Existence results for a generalization of the time-fractional diffusion equation with variable coefficients

Self Cite

“…Theorem 5.1 ( 14,30 ). Set f(t) ∈ C[0, +∞); then there exists an explicit solution of Equation (15), which is given in the integral form Proof.…”

Section: The Case With One Caputo-like Counterpart Hyper-bessel Operamentioning

confidence: 97%

Section: Gronwall-type Integral Inequalities Involving Mittag-lefflermentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Positive solution of nonlinear fractional differential equations with Caputo‐like counterpart hyper‐Bessel operators

Zhang

2019

Self Cite

“…With the initial condition u ( x ,0)= u 0 ( x ), we have the formula (see Al‐Musalhi et al and Zhang)

sans-serifu false(sans-serift, monospacex false) = true \sum_{q = 0}^{\infty} ([], E_{normalβ, 1} ((), - \frac{a_{q}}{γ^{β}} t^{normalβ normalγ}) u_{0, sans-serifq} + \frac{1}{γ^{β}} \int_{0}^{sans-serift} {false(t^{γ} - s^{γ} false)}^{normalβ - 1} E_{normalβ, normalβ} ((), - \frac{a_{q}}{γ^{β}} {false(t^{γ} - s^{γ} false)}^{β}) f_{q} false(monospaces false) d false(s^{γ} false)) {\overset{e}{true}}_{q} false(monospacex false),

where γ=1−α,

u_{0, sans-serifq} = (⟨⟩, sans-serifu false(0 false), {\overset{e}{true}}_{q})

f_{q} false(sans-serift false) = (⟨⟩, monospacef false(sans-serift, \cdot false), {\overset{e}{true}}_{q})

. By letting

sans-serift = T_{o}

, we get

{sans-serifh}_{sans-serifq} = E_{β...}

…”

Section: Preliminariesmentioning

confidence: 99%

“…The results for Equation was investigated by some recent works, such as Al‐Musalhi et al and Zhang . The authors considered two direct and inverse source problems of a fractional diffusion equation with regularized Caputo‐like counterpart hyper‐Bessel operator.…”

Section: Introductionmentioning

confidence: 99%

On a terminal value problem for a generalization of the fractional diffusion equation with hyper‐Bessel operator

Tuan

Huynh

Băleanu

et al. 2019

In this paper, we consider an inverse problem of recovering the initial value for a generalization of time‐fractional diffusion equation, where the time derivative is replaced by a regularized hyper‐Bessel operator. First, we investigate the existence and regularity of our terminal value problem. Then we show that the backward problem is ill‐posed, and we propose a regularizing scheme using a fractional Tikhonov regularization method. We also present error estimates between the regularized solution and the exact solution using two parameter choice rules.

Existence and uniqueness of analytical solution of time‐fractional Black‐Scholes type equation involving hyper‐Bessel operator

Zhang

2021

Self Cite

In this paper, we consider direct problem and inverse source problem of time‐fractional Black‐Scholes type model involving hyper‐Bessel operator. Analytical solutions to these problems are constructed based on appropriate eigenfunction expansion and Erdélyi‐Kober fractional integrals whose kernel has double singularities; then, existence and uniqueness are established. At last, the results are demonstrated by explicit solutions of some examples using appropriate choice of the given data.