02- Progress in Fractional Differentiation and Applications An International Journal

Forthcoming

 Error Analysis of the Generalized Jacobi Galerkin Method in Nonlinear Fractional Differential Equations Abstract : In this paper, the Generalized Jacobi Galerkin (GJG) method is analyzed for the numerical solution of nonlinear multi- order fractional differential equations(FDEs). We consider the generalized Jacobi functions(GJFs) with indexes corresponding to the number of homogeneous initial conditions as natural basis functions for the GJG approximation. The unique solvability of the resulting nonlinear algebraic system, as well as convergence properties of the proposed approach, are discussed. The validity of the method is demonstrated with some illustrative examples.

 Fractional Order Mathematical Modelling for Studying the Impact on the Emergence of Pollution and Biodiversity Pertaining to Incomplete Aleph functions Abstract : Civilizations are always known for utilization, destruction and ignoring the environment. Different opinions have been put at the root of our environmental-disruptive tendencies, religion, social and economic structures and the adoption of technology. It is conventional to blame environmental conditions but we do not utilize practically our knowledge and understanding of nature. It seems that we continuously blame pollution and its destruction instead of our efforts. With the use of a fractional order mathematical model involving incomplete Aleph (א) functions (IAFs) and the Caputo fractional derivative, we explore the impact of the emergence of pollution and study the effect of industrial operations on the environment of the region. In this work, we evaluate the Caputo fractional derivative of incomplete Aleph (א) functions. Here, we define the concentration C(x,t) in terms of incomplete Aleph (א) functions and obtain some particular cases by giving specific values to the parameters of our primary results and expressing them in terms of special functions, notably H-functions, incomplete H-functions, and incomplete I-functions, and demonstrating their relationship with existing results.

 Hamiltonian Analysis Formulation of Lee-Wick Field using Riemann-Liouville Fractional Derivatives Abstract : In this paper, we generalized the Hamilton formulation for continuous systems with third order derivatives and applied it to Lee-Wick generalized electrodynamics. A combined Riemann–Liouville functional fractional derivative operator was built, and a fractional variational principle was established under this formulation. The fractional Euler-Lagrange equations and fractional Hamiltons equations were created using functional fractional derivatives. We found that the Euler-Lagrange equation and the Hamiltonian equation resulted in the same outcome. We looked at one example in an effort to explain the formalism.

 Fractional spectral approaches for some fractional partial differential equations Abstract : The multi-order linear and nonlinear partial differential equations of fractional order have been solved numerically. A computational strategy based on fractional spectral operational matrices (OM) and generalized fractional Lageurre (GFL), generalized fractional shifted Legendre (GFSL), generalized fractional modified Bernstein (GFMB) are provided. Our fractional spectral methods have been used to solve nonlinear fractional partial differential equations and fractional Korteweg-de Vries-Burgers (KdVB) equation. Making use of a variety of test and application examples, the effectiveness of the numerical solution have been satisfied.

 Poisson Bracket Formulation for a Dissipative Two-Dimensional Anisotropic Harmonic Oscillator with Fractional Derivatives: Analysis and Applications Abstract : We recast the Harmonic Oscillator using fractional differential equations. to be more developed By applying the Hamiltonian formulation with fractional derivatives to the resulting Harmonic Oscillator. the canonical conjugate-momentum coordinates are defined and converted into operators that fulfill the commutation relations, which correspond to the classical theorys Poisson-bracket relations. The equations of motion are redefined in terms of the generalized brackets when these are generalized. We present a generalized dissipative two-dimensional anisotropic harmonic oscillator equation of motion with fractional derivatives. The novel method was evaluated on a single example and found to be consistent agreement with the classical fractional method.

 Analysis of Fractional Differential Equations with Antagana-Baleanu Fractional Operator Abstract : To solve fractional-order differential equations (FODEs) with Antagana-Baleanu fractional operator (ABFO), an efficient strategy based on the variational iteration method (VIM) and natural transform(NT) is given. The natural variational iteration technique is the name of this method (NVIM). This work also investigates the convergence of the solution of general FODEs obtained by the suggested method in view of the theory of fixed point and Banach spaces. Furthermore, the error analysis of the NVIM solution is also discussed. Two problems are solved to validate and efficacy demonstrate in the present. It is also demonstrated that the results obtained from the suggested technique are in excellent agreement with the exact solution. The NVIM’s numerical results reveal that the technique is simple to implement and computationally appealing.

 Generalized Hamilton-Jacobi Formulation of Damped Harmonic System Using Fractional Derivatives Abstract : In this paper, we establish a fractional Hamilton-Jacobi formulation in terms of fractional derivatives for a damped harmonic system. A fractional Riemann–Liouville fractional derivative operator is defined, and a fractional Hamilton–Jacobi function is established using this formulation. The fractional Hamilton-Jacobi function for fractional damped harmonic systems is obtained using the Hamilton-Jacobi theory. We used this new formulation to solve the Hamilton-Jacobi partial differential equation for a fractional damped harmonic system after introducing the Hamilton-Jacobi formulation using fractional derivatives. The motion equations are then defined in terms of Poisson brackets, and the Heisenberg equations are expressed in terms of commutators, the quantum counterpart of the Poisson bracket. Finally, we look at an example to help explain the findings. According to the findings of this study, fractional calculus, owing to the order of the fractional derivative and the fractional operator, allows for more flexible models than classical calculus. This property is critical for developing a novel formulation of the Hamilton-Jacobi formulation of the Damped Harmonic System equation generalized using the Riemann–Liouville derivative.

 A Computational Numerical study of Burger Equation with Fractal Fractional Derivative Abstract : In the several real world problem we can see the concept of fractal medium. In this paper we show that the concept of the fractal derivative is not only represent the fractal sharps but also describe the movement of the fluid within these media. In this paper we develop the solution of viscous Burger equation with different kernel of fractal-fractional derivative. We solve the numerical technique using Newton’s Polynomial method for solving fractional Burgers equation. To show the applicability of the present model in fractal media we present simulations for different values of fractal dimensions.

 Exploring the solvability for coupled system of nonlinear fractional Langevin equations Abstract : In this article, we investigate the existence and uniqueness of the solution to a nonlinear fractional coupled system of Langevin equations with boundary conditions. The required results are established based on fixed point theorem and fractional calculus. Finally, an example is given to illustrate the obtained results.

 Mathematical Analysis of Streptococcus Suis Infection in Pig-Human Population by Riemann-Liouville’s Fractional Operator Abstract : In the current article, we have discussed a mathematical model for a rare disease known as Streptococcus Suis which is a contaminated and infectious disease. The disease Streptococcus Suis is spread by the infection found in pigs that is transmitted in humans, later. In human transmissions, this disease takes very severe form that can result death as well with serious illness. In this paper, we have taken seven population groups (some are from human populations and some are from pig’s populations as well) under consideration. We have applied the Riemann Liouville’s fractional derivative with Laplace transformation to analysis and study of the presented mathematical model of the disease. We have also validated the presence and oneness of solution with graphical presentation of the solutions.

 An Optimal Control Problem for Dengue Fever Model using Caputo Fractional Derivatives Abstract : In this paper, we study the optimal control of the dengue disease model with the vertical transmission in terms of the Caputo fractional derivatives. We apply the control parameters like larvicide, fogging, vaccination, and isolation to stop the spread of the dengue epidemic and explore the influence of the fractional order α (0.6 ≤ α ≤ 1) on the dengue transmission model. We apply a forward-backward sweep scheme using the Adams-type predictor-corrector approach for solving the proposed control problem. Finally, the effects of optimal controls considering three different cases in the given model are discussed.

 Fractional Stochastic Modelling of an Investment Model Abstract : Financial models have become very crucial in many economics because of the uncertainties within the environment. Investors are always looking for better returns on their investments and therefore, a good financial model is important to any society. In this study, a financial mathematical model is examined in the context of fractional stochastic. The existence and uniqueness of solution of the financial model is studied. A new numerical approach based on Newton polynomial interpolation is utilised to numerically investigate the financial model’s dynamics. It is observed that fractional order derivative affects the dynamics of the financial model. The Atangana -Baleanu operator shows better prediction comparing to the Caputo and Caputo-Fabrizio operators. It is suggested that other complex dynamics can be investigated using the newly developed numerical scheme based on the Atangana-Beleanu operator.

 A Coinfection Model of Malaria and COVID-19 in the context of Conformable-order derivative Abstract : Confection infection diseases are increasing everyday on the globe which could be as a result of climate issues. Malaria and Coronavirus (COVID-19) is now common infection in the Sub-Saharan Africa and this study examines both theoretical and numerical dynamics of coinfection of malaria and COVID-19. The existence and uniqueness of solution is studied using the Fixed-point theorem and Picard iterative method. Conformable-order derivative as a mathematical tool is used to investigate the dyanamics of the coinfected malaria and COVID-19 model. It is concluded that Conformable-order derivative has a great impart on the spread of the disease.

 Intuitionistic fuzzy generalized conformable fractional derivative Abstract : The innovative idea of Atanassovs intuitionistic fuzzy sets (IFSs) is to get a more comprehensive and detailed description of the ambiguity and uncertainty by introducing a membership function and a nonmembership function. Each element in an IFS is expressed by an ordered pair, which is called an intuitionistic fuzzy number (IFN). In this paper, We introduced a new definition of the generalized conformable fractional derivative of the intuitionistic fuzzy number-valued functions, Using this definition, we prove some results and with the help of $\alpha$-cut set, the Hukuhara difference between intuitionistic fuzzy numbers are defined and proved. An intuitionistic fuzzy conformable nuclear decay equation with the initial condition given to show the new theorems and is solved under a new generalized conformable fractional derivative concept.

 Approximate Solutions of Fuzzy Fractional Differential Equations via Homotopy Analysis Method Abstract : The Homotopy Analysis Method (HAM) is an approximate-analytical method for solving linear and nonlinear problems. HAM provides the auxiliary or convergence parameter, which considered as a powerful tool to examine and analyze the precision of the approximate series solution and ensure its convergence. In this article, fuzzy set theory properties is introduced to extend and reformulate HAM for the determination of approximate series solutions for fuzzy fractional differential equations involving initial value problems. The extension and reformulation of the method corresponding to Caputo’s derivative in the fuzzy domain and the fuzzification of the method followed by the convergence analysis are presented in detail. Consequently, a new HAM for the general fractional differential equation has been developed in fuzzy domain. The difference between other types of approximate-analytical approaches and HAM is that the proposed HAM offers a better way to track the convergence region of the series solution via the convergence control parameter. The capability and accuracy of the method are illustrated by solving two examples involving linear and nonlinear fuzzy fractional differential equations. The obtained results using HAM suggested HAM is effective and simple to use when solving first order initial value problem involving a fuzzy fractional differential equation.

 Inverse Problems for a Time-Fractional Diffusion Equation with Unknown Right-Hand Side Abstract : This paper is devoted to direct and inverse source problems for a two-dimensional in space variables time-fractional diffusion equation. The direct problem is an initial-boundary value problem for this equation in a rectangular area. In inverse problem the unknown right side of the equation is assumed to have the form of a product of two functions: one of which depends only on the time variable, while the other one - depends on the spatial variables. Two inverse problems of finding these functions separately under the condition that the other function is known are investigated. For the inverse problem for determining a time-dependence function, Abels integral equation of the first kind is obtained, which is further reduced to an integral equation of the second kind with the application of fractional differentiation to it. To solve the direct problem and inverse problem of determining a spatial-dependence function, the Fourier spectral method is used. Theorems of unique solvability of the formulated problems are proved. The existence and uniqueness results are based on the Fourier method, fractional calculus and properties of Mittag-Lefller function.

 Construction by the Tikhonov method of a nonzero solution of the homogeneous Cauchy problem for one equation with a fractional derivative Abstract : The article constructs a nonzero solution to the homogeneous Cauchy problem for a homogeneous equation of even order with a fractional Caputo derivative.

 The Time evolution of the Random Genetic Drift with Random Number of Individuals and the Computational Proof of the Galtons Theory Abstract : This paper focuses in studying the genetic diffusion process of a finite population in the absence of migration and immigration. Only the natural weak selection and mutation have the main impact on the diffusion process. Kimura has mathematically formulated this model in a partial differential equation as a special case of the forward Kolmogorov-Chapmann equation and called it the random drift genetic equation of a finite population. The hereditary of talents and intelligence running through families being carried on the genes through the same families are statistically studied by Galton. Galtons formulated his studies in the so called Galtons theory and collected these statistically studies and theory in his famous book, \emph{Hereditary Genius}, being published in (1869). In this paper, we try to mathematically and computationally prove the Galtons theory by numerically solved the modified Kimuras model. The proof is done in two steps. The first step is to assume that the number of diploid individuals are randomly changed from generation to generation. The second step is to add the memory effect on the diffusion process by extending the classical random genetic drift equation with selection and mutation to time-fractional random genetic drift equation. The fourth order compact finite difference is implemented to derive the approximate solutions of the studied partial differential equation. The Gr\"{u}nwald-Letnikov scheme is implemented to find the discrete scheme of the time-fractional operator. The simulations of the approximate solutions, the stationary approximate solution, the stability of the difference scheme and the total sum of the approximate solutions are computationally calculated and interpreted. The discrete convergence of the approximate solutions is numerically studied and its asymptotic behaviors are compared with the Mittag-Leffler function.

 Fractional Model of Susceptibility Characteristics of ESBL-producing K. pneumoniae Infections for Carbapenems and Piperacillin-Tazobactam Abstract : In this paper, we provide a fractional dynamical model of extended spectrum beta lactamase (ESBL)-producing Klebsiella pneumoniae to visualise future perspective of susceptibility characteristics of K. pneumoniae infections for antibiotics; carbapenems and piperacillin-tazobactam (PTZ). Our approach is based on a system of fractional differential equations, where the fractional derivative was obtained in Caputo type. We demonstrate that the stability of the disease free equilibrium is locally stable and for the endemic equilibrium point the stability analysis is shown by Hurwitz criteria. Additionally, it is shown that the system has a unique solution due to the Lipschitz condition being provided. According to the SIS-type model, while non-ESBL K. pneumoniae infections are seemed to remain susceptible to both antibiotic groups, the opposite is true for ESBL-producing K. pneumoniae infections. For ESBL-producing K. pneumoniae infections, resistance to both carbapenems and PTZ will occur approximately 7.5 years later in line with the mathematical interpretation of the data. In addition, resistance to PTZ in ESBL-producing infections is observed to be less than carbapenems. As a result, it is seen that the formation of resistance against agents called last line antibiotics, which are preferred in the treatment of ESBL-producing infections, continues in the future, and measures to prevent this issue should be improved.

 Study of a continuous Fractional-Order Dynamical System of Fibrosis of Liver Abstract : In this script, we aim to formulate a fractional-order linear model (FOLM) of the human liver to delve into the volume of storage and transfer of bromsulphthalein at the limit of chronic alcoholic liver disease. A non-orthogonal Bernstein polynomial is used to generate operational matrices for the fractional-order coupled system of the human liver. The existence and stability of a solution were examined by some already established fractional calculus results. Based on exact clinical data, numerical examples are quoted with their graphical interpretation of the fractional order of the proposed model. A comparison of results with the integer-order is given at the end to support the validity and effectiveness of the newly constructed algorithm. The transformed matrix equation, which is the result of the fractional operational matrices, is unfolded using the mathematical software Matlab.

 Two new quadratic scheme for fractional differential equation with world population growth model Abstract : In this paper, we proposed two quadratic convergence numerical technique which is very accurate and ecient for nding the approximate solution of an initial value problem (IVP) of the linear as well as non-linear FDEs of arbitrary order ; where 0 <  1. Here our fractional derivative are taken in the Riemann sense. In this suggested work, we provides the numerical solution which is comparatively faster and ecient than the Euler method(EM), Improved Euler method (IEM) and many other linear, quadratic convergence method. Also, here we have been found the numerical solution without the help of any kind of linearisation, perturbations or any other assumptions. Illustrated example shows the numerical comparisons in sense of efficiency and accuracy between the our proposed scheme and the analytical solution, Euler method and the Improved Euler method of the IVP of FDEs. Our both the proposed scheme Raltsons method and Raltsons method of minimum local error bound has quadratic convergence which is more accurate and faster than the Euler method and Improved Euler method, while Improved Euler has also quadratic convergence rate.

 Solving Fractional Partial Difference Equations Using the Discrete Homotopy Analysis Method Abstract : In this paper, we propose the discrete homotopy analysis method(DHAM) to solve time-fractional difference equations. The fractional differences are described by Caputos sense. Several illustrative examples present the capability of DHAM for wide classes of fractional partial difference equations.

 Solvability and controllability of a retarded-type nonlocal non-autonomous fractional differential equation Abstract : This paper considers a non-autonomous retarded-type fractional differential equation involving Caputo derivative along with a nonlocal condition in a general Banach space. We present a novel approach to determine the existence-uniqueness and controllability of mild solution to the considered problem using the fixed-point technique, classical semigroup theory, and tools of fractional calculus. It is imperative to mention that the main results are established without assuming the continuity of linear operator $-A(t)$ and compactness condition on semigroup. At the end, the developed theoretical results have been applied to a nonlocal fractional order retarded elliptic evolution equation.

 An approximate study of Fisher’s equation by using a semi-analytical iterative method Abstract : In this paper, a new fractional analytical iterative technique was used to get analytical solutions of the nonlinear Fisher’s equation with time-fractional order. The novelty of the study comes from the application of the Caputo fractional operator to classical equations, which results in very accurate solutions via well-known series solutions. It also doesn’t require any presumptions for nonlinear terms. The numerical results for numerous instances of the equations are displayed in tables and graphs. The method can drastically reduce the number of analytical steps while also being efficient and convenient for solving nonlinear fractional equations.

 Oscillatory Behaviour of Solutions of Delay Differential Equations of Fractional Order Abstract : In this paper, we deal with fractional delay differential equations(FDDEs). New results have been obtained on the oscillatory behaviour of solutions of FDDEs with constant as well as variable coefficients. The oscillation criteria for FDDEs with positive and negative coefficients have also been discussed. All the outcomes have been illustrated by giving suitable examples with graphical representation. The graphs are done by using MATLAB.

 Efficient B-Spline series method for solving fractional Fokker Planck equation Abstract : A reliable method for solving the fractional Fokker Planck equation based on the B-Spline method is investigated. Theoretical and numerical results are given. The purpose of this study is to derive the B-Spline series method and to show its efficiency for solving such problem. Two examples are presented to discuss the efficiency of the proposed method. We notice that the approximate solution is very close to the exact solution and the maximum error in our approximation is about $10^{-15}$ which means the proposed method is very accurate. This study shows that the proposed method is promising method and it can be implemented to other physical problems.

 Dynamical Behavior of Fractional-Order Rumor Model Based on the Activity of Spreaders Abstract : According to the similarity between the infectious disease transmission and the rumor spreading, we introduce this manuscript. In this work, the dynamical behavior of the fractional-order rumor model (FOM) is investigated in details. Also, we determine all the equilibrium fixed points of model. Nevertheless, the stability at this equilibrium points is studied. The basic reproduction number of FOM is obtained. Some valuable and essential definitions about the Caputo fractional derivative are introduced. Various methods use to solve this model such as Generalized Mittag -Leffler Function method (GMLFM) is an approximate solution and Predictor-Corrector method (PCM) as a numerical solution. Numerical simulations are performed to confirm our analytical results and elucidates the effect of various parameters on the rumor spreading.

 Haar collocation method for nonlinear variable order fractional integro-differential equations Abstract : In this study, a computational numerical scheme is developed for solution of nonlinear variable order (VO) fractional integro-differential equations(FIDEs) using Haar collocation method (HCM). The fractional VO derivative is defined in the Caputo sense. The HCM converts the given nonlinear VO FIDEs into a system of nonlinear equations. By the Broydens technique this scheme is solved, the initial guess is taken zero and when the convergence condition is satisfied, the iteration is terminated. The convergence of the proposed HCM method is checked on selected problems and moreover Mean square root and maximum absolute errors are calculated for different number of collocation points(CPs). Finding, the comparison of approximate and exact solution is also given, and rate of convergence is calculated to be approximately equal to $2$.

 Asymptotic behavior for fractional systems with lower-order fractional derivatives Abstract : The asymptotic behavior, the decay and the boundedness of solutions are discussed for the system of fractional differential equations including two types of fractional derivatives the Caputo fractional derivative (CFD) and the Riemann-Liouville fractional derivative (RLFD). Reasonable sufficient conditions are determined ensuring that solutions for the system with nonlinear right hand sides approach a power type function, power type decay and boundedness as time goes to infinity. Our approach is based on appropriate desingularization techniques and generalized the inequality of Gronwall-Bellman. Convenient assessments and lemmas such as a fractional version of lHopitals rule are used.

 IMPORTANCE OF RG TRANSFORM AND SOME OF ITS APPLICATIONS Abstract : In this survey paper, some properties and applications of RG transform are discussed. First, some new properties are presented. Finally, we briefly explained some famous applications.

 The Degasperis-Procesi Lagrangian Density: Functional Fractional Formulation Abstract : The Degasperis-Procesi Lagrangian is reformulated in fractional form in a four-dimensional space using Riemann-Liouville fractional derivatives. The fractional Euler-Lagrange equations and Hamiltons equations are constructed in terms of functional fractional derivatives. In this study, a comparison of our results, obtained through Hamilton’s equations in terms of functional fractional derivatives, is carried out using the Euler-Lagrange equations for these systems. The Riemann–Liouville fractional derivative operator is defined, following which a fractional variational principle based on this definition is established. Notably, these functional fractional derivatives are used to derive both the fractional Euler equations and the fractional Hamilton equations. We also observed that both fractional Euler-Lagrange equations and fractional Hamiltonian equations yield the same result. Finally, we studied one example in order to illustrate the results.

 FRACTIONAL DERIVATIVE AND FINANCIAL INSTRUMENTS : WAITING TIME DISTRIBUTIONS FOR THE EXCHANGE RATE MOVEMENT OF US DOLLAR TO JAPANESE YEN Abstract : We derive a mathematical model for exchange rate movements employing random walks theory and Caputo fractional derivative operator. We also study waiting time distributions for the exchange rate movement of US Dollar to Japanese Yen, which are intimately related to the mathematical model, during February 2019. Here, we compare three types of waiting time distribution. i.e., exponential, stretched exponential, and Mittag-Leffler distributions. We obtain that Mittag-Leffler Distribution is the best distribution to approximate the empirical distribution of the exchange rate data during February 2019 except the data of February 18, 2019 approximated better by stretched exponential distribution.

 On the fractional differential equations associated with integral operator involving Aleph function in the kernel Abstract : In the present article, we introduced and explore an integral operator which consist Aleph function in the kernel with fractional calculus. In second section, we construct the characteristics of R-L fractional integral operator I_(a+)^β and derivative operator D_(a+)^β containing the Aleph-function and in third section, we develop the sumudu transform of propose integral operator. In fourth section, we find the solutions of arbitrary order differential equations which consists the Hilfer derivative operator along with propose integral operator by applying sumudu transform. We also established some fascinating corollaries and particular cases of our key results presented here in terms of a number of special functions particularly H-function, I-function, Mittag-Leffler, and generalized Bessel-Maitland function and exhibit to be their relation with certain known results. In the end of the article, we develop some graphical results to show the behavior of differential equation by assigning particular values to the parameters.

 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.

 On the existence results of a coupled system of generalized Katugampola fractional differential equations Abstract : This paper is mainly devoted to investigate the existence of solutions to a coupled system of fractional differential equations involving generalized Katugampola derivative with non local initial conditions. The existence results are carried out by using some standard fixed point theorem techniques. A suitable example is also provided to illustrate the applications on our main results.

 About Convergence and Order of Convergence of some Fractional Derivatives Abstract : In this paper we establish some convergence results for Riemann-Liouville, Caputo and Caputo-Fabrizio fractional operators when the order of differentiation approaches one. We consider some errors given by ||D^{1-\alpha}f-f||_p for p=1 and p=\infty and we prove that for both Caputo and Caputo-Fabrizio operators the order of convergence is a positive real r\in(0,1). Finally, we compare the speed of convergence between Caputo and Caputo-Fabrizio operators obtaining that they are related by the Digamma function.

 On the Explicit Solution of ψ-Hilfer Integro-Differential Nonlocal Cauchy Problem Abstract : In this paper, we derive the representation formula of the solution for ψ-Hilfer fractional differential equation with constant coefficient in the form of Mittag-Leffler function by using Picards successive approximation. Moreover, by using some properties of Mittag-Leffler function and fixed point theorems such as Banach and Schaefer, we introduce new results of some qualitative properties of solution such as existence and uniqueness. The generalized Gronwall inequality lemma is used in analyze E_{α}-Ulam-Hyers stability. Finally, one example to illustrate the obtained results.

 A general fractional-order viral infection model with cell-to-cell transmission and adaptive immunity Abstract : We propose a general fractional-order viral infection model with adaptive immune response including both productively infected cells and latently infected cells. This model incorporates two ways of infection, one by virus-to-cell and other by cell-to-cell transmissions, which are modeled by general nonlinear incidence functions. We first show that the proposed model is mathematically and biologically well-posed. Stability analysis of different steady states is explicitly performed and five threshold parameters are identifi ed which determine clearance or maintenance of infection. In addition, we examine the robustness of the model to certain parameters by examining the reproduction numbers. Finally, we present numerical simulations which confirm that our model predicts well the evolution of viral infection.

 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.

 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.