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The aim of this paper is to establish the stability results based on the approach of Linear Matrix Inequality (LMI) for the addressed mathematical model using Caputo–Fabrizio operator (CF operator). Firstly, we extend some existing results of Caputo fractional derivative in the literature to a new fractional order operator without using singular kernel which was introduced by Caputo and Fabrizio. Secondly, we have created a mathematical model to increase Cytoplasmic Incompatibility (CI) in Aedes Aegypti mosquitoes by releasing Wolbachia infected mosquitoes. By this, we can suppress the population density of A.Aegypti mosquitoes and can control most common mosquito-borne diseases such as Dengue, Zika fever, Chikungunya, Yellow fever and so on. Our main aim in this paper is to examine the behaviours of Caputo–Fabrizio operator over the logistic growth equation of a population system then, prove the existence and uniqueness of the solution for the considered mathematical model using CF operator. Also, we check the α -exponential stability results for the system via linear matrix inequality technique. Finally a numerical example is provided to check the behaviour of the CF operator on the population system by incorporating the real world data available in the known literature.

Recently, a symmetric differential operator (SDO) is attracted to studying in the field of mathematical analysis. A new formal of SDO is presented to generalize some well known differential operators in a complex domain. According to this formulation, we shall exam the boundedness and compactness of this operator in complex spaces, such as Hilbert space and Sobolev space. For this purpose, we suggest new norms to solve the fractional Beltrami equation in the open unit disk. This operator has ability to depict the analytic geometric representation of the solution of second order differential equation utilizing the concept of Schwarzian derivative in the open unit disk. Applications in imagings are given the sequel.

We consider an Klein–Gordon relativistic equation with a boundary dissipation of fractional derivative type. We study of stability of the system using semigroups theory and classical theorems over asymptotic behavior.

Linear first order Riemann-Liouville fractional differential equations are studied. These new equations unify and generalize the Riemann-Liouville, modified Caputo and Caputo fractional differential equations. The equivalences between the fractional differential equations and the corresponding perturbed Abel's integral equations are obtained. These results are useful not only to study the initial or boundary value problems for nonlinear first order Riemann-Liouville fractional differential equations but also to study the solutions of the perturbed Abel's integral equations arising in a problem of mechanics and many other physical problems. The well-known Tonelli's result on solvability of the Abel's integral equation is generalized. We exhibit that there are nonconstant equilibria for some first order Caputo fractional equations. This is different from nonlinear first order ordinary differential equations which have only constant equilibria. The equivalence results are applied to generalize the classical Mean Value Theorem to the first order Riemann-Liouville fractional derivatives.

In this paper, the space-time fractional advection-diffusion equation (STFADE) is considered in the finite domain that the time and space derivatives are the Caputo fractional derivative. At first, a quadratic interpolation with convergence order O ( τ 3 - α ) is applied to obtain the semi-discrete in time variable. Then, the Chebyshev collocation method of the fourth kind has been used to approximate the spatial fractional derivative. In addition, the energy method has been employed to show the unconditional stability and gained convergence order of the time-discrete scheme. Finally, the accuracy of the numerical method is analyzed and showed that our method is much more accurate than existing techniques.

In this article, we introduce the discrete fractional sum equations and inequalities. Some new generalized Gruss type discrete fractional sum inequalities are developed. We employ a nabla or backward difference; we employ the Riemann-Liouville definition of the fractional difference. Furthermore, the current research is a discrete variant of fundamental inequalities set up in the recent literary works and extends a few discrete version for ∇ -fractional sum specifically on time scale ℏ N , where 0 ℏ ⩽ 1 .

The analysis of Oldroyd fluids; a class of Maxwell fluids which also bears Newtonian properties under some conditions, has important implications for different scientific and engineering/industrial applications. Examples of these fluids are typically dilute polymer solutions which demonstrate both viscous and elastic behaviors when subjected to strain. For unidirectional steady flows, the Oldroyd-B (3-Constant) model is the simplest when it comes to describing these behaviors. This model is not sufficient in certain cases, for instance, convergent flow channels, pulling effects, and other extensional flows. In all such scenarios, its application may lead to having out of bound tensile stresses. Hence, in all practicality, higher constants of the Oldroyd model are required to include different tensorial invariants. In this article, we perform this analysis in fractional space on Oldroyd fluids in thin-film flow context using the Oldroyd 6-Constant model for both lifting and drainage scenarios. Solutions to the highly non-linear fractional differential equations are obtained by means of the Homotopy Perturbation Method (HPM) along with fractional calculus. Validation and convergence of the solutions are confirmed by finding residual errors. To the best of our knowledge, the given problem has not been attempted before in fractional space. Various physical aspects, such as the velocity profile, volumetric flux and average velocities are determined and analyzed both graphically and in tabulation form as part of this analysis.

In this research, Genocchi wavelets method, a quite new type of wavelet-like basis, is adopted to obtain a numerical solution for the classical and time-fractional Rosenau-Hyman or K(n, n) equation arising in the formation of patterns in liquid drops. The considered partial differential equation can be transformed into a system of non-linear algebraic equations by utilizing the wavelets method including an integral operational matrix and then discretizing the equation at the collocation points. The system can be simply solved by several traditional methods. Finally, the algorithm is implemented for some numerical examples and the numerical solutions are compared with the exact solutions using MAPLE. The obtained results are demonstrated using figures and tables. When the results are compared, it is evinced that the algorithm is quite effective and advantageous due to its easily computable algorithm, high accuracy, and less process time. (C) 2021 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University.

In this paper, a general class of nonlinear initial value problems involving a Riemann-Liouville fractional derivative and a variable-order fractional derivative is investigated. An existence result of the exact solution is established by using Weissinger's fixed point theorem and Gronwall-Bellman lemma. An hp-version spectral collocation method is presented to solve the problem in numerical frames. The collocation method employs the Legendre-Gauss interpolations to conquer the influence of the nonlinear term and variable-order fractional derivative. The most remarkable feature of the method is its capability to achieve higher accuracy by refining the mesh and/or increasing the degree of the polynomial. The error estimates under the H 1 -norm for smooth solutions on arbitrary meshes and singular solutions on quasi-uniform meshes are derived. Numerical results are given to support the theoretical conclusions.

The main objective of this paper is to develop a new technique for the supercapacitor’s parameter identification that can handle the issue of initial voltage. An effective approach is proposed using the fractional-order derivative for accurate identification of a well-known series Resistance-Capacitance (R-C) model. An expression is derived using the Caputo definition and the Haar wavelet operational matrix, which is sufficient for both charging and discharging phase data of supercapacitors. To extract impedance parameters, voltage stimulated step response is utilized and parameters are calculated despite random initial voltage stored in a supercapacitor. This operational matrix approach transforms complex fractional derivative terms into simple algebraic expressions and reduces the overall complexity. The proposed technique shows a very good agreement with experimental data that exhibit different initial voltages and different time-frames. Investigations and experiments with various supercapacitors clearly reveal the importance of the developed equation for a fractional R-C model.

In this study, for the first time, the approximate solution of Black–Scholes option pricing distributed order time-fractional partial differential equation by means of Legendre and Chebyshev wavelets is considered. The operational matrices of Legendre and Chebyshev wavelets for integer order derivative and distributed order fractional derivative are derived. Furthermore, the combination of Gauss–Legendre quadrature formula and standard Tau method along with the obtained operational matrices reduces the distributed order time-fractional Black–Scholes model (DOTFBSM) into the system of linear algebraic equations. Convergence analysis, error bounds and numerical stability of the proposed approach are discussed in detail. The presented scheme is applied on three test examples and numerical experiments confirm the theoretical results and illustrate robustness of the presented method. The results produced by current approach are found to be more accurate than some available results.

This paper proposes an advanced numerical-analytical approach for handling a class of fuzzy fractional differential equations involving Caputo-Fabrizio derivative with a non-singular kernel arsing in the medical sector. The solution methodology relies on the reproducing-kernel algorithm to generate analytical solutions in the form of a uniformly convergent series in the direct sum of the desired Hilbert spaces. The effectiveness of the method is analyzed by studying some theoretical, analytical, and stability results of the derived solutions based on the reproducing kernel theory. Numerical simulations are also provided in tables and graphs to demonstrate the reliability of this algorithm in solving fuzzy models using the new Caputo-Fabrizio fractional operator, especially for the drug pharmacokinetic model. The obtained results show the ability of the applied algorithm to solve a wide range of nonlinear fractional models emerging in pharmacology, medicine, and biochemistry.

In this article, the new approach called, the optimal auxiliary function method (OAFM) has been extended with the help of the fractional complex transform method (FCTM) to fractional order partial differential equations (PDEs). The beauty of the method is that, there is no need for small or large parameter assumption in the problem. Similarly, the proposed method gives an approximate solution after only one itearion. To test the proposed method, we take the time-fractional order Zakharov-Kuznetsov equations as a test example. The computed results of the suggested method are compared with the new iterative method (NIM), perturbation iteration method (PIA), and Residual power series method (RPSM). The accuracy and effectiveness of the proposed method can be proven by numerical and graphical results.

In this paper, a coupled system of fractional differential equations along with integral boundary conditions is discussed by means of the iterative reproducing kernel algorithm. Towards this end, a recently advanced analytical approach is proposed to obtain approximate solutions of nonclassical-types boundary value problems of fractional derivatives in Caputo sense. This approach optimizes approximate solutions based on the Gram-Schmidt process on Sobolev spaces that execute to generate Fourier expansion within a fast convergence rate, whereby the constructed kernel function fulfills homogeneous integral boundary conditions. Moreover, the solution is presented in the form of a fractional series over the entire Hilbert spaces without unwarranted assumptions on the considered models. The validity of the present algorithm is illustrated by expounding and testing two numerical examples. The achieved results indicate that the proposed algorithm is systematic, feasibility, stability, and convenient for dealing with other fractional systems emerging in the physical, technology and engineering.

In this work, we aim to propose a mathematical formalism for the human liver involving Caputo fractional derivative (CFD) with modified parameters. Regarding the proposed fractional order model (FOM), the positivity and boundedness theory of an explicit solution are established by using the fractional mean-value theorem in the sense of Caputo. In addition, the validation of the FOM is provided to ensure that it complies with the medical interpretation and we also discuss the stability of unique equilibrium point (EP) for the presented model. We apply the Generalized Mittag–Leffler function method (GMLFM) and Predictor–Corrector method (PCM) to solve this model. The GMLFM is used to give the proper results as a series of the time variable to obtain the desired solution at any time and the PCM is implemented to obtain the solution of the FOM over a long time period. The tabled results from this study are compared with the real clinical data and with other methods. The simulations approximate the real clinical data and support the obtained theoretical results. Precisely, the outcomes of the FOM by using the proposed methods coincide with most of the clinical data that means it more accurate. Ultimately, the graphical results indicate the superiority of the proposed FOM over the classical model.

In this paper, we construct and investigate an accurate numerical scheme for solving a class of variable-order (VO) fractional diffusion equation based on the Caputo–Fabrizio fractional derivative. The scheme is presented by using a finite difference method in temporal variable and a local discontinuous Galerkin method (LDG) in space. For all variable-order α ( t ) ∈ ( 0 , 1 ) , we derive the stability and L 2 convergence of proposed scheme and prove that the method is of accuracy-order O ( τ + h k + 1 ) , where τ , h and k are temporal step sizes, spatial step sizes and the degree of piecewise P k polynomials, respectively. Several numerical tests are given to validate the theoretical analysis and efficiency of the proposed algorithm.

A fully discrete ADI scheme is proposed for solving the two-dimensional time-fractional diffusion equation with weakly singular solutions, where L1 scheme on graded mesh is adopted to tackle the initial singularity. An improved discrete fractional Gronwall inequality is employed to give an α-robust H 1 -norm convergence analysis of the fully discrete ADI scheme, where the error bound does not blow up when the order of fractional derivative α → 1 − . Numerical results show that the theoretical analysis is sharp.

The use of constant order differential equations to describe the evolution of complex systems is often unable to describe some of the changing characteristics of the systems accurately. Variable order fractional derivatives provide us with new tools to solve such problems. In this paper, the accumulation and derivative orders of the classic grey model are expanded from constants to functions, and a variable order fractional grey model is established to describe the evolution process of complex systems. Firstly, this paper defines the variable order fractional accumulation generation sequence. On the basis of this sequence, a variable order fractional derivative grey model is established, the parameters of the model are estimated using the least square method, and the quantum particle swarm optimization algorithm is used to solve the order of fractional derivative and accumulation. Sadik transform and Laplace transform are adopted to obtain the analytical solution of the new model. Lastly, the effectiveness of the new model is verified through four cases. Compared with other models, the variable order fractional model can describe the development process of complex systems more accurately.

In this paper, we study the generalized time fractional Burgers equation, where the time fractional derivative is in the sense of Caputo with derivative order in ( 0 , 1 ) . If its solution u ( x , t ) has strong regularity, for example u ( ⋅ , t ) ∈ C 2 [ 0 , T ] for a given time T , then we use the L1 scheme on uniform meshes to approximate the Caputo time-fractional derivative, and use the local discontinuous Galerkin (LDG) method to approach the space derivative. However, the solution u ( x , t ) likely behaves a certain regularity at the starting time, i.e., ∂ u ∂ t and ∂ 2 u ∂ 2 t can blow up as t → 0 + albeit u ( ⋅ , t ) ∈ C [ 0 , T ] for a given time T . In this case, we use the L1 scheme on non-uniform meshes to approximate the Caputo time-fractional derivative, and use the LDG method to discretize the spatial derivative. The fully discrete schemes for both situations are established and analyzed. It is shown that the derived schemes are numerically stable and convergent. Finally, several numerical experiments are provided which support the theoretical analysis.

In this paper, we consider high order numerical methods for the solution of the initial-boundary value problem of two-sided space fractional advection-diffusion equations (SFADEs). We use the Crank-Nicolson (CN) technique to discretize the temporal derivative, and apply 3-point weighted and shifted Grunwald difference (WSGD) operators to discretize the fractional derivatives of order α ∈ ( 1 , 2 ) and fractional derivatives of order β ∈ ( 0 , 1 ) , respectively. As a result, a new family of CN-WSGD schemes for SFADEs with temporally 2nd order and spatially jth order ( j ≥ 2 ) accuracy are obtained. We then analyse the stability and the convergence of the numerical schemes. The extrapolated WSGD (EWSGD) operators are also discussed. Numerical examples are implemented to verify the theoretical results.

The analysis of diffusion-reaction equations with general two-sided fractional derivative characterized by a parameter p ∈ [ 0 , 1 ] is investigated in this paper. First, we present a Petrov-Galerkin method, derive a proper weak formulation and show the well-posedness of its weak solution. Moreover, on the basis of the two-sided Jacobi polyfractonomials, a priori error analysis of Petrov-Galerkin method is derived. Further, a posteriori error analysis is established rigorously. More precisely, we develop a novel postprocessing technique to enhance the Petrov-Galerkin method by adding a small amount of computation, and analyze asymptotically exact a-posteriori error estimators. Finally, we demonstrate the theoretical results with numerical examples.

In this work, we studied the radial point interpolation collocation method (RIPCM) for the solution of the partial differential models with fractional derivatives. In the present meshless method, polynomial basis function and two novel types local support fields were considered. The (time-)space fractional differential equations with single-term or multi-term time derivatives in irregular domain were well simulated numerically by the proposed RIPCM method. Numerical tests were carried out to verify the accuracy and demonstrate the strength of the RIPCM in solving fractional derivative models in irregular domain.

This paper presents a new predictor–corrector numerical scheme suitable for fractional differential equations. An improved explicit Atangana–Seda formula is obtained by considering the neglected terms and used as the predictor stage of the proposed method. Numerical formulas are presented that approximate the classical first derivative as well as the Caputo, Caputo–Fabrizio and Atangana–Baleanu fractional derivatives. Simulation results are used to assess the approximation error of the new method for various differential equations. In addition, a case study is considered where the proposed scheme is used to obtain numerical solutions of the Gierer–Meinhardt activator–inhibitor model with the aim of assessing the system’s dynamics.

In this paper, we study the numerical method for two dimensional fractional integro-differential equations, where the order of time fractional derivative α ∈ ( 1 , 2 ) and integral order γ ∈ ( 0 , 1 ) . To overcome the difficulty caused by the two fractional terms, we transform the original equation using the method of integration by parts. A novel high order compact alternating direction implicit (ADI) difference scheme is then proposed to solve the equivalent model. By some skills and detailed analysis, the unconditional stability and convergence in H 1 norm are proved, with the accuracy order O ( τ 2 + h 1 4 + h 2 4 ) , where τ , h 1 and h 2 are temporal and spatial step sizes, respectively. Finally, numerical results are presented to support the theoretical analysis.

This work proposes a novel Hausdorff fractional NGMC(p,n) grey prediction model based on the NGMC(1,n) model. The new grey model combines the Hausdorff fractional accumulation operator and the Grunwald-Letnikov fractional derivative with more freedom and simpler in calculations; the time response function and recurrence expressions of the new model are deduced by using forward difference; the recurrence relation of the binomial in the discrete solution to avoid calculating the Gamma function and simplifies the calculation; the Grey Wolf Optimizer (GWO) is introduced to optimize parameters of the new model for improving adaptability. To verify the efficiency of the new model, nine existing grey models are used to predict the total renewable energy production, the energy conversion efficiency and the total electricity production of China. The experimental results show that the fitting accuracy and prediction accuracy of the new model are better than those of the other nine existing grey models.

In the present paper, new analytical solutions for the space-time fractional Boussinesq and (2 + 1)-dimensional breaking soliton equations are obtained by using the simplified tan (phi(xi)/2)-expansion method. Here, fractional derivatives are defined in the conformable sense. To show the correctness of the obtained traveling wave solutions, residual error function is defined. It is observed that the new solutions are very close to the exact solutions. The solutions obtained by the presented method have not been reported in former literature. (C) 2020 Shanghai Jiaotong University. Published by Elsevier B.V.

The paper aims to study differential subordination and superordination preserving properties for certain analytic multivalent functions within the open unit disk related to a novel generalized fractional derivative operator for higher-order derivatives. As an application, we provide an explicit construction for the complex potential (the complex velocity) and the stream function of two-dimensional fluid flow problems over a circular cylinder using both vortex and source/sink. We further determine the fluid flow produced by a single source and construct a univalent function so that the image of a source is also a source for a given complex potential. Finally, we present some plot simulations that illustrate the results of this work.

This study aims to develop a mathematical model that can describe the vertical distribution of suspended sediment particles in an open channel turbulent flow under unsteady nonequilibrium condition with non-local mixing effect. A space non-local transport of sediment is considered that arises due to turbulent bursting where the hopping height of a particle is not limited to a small distance as described by classical theory of turbulence, rather it can jump upto any height. To take into account this fact unlike previous researchers who used Rouse equation, Hunt equation or traditional advection–diffusion equation as the governing equation, the present study uses fractional advection–diffusion equation (fADE) where space fractional derivative for the local diffusive flux has been used that counts the effect of non-locality. The fractional derivative is considered in Caputo sense and the order α of the fractional derivative corresponds to α -order L e ´ vy stable distribution with value 1 α ≤ 2 . The non-local effect has been considered in the boundary conditions as well as in the non-linear models of sediment diffusivity. Four types of sediment diffusivity are considered which are generalized non-linear models for broad applicability of the study. The fADE together with the boundary and initial conditions, is solved by Chebyshev collocation method and Euler backward method. This proposed method is unconditionally convergent and converges more rapidly than other previous methods used in similar studies. Effects of non-local mixing have been investigated for transient and bottom concentration distributions. Proposed models have been validated for sediment distribution under steady and unsteady condition with existing experimental data and satisfactory results are obtained for all choices of sediment diffusivity. The variation of transient and bottom concentration with non-local effects are physically justified. Validation results show that the model can be applied to describe vertical concentration distribution in unsteady and steady turbulent flows in practical situation.

In this paper, a new type of wavelet method to solve fractional differential equations (linear or nonlinear) is proposed. The proposed method is based on the generalized Gegenbauer–Humbert polynomial. First, we derived the operational matrices for integer and fractional order derivatives. Then, using these operational matrices with the proposed method, we transformed the given problem into a system of algebraic equations. Then, some linear and nonlinear examples were considered and discussed to confirm the efficiency and accuracy of the proposed method. 2021 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/ 4.0/).

In this paper, we present a numerical scheme and mathematical analysis for the famous three-dimensional quadratic autonomous self-govern system that happens to be chaotic with the coexistence of multi-scroll attractors. The scheme is based on the Atangana-Baleanu fractional derivative in the Caputo sense. The formulation of these schemes introduces the non-local and non-singular kernel to the fractional derivatives. The fractional derivative is then approximated using the family of the Adams–Bashforth schemes. The results are presented in both numerical and graphical as the fractional order β varies between 0 β ⩽ 1 . We study the proposed model in the both generalized case that is 0 β 1 and the case where β = 1 , which is the integer standard case. Due to the impact of the generalized case, the proposed model is able to maintain the coexistence of multi-scroll attractors.

The objectives of this research describes a novel meshless technique for solving space–time fractional Burgers’ equation by using concept of weighted and Caputo fractional derivative type. An efficient and accurate meshfree method based on backward substitution method and finite difference method is applied for problem on the various computational domain with scattering nodes. According to proposal method, Crank–Nicolson techniques is exploited to find the approximation in time level and the backward substitution method is utilized to compute node on the boundary. Subsequently, we obtain the corresponding weighted parameters the governing equations, we implemented collocation approach based on radial basis function. To eliminate nonlinearity, quasilinearization technique is applied to transform nonlinear source term into a linear source term. For investigation accuracy and reliably, this paper is compared with other previous researches. It is found that proposed scheme outperforms compared with existing numerical method.

In deep coal and rock mass engineering, extensive initial damage problems caused by excavation and unloading are critical to the long-term deformation and stability of underground projects. An innovative initial damage coal sample test method was proposed, and coal samples with different initial damage were prepared by the MTS pressure tester's circular displacement cyclic loading and unloading control. The creep characteristics of coal samples with different initial damages were tested by the staged loading creep test method. With the increase of initial damage, the coal samples’ creep time became shorter, and the rate of accelerated creep damage increased sharply. The relationship between fractional viscosity coefficient and stress was introduced to describe the non-linear relationship of creep strain increase at each stage. The creep model in the acceleration phase was improved by establishing the relationship between the damage variable and the initial damage. Based on the fractional calculus theory, a viscoelastic-plastic model considering the initial damage of coal samples was established, and the fractional order model was verified. The coefficient of correlation between the experimental data and the creep model was higher than 0.95, indicating that the fractional creep model can better describe the creep characteristics of coal samples with initial damage.

This paper provides an effective method for a class of 2-dim nonlinear variable order fractional optimal control problems (2DNVOFOCP). The technique is based on the new class of basis functions namely the generalized shifted Legendre polynomials. The dynamic constraint is described by a nonlinear variable order fractional differential equation where the fractional derivative is in the sense of Caputo. The 2-dim Gauss-Legendre quadrature rule together with the Lagrange multipliers method are utilized to find the solutions of the given 2DNVOFOCP. The convergence analysis of the presented method is investigated. The examined numerical examples manifest highly accurate results.

To capture more complexities associated to the spread of Covid-19 within a given population, we considered a system of nine differential equations that include a class of susceptible, 5 sub-classes of infected population, recovered, death and vaccine. The mathematical model was suggested with a lockdown function such that after the lockdown, the function follows a fading memory rate, a concept that is justified by the effect of social distancing that suggests, susceptible class should stay away from infected objects and humans. We presented a detailed analysis that includes reproductive number and stability analysis. Also, we introduced the concept of fractional Lyapunov function for Caputo, Caputo-Fabrizio and the Atangana-Baleanu fractional derivatives. We established the sign of the fractional Lyapunov function in all cases. Additionally we proved that, if the fractional order is one, we recover the results Lyapunov for the model with classical differential operators. With the nonlinearity of the differential equations depicting the complexities of the Covid-19 spread especially the cases with nonlocal operators, and due to the failure of existing analytical methods to provide exact solutions to the system, we employed a numerical method based on the Newton polynomial to derive numerical solutions for all cases and numerical simulations are provided for different values of fractional orders and fractal dimensions. Collected data from Turkey case for a period of 90 days were compared with the suggested mathematical model with Atangana-Baleanu fractional derivative and an agreement was reached for alpha 1.009 .

The aim of the present work is to apply the Caputo-Fabrizio fractional derivative to the heat transformation of a second grade fluid. The flow is analyzed under the effect of magneto hydrodynamic together with heat transfer as well as Darcy’s law. The infinite vertical plate is subjected to a more general time dependent motion. Moreover, on the thermal aspects of the infinite plate, we are taking into account the exponential heat phenomena. Exact analytical solutions are obtained by using Laplace transform method to the set of dimensionless fractional governing equations, containing the momentum and energy equations subjected to the boundary and initial conditions. Solutions are plotted graphically to analyze the effect of different parameters i.e. fractional parameter α , Grashof number and Prandtl number on the physical behavior of the fluid flow and temperature.

The cur­rent pan­demic sit­u­a­tion caused by COVID-19 has affected human life globally at the economic, social and men­tal health levels. Specifically, ten­sion has led an in­creas­ing number of people to the consumption of various types of to­bacco.. In this work, an existing tobacco smoking model with a specific class of tobacco snuffing is converted into a fractional order as many applications of fractional derivatives to recall the past history of smokers in the present model. For this purpose, we use fractional derivative in Caputo sense to study the model in the form of fractional order. Then Positivity, boundness and dynamics of the proposed model are investigated. For numerical results, the generalized “Adams–Bashforth–Moulton Method (GABMM) and fourth-order Runge–Kutta (RK4) method” are used to solve the proposed model and Matlab numerical computing environment is the current software used.

This paper introduces a Fractional Order Field Oriented Control (FO-FOC) with sensorless Model Reference Adaptive System (MRAS) for the Permanent Magnet Synchronous Motor (PMSM). In this work, the conventional integer controllers of the vector control method are replaced with the Fractional Order PI (FOPIα) controllers. Particle Swarm Optimization technique (PSO) is employed to tune the gain, integration parameter and the fractional order parameter for each of the speed, current and MRAS controllers to get the optimum values. The presented methodology is tested using MATLAB Simulink simulation and the results show improved performance of the field oriented control technique by the employment of the fractional order controllers instead of the conventional integer at different operating points. Furthermore, the additional degree of freedom in the fractional controllers helps the PSO technique to get the optimum cost at a very low number of iterations compared with that of the integer one.

Fractional calculus as was predicted by Leibniz to be an anecdote, has nowadays evolved to become a centre of interest for many researchers from various backgrounds. Their works have made signi…cant contribution to the general topic of fractional calculus and made it to be widely applicable in their respective …elds. This has permitted every researcher to formulate their own topic of interest and lay down their own foundation thereof. As a result, multiple innovative ideas had emerged; which caused signi…cant divisions regarding fractional calculus in the past three years. Some researchers believed fractional di¤erential operators should not have a nonsingular kernel; while others strongly believed that due to the complexity of nature, fractional di¤erential operators can have either singular or non-singular kernels. This contradiction in thoughts had led to the publication of few papers which are against di¤erential operators with non-singular kernels, causing some negative impacts. Thus, publishers and some editors in chief are concerned about the future of fractional calculus; which generally brought confusion among vibrant and innovative young researchers who desire to apply fractional calculus within their respective …elds. Therefore, this work is aimed at developing a mathematical model that could be used to depict the survival of fractional calculus. Six classes are herein considered to construct a mathematical model with six ordinary di¤erential equations. All elementary analysis, equilibrium points, stability analysis of equilibrium points, existence and uniqueness of system solutions are performed. Additionally, a new analysis including strength number that accounts for the accelerative information of nonlinear and linear parts of a given epidemiological model is introduced to better understand the system. This new number is deemed more informative than the well-known reproductive number. Therefore, an analysis of the second derivative of the Lyapunov function as well as an analysis of the second derivative of each class is applied to assess how a wave could be detected. It is strongly believed that this new analysis will particularly open new doors within the …eld of epidemiological modelling; which will aid researchers to better understand the spread of infectious diseases. The stochastic version of the suggested model was also investigated and numerical simulations were performed. The obtained reproductive number, strength number, extinction of criticism together with numerical simulation, revealed that the …eld of fractional calculus will be stable. The present criticism 1 will therefore have no signi…cant e¤ect in the near future. The outcome of this work further predicted that more researchers will join the …eld of fractional calculus, and will utilize both di¤erential operators with singular and non-singular kernels.

In this paper, we devoted to deal with the rotational periodic problem of some fractional iterative systems in the sense of Caputo fractional derivative. Under one sided-Lipschtiz condition on nonlinear term, the existence and uniqueness of solution for a fractional iterative equation is proved by applying the Leray-Schauder fixed point theorem and topological degree theory. Furthermore, the well posedness for a nonlinear control system with iteration term and a multivalued disturbance is established by using set-valued theory. The existence of solutions for a iterative neural network system is demonstrated at the end.

In this paper, we introduce new subclasses of analytic and p-valent functions related to starlike, convex, close-to-convex, and quasi-convex functions by using a p-valent analog of the Tremblay fractional derivative operator. Inclusion relationships for these subclasses are established.

The authors of this paper solve the fractional space-time advection-dispersion equation (ADE). In the advection-dispersion process, the solute movement being nonlocal in nature and the velocity of fluid flow being nonuniform, it leads to form a heterogeneous system which approaches to model the same by means of a fractional ADE which generalizes the classical ADE, where the time derivative is substituted through the Caputo fractional derivative. For the study of such fractional models, various numerical techniques are used by the researchers but the nonlocality of the fractional derivative causes high computational expenses and complex calculations so the challenge is to use an efficient method which involves less computation and high accuracy in solving such models numerically. Here, in order to get the FADE solved in the form of convergent infinite series, a novel method NHPM (natural homotopy perturbation method) is applied which couples Natural transform along with the homotopy perturbation method. The homotopy peturbation method has been applied in mathematical physics to solve many initial value problems expressed in the form of PDEs. Also, the HPM has an advantage over the other methods that it does not require any discretization of the domains, is independent of any physical parameters, and only uses an embedding parameter p ∈ 0 , 1 . The HPM combined with the Natural transform leads to rapidly convergent series solutions with less computation. The efficacy of the used method is shown by working out some examples for time-fractional ADE with various initial conditions using the NHPM. The Mittag-Leffler function is used to solve the fractional space-time advection-dispersion problem, and the impact of changing the fractional parameter α on the solute concentration is shown for all the cases.

This article studies modeling of a population growth by logistic equation when the population carrying capacity K tends to infinity. Results are obtained using fractional calculus theories. A fractional derivative known as psi-Caputo plays a substantial role in the study. We proved existence and uniqueness of the solution to the problem using the psi-Caputo fractional derivative. The Chinese population, whose carrying capacity, K, tends to infinity, is used as evidence to prove that the proposed approach is appropriate and performs better than the usual logistic growth equation for a population with a large carrying capacity. A psi-Caputo logistic model with the kernel function x + 1 performed the best as it minimized the error rate to 3.20% with a fractional order of derivative α = 1.6455.

In the paper, we discuss necessary and sufficient conditions to obtain the existence, uniqueness and stability of solutions of fractional impulsive functional differential equations towards the $$\psi $$ -Liouville–Caputo fractional derivative, through fixed point theorem, Arzela–Ascoli theorem and multivalued analysis theory.

In this paper, we have studied the time-fractional Zakharov-Kuznetsov equation (TFZKE) via natural transform decomposition method (NTDM) with nonsingular kernel derivatives. The fractional derivative considered in Caputo-Fabrizio (CF) and Atangana-Baleanu derivative in Caputo sense (ABC). We employed natural transform (NT) on TFZKE followed by inverse natural transform, to obtain the solution of the equation. To validate the method, we have considered a few examples and compared with the actual results. Numerical results are in accordance with the existing results.

Diffusion equations with Riemann–Liouville fractional derivatives are Volterra integro-partial differential equations with weakly singular kernels and present fundamental challenges for numerical computation. In this paper, we make a convergence analysis of the Schwarz waveform relaxation (SWR) algorithms with Robin transmission conditions (TCs) for these problems. We focus on deriving good choice of the parameter involved in the Robin TCs, at the continuous and fully discretized level. Particularly, at the space-time continuous level, we show that the derived Robin parameter is much better than the one predicted by the well-understood equioscillation principle. At the fully discretized level, the problem of determining a good Robin parameter is studied in the convolution quadrature framework, which permits us to precisely capture the effects of different temporal discretization methods on the convergence rate of the SWR algorithms. The results obtained in this paper will be preliminary preparations for our further study of the SWR algorithms for integro-partial differential equations.

This paper aims to implement an analytical method, known as the Laplace homotopy perturbation transform technique, for the result of fractional-order Whitham–Broer–Kaup equations. The technique is a mixture of the Laplace transformation and homotopy perturbation technique. Fractional derivatives with Mittag-Leffler and exponential laws in sense of Caputo are considered. Moreover, this paper aims to show the Whitham–Broer–Kaup equations with both derivatives to see their difference in a real-world problem. The efficiency of both operators is confirmed by the outcome of the actual results of the Whitham–Broer–Kaup equations. Some problems have been presented to compare the solutions achieved with both fractional-order derivatives.

A reaction-diffusion system can be represented by the Gray-Scott model. In this study, we discuss a one-dimensional time-fractional Gray-Scott model with Liouville-Caputo, Caputo-Fabrizio-Caputo, and Atangana-Baleanu-Caputo fractional derivatives. We utilize the fractional homotopy analysis transformation method to obtain approximate solutions for the time-fractional Gray-Scott model. This method gives a more realistic series of solutions that converge rapidly to the exact solution. We can ensure convergence by solving the series resultant. We study the convergence analysis of fractional homotopy analysis transformation method by determining the interval of convergence employing the ℏ u , v -curves and the average residual error. We also test the accuracy and the efficiency of this method by comparing our results numerically with the exact solution. Moreover, the effect of the fractionally obtained derivatives on the reaction-diffusion is analyzed. The fractional homotopy analysis transformation method algorithm can be easily applied for singular and nonsingular fractional derivative with partial differential equations, where a few terms of series solution are good enough to give an accurate solution.

In this manuscript, a semianalytical solution of the time-fractional Navier-Stokes equation under Caputo fractional derivatives using Optimal Homotopy Asymptotic Method (OHAM) is proposed. The above-mentioned technique produces an accurate approximation of the desired solutions and hence is known as the semianalytical approach. The main advantage of OHAM is that it does not require any small perturbations, linearization, or discretization and many reductions of the computations. Here, the proposed approach’s reliability and efficiency are demonstrated by two applications of one-dimensional motion of a viscous fluid in a tube governed by the flow field by converting them to time-fractional Navier-Stokes equations in cylindrical coordinates using fractional derivatives in the sense of Caputo. For the first problem, OHAM provides the exact solution, and for the second problem, it performs a highly accurate numerical approximation of the solution compare with the exact solution. The presented simulation results of OHAM comparison with analytical and numerical approaches reveal that the method is an efficient technique to simulate the solution of time-fractional types of Navier-Stokes equation.

In this paper, we construct transient electro-osmotic flow of Burgers’ fluid with Caputo fractional derivative in a micro-channel, where the Poisson–Boltzmann equation described the potential electric field applied along the length of the microchannel. The analytical solution for the component of the velocity profile was obtained, first by applying the Laplace transform combined with the classical method of partial differential equations and, second by applying Laplace transform combined with the finite Fourier sine transform. The exact solution for the component of the temperature was obtained by applying Laplace transform and finite Fourier sine transform. Further, due to the complexity of the derived models of the governing equations for both velocity and temperature, the inverse Laplace transform was obtained with the aid of numerical inversion formula based on Stehfest's algorithms with the help of MATHCAD software. The graphical representations showing the effects of the time, retardation time, electro-kinetic width, and fractional parameters on the velocity of the fluid flow and the effects of time and fractional parameters on the temperature distribution in the micro-channel were presented and analyzed. The results show that the applied electric field, electro-osmotic force, electro-kinetic width, and relaxation time play a vital role on the velocity distribution in the micro-channel. The fractional parameters can be used to regulate both the velocity and temperature in the micro-channel. The study could be used in the design of various biomedical lab-on-chip devices, which could be useful for biomedical diagnosis and analysis.

In Continuous Stirred Tank Reactor (CSTR) have Fractional order PID with the nominal order PID controller has been used to Multi-Criteria Decision Making (MCDM) and EMO (Evolutionary Multi-objective Optimization) by adjustment of control parameters like Hybrid methods in Multi objective optimization. But, this Fractional order PID with the nominal PID controller has maximum performance estimation. Proposed research work focused the Flower Pollination Algorithm based on Multi objective optimization with Genetic evaluation and Fractional order PID with the nominal PID controller is provides CSTR results. When a flower is displayed to maximum variations in this practical state, the Genetic evaluation has been used to identify the variations. The FPID (Flower Pollination Integral Derivative) is used for tuning the parameters of a Fractional order PID with the nominal PID controller for each region to improve the multi-criteria decision making. FPID also denoted as Flower Optimization Integral Derivative (FOID). The Genetic evaluation scheduler has been combined with multiple local linear Fractional order PID with the nominal PID controller to check the stability of loop for entire regions with various levels of temperatures. MATLAB results demonstrate that the feasibility of using the proposed Fractional order PID with the nominal PID controller compared than the existing PID controller, and it shows the FOID attained better results.