02- Progress in Fractional Differentiation and Applications An International Journal

Forthcoming

 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.

 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.

 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.

 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.

 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.

 A Fractional Variable Order Model of COVID-19 Pandemic Abstract : 2020 has witnessed a rapidly spread pandemic COVID-19 which is one of the worst in the history of mankind. Scientists believe that COVID-19 spreads mainly from a person to another. Recent researches consider bats as a vector for COVID-19. This paper suggests a variable fractional order model for COVID-19 to figure out how bats and hosts interact, and how the seafood market affect people. The proposed model assumes that infection cannot be recovered. The basic reproduction number R0 for real data on reported cases in Wuhan China was computed. Disease-free equilibrium points and proposed model stability are studied.

 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.

 Dynamics of fractional order bio-regulatory system Abstract : In this paper a fractional-order bio-regulatory system is proposed. Stability and Hopf bifurcation of the systems have been investigated. Several numerical examples are demonstrated to validate the theoretical 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.

 k-Generalized Space-Time Fractional Ultra-Hyperbolic Diffusion-Wave Equation with the Prabhakar Integral Operator. Abstract : The objective of the present work is to study a generalization of the ultra-hyperbolic diffusion-wave equation introduced in \cite{Dorrego} using the $k$-Prabhakar derivative introduced in \cite{Dorrego 3} in the time variable and a fractional power of ultra-hyperbolic operator on the space variable.

 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.

 Existence solutions for a nonlinear Langevin fractional q-difference system in Banach Space Abstract : In this paper, we study the existence of solutions for a new class of a nonlinear Langevin fractional q-difference system in Banach space. Our analysis relies on the Monchs fixed-point theorem combined with the technique of kuratowski measures of noncompactness. We conclude with an illustrative example.

 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.