Analytical Technique for (2+1) Fractional Diffusion Equation with Nonlocal Boundary Conditions

Nuruddeen, Rahmatullah Ibrahim; Garba, Bashir Danladi

doi:10.30538/oms2018.0035

Cited by 11 publications

(11 citation statements)

References 16 publications

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“…This derivative is indeed a very special one since it interpolates between the well‐known Caputo and Riemann–Liouville fractional derivatives. Riemann–Liouville fractional integral Let

0 \leq t < \infty .

The Riemann–Liouville fractional integral of a function

w (t)

for

0 < α \leq 1

is defined by previous research 22,23

J_{0 +}^{α} w (t) = \frac{1}{Γ (α)} \int_{0}^{t} {(t - τ)}^{α - 1} w (τ) d τ,

where

Γ (.)

is the famous gamma function. Riemann–Liouville fractional derivative Based on the above definition of Riemann–Liouville fractional integral, the definition the Riemann–Liouville fractional derivative of the function

w (t)

for

0 < α \leq 1

is as follows:

^{R} D_{0 +}^{α} w (t) = \frac{d}{d t} J_{0 +}^{1 - α} w (t) .

Caputo fractional derivative Accordingly, we define the Caputo fractional derivative of the function

w (t)

for

0 < α \leq 1

as 15

^{C} D_{0 +}^{α} w (t) = J_{0 + ...}

…”

Section: About Fractional Derivatives and Methodologymentioning

confidence: 99%

“…Accordingly, we define the Caputo fractional derivative of the function

w (t)

for

0 < α \leq 1

as 15

^{C} D_{0 +}^{α} w (t) = J_{0 +}^{1 - α} \frac{d}{d t} w (t) .

…”

Section: About Fractional Derivatives and Methodologymentioning

confidence: 99%

“…Also, from Equation (2.7), we express as a consequence the Laplace transform with regard to the Hilfer fractional derivative

D_{0 +}^{α, β}

of the function

w (t)

t

L {D_{0 +}^{α, β} w (t)} = s^{α} w^{*} (s) - s^{- β (1 - α)} J_{0 +}^{(1 - β) (1 - α)} w (0),

where

w (0)

is the value of

w (t)

t = 0

. Similarly, the Laplace transform with regard to the Caputo fractional derivative

^{C} D_{0 +}^{α}

of the function

w (t)

t

for

n - 1 < α \leq n

is expressed as 14,15

L {^{C} D_{0 +}^{α} w (t)} = s^{α} w^{*} (s) - {falsefalse}_{\sum}^{k = 0} s^{α - k<...}

…”

Section: Methodsmentioning

confidence: 99%

“…Various approximate and analytical techniques have been utilized in the recent literature to investigate different types of fractional, ordinary, as well as partial differential equations; see previous studies. [3][4][5][6][7][8][9][10] In line with this, and due to the evolving relevance of the fractional derivative operators in real-life applications [11][12][13][14][15] including the design of modern electric circuits and devices for the betterment of humanity, we, however, aim in the present examination to investigate the fractional version of two electric circuit models in the presence of various voltage source functions by pairing the Laplace integral transform and the negative binomial formula. 16 These models play a vital part in modern electronics and electrical engineering applications.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Investigating the dynamics of Hilfer fractional operator associated with certain electric circuit models

Nuruddeen¹,

Gómez-Aguilar

Ahmad³

et al. 2022

Circuit Theory & Apps

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The present manuscript investigates the action of the Hilfer fractional operator on the dynamics of resistor-capacitor (RC) and certain resistor-inductorcapacitor (RLC) electric circuits using the Laplace transform method alongside utilizing the negative binomial formula. The choice of Hilfer's operator here was basically based on its interpolating nature between the well-known Caputo and Riemann-Liouville fractional derivatives. Meanwhile, this operator has not been adequately scrutinized with regard to its effects on electric circuits in the literature. Thus, respective approximate exact solutions are determined convolutionally in the presence of certain periodic voltage source functions. Lastly, the significance of the operator on the dynamics of these models was demonstrated graphically for different values of the fractionalorder α and type β. More so, the present study will be beneficial in designing modern electric devices involving temporal delays and memory record.

show abstract

“…This derivative is indeed a very special one since it interpolates between the well‐known Caputo and Riemann–Liouville fractional derivatives. Riemann–Liouville fractional integral Let

0 \leq t < \infty .

The Riemann–Liouville fractional integral of a function

w (t)

for

0 < α \leq 1

is defined by previous research 22,23

J_{0 +}^{α} w (t) = \frac{1}{Γ (α)} \int_{0}^{t} {(t - τ)}^{α - 1} w (τ) d τ,

where

Γ (.)

w (t)

for

0 < α \leq 1

is as follows:

^{R} D_{0 +}^{α} w (t) = \frac{d}{d t} J_{0 +}^{1 - α} w (t) .

Caputo fractional derivative Accordingly, we define the Caputo fractional derivative of the function

w (t)

for

0 < α \leq 1

as 15

^{C} D_{0 +}^{α} w (t) = J_{0 + ...}

…”

Section: About Fractional Derivatives and Methodologymentioning

confidence: 99%

“…Accordingly, we define the Caputo fractional derivative of the function

w (t)

for

0 < α \leq 1

as 15

^{C} D_{0 +}^{α} w (t) = J_{0 +}^{1 - α} \frac{d}{d t} w (t) .

…”

Section: About Fractional Derivatives and Methodologymentioning

confidence: 99%

“…Also, from Equation (2.7), we express as a consequence the Laplace transform with regard to the Hilfer fractional derivative

D_{0 +}^{α, β}

of the function

w (t)

t

L {D_{0 +}^{α, β} w (t)} = s^{α} w^{*} (s) - s^{- β (1 - α)} J_{0 +}^{(1 - β) (1 - α)} w (0),

where

w (0)

is the value of

w (t)

t = 0

. Similarly, the Laplace transform with regard to the Caputo fractional derivative

^{C} D_{0 +}^{α}

of the function

w (t)

t

for

n - 1 < α \leq n

is expressed as 14,15

L {^{C} D_{0 +}^{α} w (t)} = s^{α} w^{*} (s) - {falsefalse}_{\sum}^{k = 0} s^{α - k<...}

…”

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Investigating the dynamics of Hilfer fractional operator associated with certain electric circuit models

Nuruddeen¹,

Gómez-Aguilar

Ahmad³

et al. 2022

Circuit Theory & Apps

View full text Add to dashboard Cite

show abstract

“…Yan et al [10] proposed and utilized a novel series method to examine the fractional diffusion equation; while Al-Khaled and Momani [11] investigated the diffusion-wave equation with a fractional-order via the use of the approximate decomposition approach. Other related studies include the investigation of the nonlinear heat diffusion model, comparative study of the Caputo and conformable fractional-order derivatives in relation to the heat conduction equation, tackling heat equation with nonlocal conditions, a method for the fractional conduction models, and the treatment of the nonlinear conduction process were all studied using methods based on the Adomian's approach [12][13][14][15][16][17][18]. Other approaches are comprised of the Lie's symmetry method for the reduction of the heat conduction equation [19] and the eigenfunctions expansion approach to a layered conduction slab [20] to mention a few.…”

Section: Introductionmentioning

confidence: 99%

Approximate analytical solution to the Cattaneo heat conduction model with various laser sources

Nuruddeen¹

2022

J. Appl. Math. Comput. Mech.

Self Cite

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The present manuscript investigates the role being played by various laser short heating sources in a conduction process of a metallic substrate. The Cattaneo heat conduction model is considered in favour of its finiteness of conduction speed. The analytical solutions for the temperature fields are determined via the application of the Laplace integral transform. Finally, we sought a numerical Laplace inversion scheme where the analytical inversion failed and graphically examined the significance of the heating parameters on the temperature fields.

show abstract

Analytical investigation of the coupled fractional models for immersed spheres and oscillatory pendulums

Emadifar

Nonlaopon

Shoaib

et al. 2023

Chaos, Solitons & Fractals

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Analytical Technique for (2+1) Fractional Diffusion Equation with Nonlocal Boundary Conditions

Cited by 11 publications

References 16 publications

Investigating the dynamics of Hilfer fractional operator associated with certain electric circuit models

Investigating the dynamics of Hilfer fractional operator associated with certain electric circuit models

Approximate analytical solution to the Cattaneo heat conduction model with various laser sources

Analytical investigation of the coupled fractional models for immersed spheres and oscillatory pendulums

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