFO) and solved by Homotopy Perturbation Method with triangular fuzzy initial conditions. The results for both models in terms of non-linearity of DTM and CDFO are analyzed and compared numerically.

 A New Sumudu Type Integral Transform and Its Applications Abstract : In this paper, we introduce new Sumudu type integral transform to investigate conformable fractional derivative, convolution, commutative semi group property and to obtain solution of conformable heat transfer problems.

 Theory of stochastic pantograph differential equations with psi-Caputo fractional derivative Abstract : In this paper, we mainly study the existence of analytical solution of stochastic pantograph differential equations. The standard Picard’s iteration method is used to obtain the theory.

 A competitive version of equivalence scheme based on Atangana-Baleanu fractional derivative for fractional differential equations Abstract : In this letter, we give some extensions of the recent result of Kilbas, Srivastava and Trujillo ( [1], Chapter 3 ) on the fundamental existence and uniqueness theorems for ordinary fractional differential equation based on the reducing Cauchy typeproblems and Atangana-Baleanu fractional derivative in Caputo sense. The obtained results show that, the final output of [2] is a special case of this study.

 Some Results On Multipoint Integral Boundary Value Problems For Fractional Integro-Differential Equations Abstract : This paper is devoted to the study of nonlinear fractional integro-differential equation involving caputo fractional derivative with non local multi point integral boundary conditions. Main results are obtained by applying the Banach contraction principle and Krasnoselskii fixed point theorems. Example are given to illustrate the results.

 Trapezium-type inequalities for the AB-fractional integrals via generalized convex and quasi convex functions Abstract : The authors have proved Hermite-Hadamard inequality and an identity for the AB-fractional integrals via generalized φ-convex functions. By applying the established identity using generalized φ-convex function we give some integral inequalities connected with the right hand side of Hermite-Hadamard type inequalities for the AB-fractional integrals. Various special cases have been identiﬁed. Also, it is introduced the concept of generalized quasi φ−convex functions and it is established some fractional inequalities related with the aforementioned inequality type. The ideas and techniques of this paper may stimulate further research in the ﬁeld of integral inequalities.

 Inverse source problems for degenerate time-fractional PDE Abstract : In this paper, we investigate two inverse source problems for degenerate time-fractional partial differential equation in rectangular domains. The rst problem involve a space-degenerate partial di erential equation and the second one involve a time-degenerate partial differential equation. Solutions to both problem are expressed in series expansions. For the fi rst problem, we obtained solutions in the form of Fourier-Legendre series. Convergence and uniqueness of solutions have been discussed. Solutions to the second problem are expressed in the form of Fourier-Sine series and they involve a generalized Mittag- Leffler type function. Moreover, we have established a new estimate for this generalized Mittag-Leffler type function. The obtained general solutions are also illustrated by example solutions using appropriate choices of the given conditions.

 Approximate Analytical and Numerical Solutions for time-factional generalized nonlinear Huxley equation Abstract : In this work, multiple traveling wave solutions for one kind of nonlinear partial differential equations of fractional order using tanh-function method are investigated. Namely, time fractional generalized nonlinear Huxley equation is studied. The proposed method is shown to be useful for handling other related general form of fractional nonlinear partial differential equation. The analytic solutions behavior is illustrated graphically. In addition, a numerical treatment for the same problem is proposed using cubic spline function, of nonlinear form, method. The stability of the method is investigated based on Von Neumann concept. The method is proved to be conditionally stable. A numerical example is given to assert the proposed algorithm is highly effective. The obtained results confirm that the proposed technique is effective and accurate.

 Impulsive mixed fractional differential equations with delay Abstract : We use Krasnoselskiis fixed point theorem to investigate the existence of solutions for impulsive differential equations involving left and right Caputo fractional derivatives with multi-base points and in presence of a delay. The results obtained are new and compliment the existing ones.

 Incomplete Fractional Calculus Operators Abstract : Here we shall define a ceratin integral operators involving the incomplete hypergeometric function due to Srivastava Chaudhry and Agarwal. The considered generalized fractional integration and differentiation operators contain the incomplete hypergeometric function as a kernel. Some illustrative examples are given revealing the effectiveness and conveniences of the method.

 Higher Order Nonlinear Multi-Point Fractional Boundary Value Problems Abstract : In this study, we investigate the conditions for the existence of at least one and three positive solutions to nonlinear higher order multi-point fractional boundary value problems by using Krasnoselskii fixed point theorem and the five functionals fixed point theorem, respectively.