New Numerical Algorithm to Solve Variable-Order Fractional Integrodifferential Equations in the Sense of Hilfer-Prabhakar Derivative

Derakhshan, Mohammadhossein

doi:10.1155/2021/8817794

Cited by 7 publications

(5 citation statements)

References 34 publications

(38 reference statements)

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“…The Prabhakar integral operator is defined for analytic function ϕðzÞ ∈ H ½0, 1 = fϕ∈∪ : ϕ 1 z + ϕ 2 z 2 +⋯g by the formula [14][15][16][17][18][19][20]…”

Section: Complex Prabhakar Operator (Cpo)mentioning

confidence: 99%

Analytic Normalized Solutions of 2D Fractional Saint-Venant Equations of a Complex Variable

Alarifi

Ibrahim

2021

Journal of Function Spaces

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Saint-Venant equations describe the flow below a pressure surface in a fluid. We aim to generalize this class of equations using fractional calculus of a complex variable. We deal with a fractional integral operator type Prabhakar operator in the open unit disk. We formulate the extended operator in a linear convolution operator with a normalized function to study some important geometric behaviors. A class of integral inequalities is investigated involving special functions. The upper bound of the suggested operator is computed by using the Fox-Wright function, for a class of convex functions and univalent functions. Moreover, as an application, we determine the upper bound of the generalized fractional 2-dimensional Saint-Venant equations (2D-SVE) of diffusive wave including the difference of bed slope.

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“…The Prabhakar integral operator is defined for analytic function ϕðzÞ ∈ H ½0, 1 = fϕ∈∪ : ϕ 1 z + ϕ 2 z 2 +⋯g by the formula [14][15][16][17][18][19][20]…”

Section: Complex Prabhakar Operator (Cpo)mentioning

confidence: 99%

Analytic Normalized Solutions of 2D Fractional Saint-Venant Equations of a Complex Variable

Alarifi

Ibrahim

2021

Journal of Function Spaces

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“…However, due to the presence of more parameters in the Prabhakar operator, existing numerical schemes may not be useful to solve fractional differential equations defined in the Prabhakar sense. While certain studies do exist wherein numerical approximations of fractional differential equations defined in the Prabhakar sense are provided [38][39][40][41], these resources remain inadequate in number. In [38], the Lagrange multiplier technique and Chebyshev cardinal functions have been used in order to approximate the variable-order fractional integro-differential equations with Hilfer-Prabhakar fractional derivatives.…”

Section: Introductionmentioning

confidence: 99%

“…While certain studies do exist wherein numerical approximations of fractional differential equations defined in the Prabhakar sense are provided [38][39][40][41], these resources remain inadequate in number. In [38], the Lagrange multiplier technique and Chebyshev cardinal functions have been used in order to approximate the variable-order fractional integro-differential equations with Hilfer-Prabhakar fractional derivatives. Singh et al [39] provided the approximations of the Caputo-Prabhakar derivative and numerically treated the Caputo-Prabhakar fractional advection-diffusion equations, where a central difference scheme is used in space.…”

Section: Introductionmentioning

confidence: 99%

On new approximations of Caputo–Prabhakar fractional derivative and their application to reaction–diffusion problems with variable coefficients

Singh,

Kumar,

Vigo‐Aguiar

2023

Math Methods in App Sciences

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This article is devoted to constructing and analyzing two new approximations (CPL2‐1 and CPL‐2 formulas) for the Caputo–Prabhakar fractional derivative. The error bounds for the CPL2‐1 and CPL‐2 formulas are proved to be of order and , respectively, where is the order of time‐fractional derivative. The newly developed approximations are then used in the numerical treatment of a reaction–diffusion problem with variable coefficients defined in the Caputo–Prabhakar sense. Moreover, the space variable in the developed numerical schemes, CFD1 and CFD2, is discretized using a fourth‐order compact difference operator. Both schemes' stability and convergence analysis are demonstrated thoroughly using the discrete energy method. It is shown that the convergence orders of CFD1 and CFD2 schemes are and , respectively, where and represent the mesh spacing in time and space directions, respectively. In addition, numerical results are obtained for three test problems to confirm the theory and demonstrate the efficiency and superiority of the proposed schemes.

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“…Prabhakar fractional operator used to explain certain odd habits in disordered people materials that are nonlinear and nonlocal. The need for Prabhakar operators to specific fractional coefficients may be a useful tool for determining an appropriate statistical method that produced a strong agreement between theoretically and experimentally findings in [29][30][31][32][33]. For the mathematical model of variable-order fractional Volterra integral, a numerical approach focused on CCFs and the Lagrange technique is shown in [34,35].…”

Section: Introductionmentioning

confidence: 99%

Advances in transport phenomena with nanoparticles and generalized thermal process for vertical plate

2021

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In this paper, a quantitative analysis is performed to investigate the convective flow of Maxwell fluid along with an erect heated plate via Prabhakar-like energy transport. The governing equations for this mathematical model are attained by Prabhakar fractional derivative. Laplace transform is applied to obtain the generalized results for dimensionless velocity and temperature profiles. By applying the conditions of nanofluid flow, we develop the constantly accelerated, variables accelerated, and non-uniform accelerated solution of the model, respectively. Prabhakar fractional derivative for Maxwell fluid based on generalized Fourier's thermal flux is determined for heat transfer. For volume fraction j and fractional parameters α, β, and γ, different structures of graphs are obtained. Furthermore, fluid temperature and velocity can be increased by enhancing the fractional parameters' values. The comparison of nanoparticles Ag and nanoparticles Cu with water-based nanoparticles for velocity and temperature profile is also examined. The behavior of second grade and Maxwell fluid results to become a viscous fluid is justified. NomenclatureRECEIVED

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New Numerical Algorithm to Solve Variable-Order Fractional Integrodifferential Equations in the Sense of Hilfer-Prabhakar Derivative

Cited by 7 publications

References 34 publications

Analytic Normalized Solutions of 2D Fractional Saint-Venant Equations of a Complex Variable

Analytic Normalized Solutions of 2D Fractional Saint-Venant Equations of a Complex Variable

On new approximations of Caputo–Prabhakar fractional derivative and their application to reaction–diffusion problems with variable coefficients

Advances in transport phenomena with nanoparticles and generalized thermal process for vertical plate

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