Fractional Partial Derivatives Research Articles

Riemann-Liouville derivatives of abstract functions and Sobolev spaces

We introduce and study fractional Sobolev spaces of functions taking their values in a Banach space. Our approach is based on Riemann-Liouville derivative. In this regard, paper is a continuation of the paper [D. Idczak, S. Walczak, Fractional Sobolev spaces via Riemann-Liouville derivatives, J. of Function Spaces and Appl. 2013 (2013), Art. ID 128043, 15 pp.], where real-valued functions are investigated. Moreover, we study the relation between the fractional derivative of an abstract function of one variable and partial fractional derivative of a real-valued function of two variables.

The Boundary Value Problem with Stationary Inhomogeneities for a Hyperbolic-Type Equation with a Fractional Derivative

The paper presents an analytical solution of a partial differential equation of hyperbolic-type, containing both second-order partial derivatives and fractional derivatives of order below the second. Examples of applying the solution of a boundary value problem with stationary inhomogeneities for a hyperbolic-type equation with a fractional derivative in modeling the behavior of polymer concrete under the action of loads are considered.

Numerical Analysis of Time-Fractional Whitham-Broer-Kaup Equations with Exponential-Decay Kernel

This paper presents the semi-analytical analysis of the fractional-order non-linear coupled system of Whitham-Broer-Kaup equations. An iterative process is designed to analyze analytical findings to the specified non-linear partial fractional derivatives scheme utilizing the Yang transformation coupled with the Adomian technique. The fractional derivative is considered in the sense of Caputo-Fabrizio. Two numerical problems show the suggested method. Moreover, the results of the suggested technique are compared with the solution of other well-known numerical techniques such as the Homotopy perturbation technique, Adomian decomposition technique, and the Variation iteration technique. Numerical simulation has been carried out to verify that the suggested methodologies are accurate and reliable, and the results are revealed using graphs and tables. Comparing the analytical and actual solutions demonstrates that the proposed approaches effectively solve complicated non-linear problems. Furthermore, the proposed methodologies control and manipulate the achieved numerical solutions in a vast acceptable region in an extreme manner. It will provide us with a simple process to control and adjust the convergence region of the series solution.

On a Nonlocal Problem for Mixed-Type Equation with Partial Riemann-Liouville Fractional Derivative

The present paper presents a study on a problem with a fractional integro-differentiation operator in the boundary condition for an equation with a partial Riemann-Liouville fractional derivative. The unique solvability of the problem is proved. In the hyperbolic part of the considered domain, the functional equation is solved by the iteration method. The problem is reduced to solving the Volterra integro-differential equation.

On the objectivity of mathematical description of ion transport processes using general temporal Caputo and Riemann-Liouville fractional partial derivatives

It is shown that the mathematical description of the ion transport across passive or active biological neuron membrane, voltage propagation along neuron axons and dendrites having passive or active membrane and ion transport in biological neuron networks, using general temporal Caputo or general temporal Riemann-Liouville fractional order derivatives, is objective. The basic idea is that different observers using this type of description obtain results which can be reconciled, i.e., transformed into each other using only formulas that link the coordinates of a point in two fixed orthogonal reference frames and the numbers representing a moment of time in two different choices of the origin of time measurement. This requirement was pointed out by Galileo Galilee (1564–1642), Isaac Newton (1643–1727), Albert Einstein (1879–1955) in the context of mathematical description of mechanical movement: “The mechanical event is independent on the observer “.

Riemann-Liouville fractional derivatives of hidden variable recurrent fractal interpolation functions with function scaling factors and box dimension

In [M.G. Ri and C.H. Yun, Smoothness and fractional integral of hidden variable recurrent fractal interpolation function with function vertical scaling factors, Fractals 29(6) (2021) 2150136], the authors proved that the partial and mixed Riemann-Liouville fractional integrals of bivariable hidden variable recurrent fractal interpolation function(HVRFIF) with function scaling factors are HVRFIFs. As its continuation, in this paper, we show that Riemann-Liouville fractional derivatives of one variable and bivariable HVRFIFs are HVRFIFs under certain conditions. We also derive the relationship between the order of fractional calculus and the upper box dimension of its graph. To do it, firstly, we prove that Riemann-Liouville fractional derivative of one variable HVRFIF is HVRFIF. Secondly, we obtain estimation of the upper box dimension of the graph of one variable HVRFIF and derive the relationship between the upper box dimension and the order of fractional calculus. Finally, in the similar way to one variable, we show that the partial and mixed Riemann-Liouville fractional derivatives of bivarible HVRFIF are HVRFIFs.

Fractional model of MHD blood flow in a cylindrical tube containing magnetic particles

In recent years, the use of magnetic particles for biomedicine and clinical therapies has gained considerable attention. Unique features of magnetic particles have made it possible to apply them in medical techniques. These techniques not only provide minimal invasive diagnostic tools but also transport medicine within the cell. In recent years, MRI, drug supply to infected tissue, Hyperthermia are more enhanced by the use of magnetic particles. The present study aims to observe heat and mass transport through blood flow containing magnetic particles in a cylindrical tube. Furthermore, the magnetic field is applied vertically to blood flow direction. The Caputo time fractional derivative is used to model the problem. The obtained partial fractional derivatives are solved using Laplace transform and finite Hankel transform. Furthermore, the effect of various physical parameters of our interest has also been observed through various graphs. It has been noticed that the motion of blood and magnetic particles is decelerated when the particle mass parameter and the magnetic parameter are increased. These findings are important for medicine delivery and blood pressure regulation.

A novel numerical method for solution of fractional partial differential equations involving the $ \psi $-Caputo fractional derivative

<abstract><p>In this study, the $ \psi $-Haar wavelets operational matrix of integration is derived and used to solve linear $ \psi $-fractional partial differential equations ($ \psi $-FPDEs) with the fractional derivative defined in terms of the $ \psi $-Caputo operator. We approximate the highest order fractional partial derivative of the solution of linear $ \psi $-FPDE using Haar wavelets. By combining the operational matrix and $ \psi $-fractional integration, we approximate the solution and its other $ \psi $-fractional partial derivatives. Then substituting these approximations in the given $ \psi $-FPDEs, we obtained a system of linear algebraic equations. Finally, the approximate solution is obtained by solving this system. The simplicity and effectiveness of the proposed method as a mathematical tool for solving $ \psi $-Fractional partial differential equations is one of its main advantages. The sparse nature of the operational matrices improves the ability of the proposed method to execute with less computation complexity. Numerical examples are provided to show the efficiency and effectiveness of the method.</p></abstract>

A New Class of Coupled Systems of Nonlinear Hyperbolic Partial Fractional Differential Equations in Generalized Banach Spaces Involving the ψ–Caputo Fractional Derivative

The current study is devoted to investigating the existence and uniqueness of solutions for a new class of symmetrically coupled system of nonlinear hyperbolic partial-fractional differential equations in generalized Banach spaces in the sense of ψ–Caputo partial fractional derivative. Our approach is based on the Krasnoselskii-type fixed point theorem in generalized Banach spaces and Perov’s fixed point theorem together with the Bielecki norm, while Urs’s approach was used to prove the Ulam–Hyers stability of solutions of our system. Finally, some examples are provided in order to illustrate our theoretical results.

Fractional improved Legendre polynomials in solving differential equations with fractional partial derivatives with time delay

Fractional improved Legendre polynomials in solving differential equations with fractional partial derivatives with time delay

Long time behavior of Robin boundary sub-diffusion equation with fractional partial derivatives of Caputo type in differential and difference settings

A Robin boundary sub-diffusion equation is considered with fractional partial derivatives of the Caputo type. The model is an extension of various well-known equations from mathematical physics, biology, and chemistry. Initial–boundary data are imposed upon a closed and bounded spatial domain. We state and prove two main theorems in differential and difference settings to ensure the algebraic decay rate of the long-time behavior for that kind of problem. The dissipation of the continuous solution for such a problem is discussed in the first theorem based on energy inequalities and by the aid of Grönwall inequalities. It demonstrates that with an L2(Ω)-bounded absorbing set, the solution is dissipated with respect to time. The numerical dissipativity is proved in the second theorem by using discrete energy inequalities and the discrete Paley–Wiener inequality. Finally, an example is provided to illustrate the main outcomes.

An efficient matrix approach for the numerical solutions of electromagnetic wave model based on fractional partial derivative

In this article, we construct an efficient numerical method by combining the matrix and Kronecker multiplication to solve a fractional partial differential equation (FPDE) arising from electromagnetic waves in dielectric media (EMWDM). We adopted the operational matrix method (OMM) based on Bernoulli and Hermite wavelets for approximating the space and time derivatives. The proposed method transform FPDE into a system of an algebraic equation. For a better understanding of the method, numerical algorithms are also provided for the considered problems. Furthermore, the convergence of approximation and theoretical unconditional stability have been proved with respect to squire integrable norm. Finally, two numerical experiments with different fractional-order derivatives have been performed and compared with the analytical solutions to illustrate the accuracy and efficiency of the method. It is observed that the OMM is simple, easy to implement, yields highly accurate results at a small number of basis functions.

Fractional isospectral and non-isospectral AKNS hierarchies and their analytic methods for N-fractal solutions with Mittag-Leffler functions

Ablowitz–Kaup–Newell–Segur (AKNS) linear spectral problem gives birth to many important nonlinear mathematical physics equations including nonlocal ones. This paper derives two fractional order AKNS hierarchies which have not been reported in the literature by equipping the AKNS spectral problem and its adjoint equations with local fractional order partial derivative for the first time. One is the space-time fractional order isospectral AKNS (stfisAKNS) hierarchy, three reductions of which generate the fractional order local and nonlocal nonlinear Schrödinger (flnNLS) and modified Kortweg–de Vries (fmKdV) hierarchies as well as reverse-t NLS (frtNLS) hierarchy, and the other is the time-fractional order non-isospectral AKNS (tfnisAKNS) hierarchy. By transforming the stfisAKNS hierarchy into two fractional bilinear forms and reconstructing the potentials from fractional scattering data corresponding to the tfnisAKNS hierarchy, three pairs of uniform formulas of novel N-fractal solutions with Mittag-Leffler functions are obtained through the Hirota bilinear method (HBM) and the inverse scattering transform (IST). Restricted to the Cantor set, some obtained continuous everywhere but nondifferentiable one- and two-fractal solutions are shown by figures directly. More meaningfully, the problems worth exploring of constructing N-fractal solutions of soliton equation hierarchies by HBM and IST are solved, taking stfisAKNS and tfnisAKNS hierarchies as examples, from the point of view of local fractional order derivatives. Furthermore, this paper shows that HBM and IST can be used to construct some N-fractal solutions of other soliton equation hierarchies.

Fractional Rogue Waves with Translational Coordination, Steep Crest, and Modified Asymmetry

To construct fractional rogue waves, this paper first introduces a conformable fractional partial derivative. Based on the conformable fractional partial derivative and its properties, a fractional Schrödinger (NLS) equation with Lax integrability is then derived and first- and second-order fractional rogue wave solutions of which are finally obtained. The obtained fractional rogue wave solutions possess translational coordination, providing, to some extent, the degree of freedom to adjust the position of the rogue waves on the coordinate plane. It is shown that the obtained first- and second-order fractional rogue wave solutions are steeper than those of the corresponding NLS equation with integer-order derivatives. Besides, the time the second-order fractional rogue wave solution undergoes from the beginning to the end is also short. As for asymmetric fractional rogue waves with different backgrounds and amplitudes, this paper puts forward a way to construct them by modifying the obtained first- and second-order fractional rogue wave solutions.

Conformable fractional derivative and its application to partial fractional derivatives

Recently, the study on conformable fractional derivative has been gaining more attention from various researchers. This is due to its importance in numerous practical applications. In this paper, we introduce new definition of conformable fractional derivation to find partial fractional derivatives. The fractional derivatives are taken in the sense of Conformable fractional derivative. This approach is considered as a powerful tool for solving linear or nonlinear equations. The new formulation of fractional calculus is applied to solve many applied problems including fractional partial derivatives. The results obtained through the conformable fractional approach have been evaluated and presented to illustrate the efficiency and good accuracy. It has been observed that the proposed approach is prevailing for the solutions of partial fractional derivatives.

OPIAL-TYPE INEQUALITY ABOUT CONFORMABLE FRACTIONAL INTEGRALS AND THE APPLICATION

In this paper, we define partial conformable fractional derivative (PCFD) which is based on the fractional derivative definition proposed by Abdeljawad. At the same time, we present some simple properties about the definition, the properties are useful for us when we carry out further research. In addition, taking advantage of Hölder’s inequality, we establish Opial-type inequalities for the partial conformable fractional (PCF) integral. As an application, the uniqueness of the partial system with initial value is proved.

Line Soliton Interactions for Shallow Ocean Waves and Novel Solutions with Peakon, Ring, Conical, Columnar, and Lump Structures Based on Fractional KP Equation

It is well known that the celebrated Kadomtsev-Petviashvili (KP) equation has many important applications. The aim of this article is to use fractional KP equation to not only simulate shallow ocean waves but also construct novel spatial structures. Firstly, the definitions of the conformable fractional partial derivatives and integrals together with a physical interpretation are introduced and then a fractional integrable KP equation consisting of fractional KPI and KPII equations is derived. Secondly, a formula for the fractional n -soliton solutions of the derived fractional KP equation is obtained and fractional line one-solitons with bend, wavelet peaks, and peakon are constructed. Thirdly, fractional X-, Y- and 3-in-2-out-type interactions in the fractional line two- and three-soliton solutions of the fractional KPII equation are simulated for shallow ocean waves. Besides, a falling and spreading process of a columnar structure in the fractional line two-soliton solution is also simulated. Finally, a fractional rational solution of the fractional KP equation is obtained including the lump solution as a special case. With the development of time, the nonlinear dynamic evolution of the fractional lump solution of the fractional KPI equation can change from ring and conical structures to lump structure.

Дифференциальные игры дробного порядка с распределенными параметрами

Изучена задача преследования в дифференциальных играх дробного порядка с распределенными параметрами. Частные дробные производные по времени и пространственные переменные рассматриваются в смысле Римана-Лиувилля, при аппроксимации применяется формула Грюн-вальда-Летникова. Рассматривается задача попадания в некоторой положительной окраине терминального множества. Для решения применяется способ конечных различий. Аппроксимируются дробные производные Римана-Лиувилля по пространственным переменным на отрезке с помощью формулы Грюнвальда-Летникова. По достаточному признаку существования дробной производной получена разностная аппроксимация производной дробного порядка по времени. Аппроксимируя дифференциальную игру на явную разностную, получаем дискретную игру. Сформулирована соответствующая задача преследования для дискретной игры, полученная с помощью аппроксимации непрерывной игры. Определено понятие возможности завершения преследования, дискретной игры в смысле точного поимки. Получены достаточные условия для завершения преследования. При этом показано, что порядок аппроксимации по времени равен единице, а по пространственным переменным — двум. Доказано, если в дискретной игре по заданному начальному положению возможно завершение преследования в смысле точного поимки, то в непрерывной игре из соответствующего начального положения возможно завершение преследования. Предложена структура построения управлений преследования, которая обеспечит завершение игры за конечное время. Методы, применяемые для этой задачи, могут быть использованы для изучения дифференциальных игр, описываемых более общими уравнениями дробного порядка.

Conformable fractional derivative and its application to partial fractional derivatives

Recently, the study on conformable fractional derivative has been gaining more attention from various researchers. This is due to its importance in numerous practical applications. In this paper, we introduce new definition of conformable fractional derivation to find partial fractional derivatives. The fractional derivatives are taken in the sense of Conformable fractional derivative. This approach is considered as a powerful tool for solving linear or nonlinear equations. The new formulation of fractional calculus is applied to solve many applied problems including fractional partial derivatives. The results obtained through the conformable fractional approach have been evaluated and presented to illustrate the efficiency and good accuracy. It has been observed that the proposed approach is prevailing for the solutions of partial fractional derivatives.

Numerical solution of variable-order space-time fractional KdV–Burgers–Kuramoto equation by using discrete Legendre polynomials

In this paper, a new version of the nonlinear space-time fractional KdV–Burgers–Kuramoto equation has been generated via the variable-order (VO) fractional derivatives defined in the Caputo type. A numerical method has been developed based on the discrete Legendre polynomials (LPs) and the collocation scheme for solving this equation. First, the solution of the problem is expanded in terms of the shifted discrete LPs. Then, this expansion and its derivatives, including the classical partial derivatives and the VO fractional partial derivatives are replaced in the equation. Eventually, the operational matrices of the shifted discrete LPs, including the classical derivatives and the VO fractional derivatives (which are derived in this study), and the collocation method are employed to convert the approximated problem into an algebraic system of equations. Some numerical results are given to illustrate the accuracy of the method.