Fractional differential equation pertaining to an integral operator involving incomplete <i>H</i>‐function in the kernel

Bansal, Manish Kumar; Lal, Shiv; Kumar, Devendra; Kumar, Sunil; Singh, Jagdev

doi:10.1002/mma.6670

Cited by 20 publications

(24 citation statements)

References 34 publications

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“…Because of these novel surprising insights, the biological systems with fractional derivatives have become more attractive in recent years. The elementary theory and some applications of fractional differential equations are widely covered previously, [30][31][32][33] and for the books associated with fractional differential equations, see previous studies. [34][35][36] The existence and uniqueness of mild solutions of generalized time fractional Navier-Stokes equation…”

Section: Introductionmentioning

confidence: 99%

“…Because of these novel surprising insights, the biological systems with fractional derivatives have become more attractive in recent years. The elementary theory and some applications of fractional differential equations are widely covered previously, 30‐33 and for the books associated with fractional differential equations, see previous studies 34‐36 . The existence and uniqueness of mild solutions of generalized time fractional Navier‐Stokes equation

\begin{matrix} ^{C} D^{α} u - μ normalΔ u + false(u \cdot \nabla false) u + \nabla p & = f, x \in ℝ^{N}, t > 0, \\ \nabla \cdot u & = 0, x \in ℝ^{N}, t > 0, \\ u false(x, 0 false) & = u_{0}, x \in ℝ^{N}, \end{matrix}

have been investigated by de Carvalho‐Neto and Planas 37 by using the semigroup theory.…”

Section: Introductionmentioning

confidence: 99%

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Nonnegative solutions to time fractional Keller–Segel system

Akilandeeswari

Tyagi

2020

Math Methods in App Sciences

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We establish the existence of nonnegative weak solutions to time fractional Keller-Segel system with Dirichlet boundary condition in a bounded domain with smooth boundary. Since the considered system has a cross-diffusion term and the corresponding diffusion matrix is not positive definite, we first regularize the system. Then under suitable assumptions on the initial conditions, we establish the existence of solutions to the system by using the Galerkin approximation method. The convergence of solutions is proved by means of compactness criteria for fractional partial differential equations. The nonnegativity of solutions is proved by the standard arguments. Furthermore, the existence of the weak solution to the system with Neumann boundary condition is discussed.

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Section: Introductionmentioning

confidence: 99%

“…Because of these novel surprising insights, the biological systems with fractional derivatives have become more attractive in recent years. The elementary theory and some applications of fractional differential equations are widely covered previously, 30‐33 and for the books associated with fractional differential equations, see previous studies 34‐36 . The existence and uniqueness of mild solutions of generalized time fractional Navier‐Stokes equation

\begin{matrix} ^{C} D^{α} u - μ normalΔ u + false(u \cdot \nabla false) u + \nabla p & = f, x \in ℝ^{N}, t > 0, \\ \nabla \cdot u & = 0, x \in ℝ^{N}, t > 0, \\ u false(x, 0 false) & = u_{0}, x \in ℝ^{N}, \end{matrix}

have been investigated by de Carvalho‐Neto and Planas 37 by using the semigroup theory.…”

Section: Introductionmentioning

confidence: 99%

Nonnegative solutions to time fractional Keller–Segel system

Akilandeeswari

Tyagi

2020

Math Methods in App Sciences

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“…The study of fractional differential equations has gained much importance in recent years due to their applications in numerous diverse fields of science and engineering. Some of the areas of applications of fractional differential equations include electrical network, viscoelasticity, optics, signal processing, finance, electromagnetics, control theory, acoustics, material science, nuclear reactor dynamics, biological systems and fluid mechanics, see in References [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Many processes in applied sciences and engineering can be modeled more accurately by fractional derivatives than integer order derivatives.…”

Section: Introductionmentioning

confidence: 99%

A high order numerical scheme for solving a class of non‐homogeneous time‐fractional reaction diffusion equation

Roul

Goura

2020

Numerical Methods Partial

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This paper is concerned with the development of a high order numerical technique for solving time‐fractional reaction–diffusion equation. The fractional derivative in the governing equation is described in the Caputo sense and a collocation method based on quintic B‐spline basis function is used to discretize the space variable. The stability and convergence analysis of the method are investigated, and it is shown that the proposed method converges to the exact solution of the problem with order of convergence O(Δt2 − α + Δx4), where α is the order of fractional derivative (0 < α < 1). Three test problems are considered to illustrate the efficiency and accuracy of present numerical scheme and to verify the theoretical result. It is shown that the order of fractional derivative has a profound effect on the solution of time‐fractional reaction–diffusion equation.

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“…Kumar et al 25 applied MHATM and RPSM for solving time‐fractional coupled SWEs. The solutions for FDEs including the generalized composite fractional derivative (GCFD) and integral operator associated with the IHF were derived successfully by using the Laplace transform technique 26 . Singh et al 27 employed the AB fractional operator to extend the fish farm model, and its solution was examined with the aid of HATM.…”

Section: Introductionmentioning

confidence: 99%

“…The solutions for FDEs including the generalized composite fractional derivative (GCFD) and integral operator associated with the IHF were derived successfully by using the Laplace transform technique. 26 Singh et al 27 employed the AB fractional operator to extend the fish farm model, and its solution was examined with the aid of HATM. Additionally, a comparative analysis was carried out for the ER model having fixed source of heat in the porous media via Caputo, CF, and AB theories.…”

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confidence: 99%

A new algorithm for fractional differential equation based on fractional order reproducing kernel space

Zhang

Yang

2020

Math Methods in App Sciences

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This paper develops an effective and new method to solve a class of fractional differential equations. The method is based on a fractional order reproducing kernel space. First, depending on some theories, a fractional order reproducing kernel space W [0, 1] is constructed. The fractional order reproducing kernel space is a very suitable space to solve a class of fractional differential equations. Then, we calculate the reproducing kernel R y (x) of the space W [0, 1] skilfully in §3. And convergence order and time complexity of this algorithm are discussed. We prove that the approximate solution u n of (1.1) converges to its exact solution u is not less than the second order. The time complexity of the algorithm is equal to the polynomial time of the third degree. Finally, three experiments support the algorithm strongly from the aspect of theory and technique. KEYWORDS complexity analysis, convergence order, fractional reproducing kernel space, initial value problem of fractional equations MSC CLASSIFICATION 34A45; 65L05; 65L20 1 INTRODUCTION Fractional differential equations(FDEs) have important applications in signal processing, physics, controller design, electric circuits, and so forth. Fractional order modelling minimizes the inaccuracy that arises from the ignorance of significant real parameters. It permits a greater degree of freedom in the model compared to an integer-order system. Thus, in the last two decades, the theory and numerical analysis of FDEs are receiving more and more attention. Many schemes such as fractional linear multistep method, 1 operational matrix method, 2,3 homotype analysis method, 4,5 moving least square reproducing kernel method, 6 predictor-corrector method, 7 dual reciprocity boundary integral equation technique, 8 finite difference method, 9 Adomian decomposition method, 10 extrapolation method, 11 Chebyshev wavelets method, 12 modified fractional Euler method, 13 variational iteration method, 14 and Laplace transform 15 have been developed to determine the numerical solutions of several classes of fractional ordinary differential equations and partial differential equations (PDEs). Recently, Kumar et al 16 used Bernstein wavelet and Euler methods to handle nonlinear fractional predator-prey biological model. In 2020, Kumar et al 17 utilized residual power series and perturbation methods to handle Fokker-Plank equation. Veeresha et al 18 used the q-HATM to acquire the solution of FGNLS equation. Kumar et al 19 applied new amagalation of homotopy analysis method, Laplace transform method, and Adomian polynomial's for fractional discreate KdV equations. In addition, Kumar et al 20 employed homotopy perturbation transform method to obtain approximate analytical solutions of the time-fractional PDEs. Odibat et al 21 analyzed systems of nonlinear FDEs by using homotopy asymptotic method (HAM). Kumar et al 22 suggested two new fractional derivatives, namely, Yang-Gao-Tenreiro

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Fractional differential equation pertaining to an integral operator involving incomplete H‐function in the kernel

Cited by 20 publications

References 34 publications

Nonnegative solutions to time fractional Keller–Segel system

Nonnegative solutions to time fractional Keller–Segel system

A high order numerical scheme for solving a class of non‐homogeneous time‐fractional reaction diffusion equation

A new algorithm for fractional differential equation based on fractional order reproducing kernel space

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