On time fractional pseudo‐parabolic equations with non‐local in time condition

Can, Nguyen Huu; Kumar, Devendra; Viet, Tri Vo; Nguyễn, Anh Tuấn

doi:10.1002/mma.7196

Cited by 10 publications

(3 citation statements)

References 36 publications

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“…Te main techniques and methods frequently used are the modifed Lavrentiev regularization method and the Fourier truncated regularization method. To the fractional pseudoparabolic equation, sometimes, the inverse source problem is also discussed [15][16][17].…”

Section: Introductionmentioning

confidence: 99%

On the Convergence Result of the Fractional Pseudoparabolic Equation

Van Tien,

Saadati

2023

Journal of Mathematics

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In this paper, we consider the nonlinear fractional Laplacian pseudoparabolic equation (NFLPPE). We mainly focus on the convergence of mild solutions with respect to the order of fractional Laplacian. By using many techniques, we obtain the result that the mild solution will converge when the fractional order of the Laplacian tends to 1 − . The proof of convergent result relies on sharp techniques of evaluating the exponential terms, the Sobolev embeddings, and weakly singular Gronwall inequalities.

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Section: Introductionmentioning

confidence: 99%

On the Convergence Result of the Fractional Pseudoparabolic Equation

Van Tien,

Saadati

2023

Journal of Mathematics

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“…Therefore, these types of equations have a significant role in applied science, and it is important to obtain exact and approximate numerical solutions for either integer or non-integer order. In the literature there are several methods studied to solve these equations both analytically and numerically [18][19][20][21][22][23][24][25]. For solving the time-fractional Burger--Huxley equation inside the Caputo type fractional derivative, a simple and powerful numerical technique was provided [26].…”

Section: Introductionmentioning

confidence: 99%

Finite difference method for the fractional order pseudo telegraph integro-differential equation

Modanlı¹,

Ozbag²,

Akgülma³

2022

J. Appl. Math. Comput. Mech.

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The main goal of this paper is to investigate the numerical solution of the fractional order pseudo telegraph integro-differential equation. We establish the first order finite difference scheme. Then for the stability analysis of the constructed difference scheme, we give theoretical statements and prove them. We also support our theoretical statements by performing numerical experiments for some fractions of order α.

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“…However, there are many phenomena that may not depend only on the time moment but also on the former time history, which cannot be modeled utilizing the classical derivatives. For this reason, many authors try to replace the classical derivatives with the so-called fractional derivatives in numerous contributions [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17], because it has been proven that this last kind of derivatives is a very good way to describe processes with memory. According to the literature of fractional calculus, it is remarkable that there are many approaches to defining fractional derivatives, and each definition has advantages compared to others [18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

On the existence of mild solutions for nonlocal differential equations of the second order with conformable fractional derivative

Atraoui

Bouaouid

2021

Adv Differ Equ

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In the work (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019), the authors have used the Krasnoselskii fixed point theorem for showing the existence of mild solutions of an abstract class of conformable fractional differential equations of the form: $\frac{d^{\alpha }}{dt^{\alpha }}[\frac{d^{\alpha }x(t)}{dt^{\alpha }}]=Ax(t)+f(t,x(t))$ d α d t α [ d α x ( t ) d t α ] = A x ( t ) + f ( t , x ( t ) ) , $t\in [0,\tau ]$ t ∈ [ 0 , τ ] subject to the nonlocal conditions $x(0)=x_{0}+g(x)$ x ( 0 ) = x 0 + g ( x ) and $\frac{d^{\alpha }x(0)}{dt^{\alpha }}=x_{1}+h(x)$ d α x ( 0 ) d t α = x 1 + h ( x ) , where $\frac{d^{\alpha }(\cdot)}{dt^{\alpha }}$ d α ( ⋅ ) d t α is the conformable fractional derivative of order $\alpha \in\, ]0,1]$ α ∈ ] 0 , 1 ] and A is the infinitesimal generator of a cosine family $(\{C(t),S(t)\})_{t\in \mathbb{R}}$ ( { C ( t ) , S ( t ) } ) t ∈ R on a Banach space X. The elements $x_{0}$ x 0 and $x_{1}$ x 1 are two fixed vectors in X, and f, g, h are given functions. The present paper is a continuation of the work (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019) in order to use the Darbo–Sadovskii fixed point theorem for proving the same existence result given in (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019) [Theorem 3.1] without assuming the compactness of the family $(S(t))_{t>0}$ ( S ( t ) ) t > 0 and any Lipschitz conditions on the functions g and h.

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On time fractional pseudo‐parabolic equations with non‐local in time condition

Cited by 10 publications

References 36 publications

On the Convergence Result of the Fractional Pseudoparabolic Equation

On the Convergence Result of the Fractional Pseudoparabolic Equation

Finite difference method for the fractional order pseudo telegraph integro-differential equation

On the existence of mild solutions for nonlocal differential equations of the second order with conformable fractional derivative

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