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Existence of a unique weak solution to a non-autonomous time-fractional diffusion equation with space-dependent variable order

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Abstract
In this contribution, we investigate an initial-boundary value problem for a fractional diffusion equation with Caputo fractional derivative of space- dependent variable order where the coefficients are dependent on spatial and time variables. We consider a bounded Lipschitz domain and a homogeneous Dirichlet boundary condition. Variable-order fractional differential operators originate in anomalous diffusion modelling. Using the strongly positive definiteness of the governing kernel, we establish the existence of a unique weak solution in u is an element of L-infinity((0, T), H-0(1)(Omega)) to the problem if the initial data belongs to H-0(1)(Omega). We show that the solution belongs to C([0, T], H-0(1)(Omega)*) in the case of a Caputo fractional derivative of constant order. We generalise a fundamental identity for integro-differential operators of the form d/dt (k * v)(t) to a convolution kernel that is also space-dependent and employ this result when searching for more regular solutions. We also discuss the situation that the domain consists of separated subdomains.
Keywords
BOUNDARY-VALUE-PROBLEMS, INTEGRODIFFERENTIAL EQUATIONS, DIFFERENTIAL-EQUATIONS, INTEGRAL-EQUATIONS, MODELS, SCALES, Time-fractional diffusion equation, Anomalous diffusion, Non-autonomous, Time discretization, Existence, Uniqueness

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MLA
Van Bockstal, Karel. “Existence of a Unique Weak Solution to a Non-Autonomous Time-Fractional Diffusion Equation with Space-Dependent Variable Order.” ADVANCES IN DIFFERENCE EQUATIONS, vol. 2021, no. 1, 2021, doi:10.1186/s13662-021-03468-9.
APA
Van Bockstal, K. (2021). Existence of a unique weak solution to a non-autonomous time-fractional diffusion equation with space-dependent variable order. ADVANCES IN DIFFERENCE EQUATIONS, 2021(1). https://doi.org/10.1186/s13662-021-03468-9
Chicago author-date
Van Bockstal, Karel. 2021. “Existence of a Unique Weak Solution to a Non-Autonomous Time-Fractional Diffusion Equation with Space-Dependent Variable Order.” ADVANCES IN DIFFERENCE EQUATIONS 2021 (1). https://doi.org/10.1186/s13662-021-03468-9.
Chicago author-date (all authors)
Van Bockstal, Karel. 2021. “Existence of a Unique Weak Solution to a Non-Autonomous Time-Fractional Diffusion Equation with Space-Dependent Variable Order.” ADVANCES IN DIFFERENCE EQUATIONS 2021 (1). doi:10.1186/s13662-021-03468-9.
Vancouver
1.
Van Bockstal K. Existence of a unique weak solution to a non-autonomous time-fractional diffusion equation with space-dependent variable order. ADVANCES IN DIFFERENCE EQUATIONS. 2021;2021(1).
IEEE
[1]
K. Van Bockstal, “Existence of a unique weak solution to a non-autonomous time-fractional diffusion equation with space-dependent variable order,” ADVANCES IN DIFFERENCE EQUATIONS, vol. 2021, no. 1, 2021.
@article{8717465,
  abstract     = {{In this contribution, we investigate an initial-boundary value problem for a fractional diffusion equation with Caputo fractional derivative of space- dependent variable order where the coefficients are dependent on spatial and time variables. We consider a bounded Lipschitz domain and a homogeneous Dirichlet boundary condition. Variable-order fractional differential operators originate in anomalous diffusion modelling. Using the strongly positive definiteness of the governing kernel, we establish the existence of a unique weak solution in u is an element of L-infinity((0, T), H-0(1)(Omega)) to the problem if the initial data belongs to H-0(1)(Omega). We show that the solution belongs to C([0, T], H-0(1)(Omega)*) in the case of a Caputo fractional derivative of constant order. We generalise a fundamental identity for integro-differential operators of the form d/dt (k * v)(t) to a convolution kernel that is also space-dependent and employ this result when searching for more regular solutions. We also discuss the situation that the domain consists of separated subdomains.}},
  articleno    = {{314}},
  author       = {{Van Bockstal, Karel}},
  issn         = {{1687-1847}},
  journal      = {{ADVANCES IN DIFFERENCE EQUATIONS}},
  keywords     = {{BOUNDARY-VALUE-PROBLEMS,INTEGRODIFFERENTIAL EQUATIONS,DIFFERENTIAL-EQUATIONS,INTEGRAL-EQUATIONS,MODELS,SCALES,Time-fractional diffusion equation,Anomalous diffusion,Non-autonomous,Time discretization,Existence,Uniqueness}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{43}},
  title        = {{Existence of a unique weak solution to a non-autonomous time-fractional diffusion equation with space-dependent variable order}},
  url          = {{http://dx.doi.org/10.1186/s13662-021-03468-9}},
  volume       = {{2021}},
  year         = {{2021}},
}

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