Year 2024

• Diffusion equations with spatially dependent coefficients and fractal Cauer-type networks

  Publication

  • J. Cresson
  • A. Szafrańska

  - Fractional Calculus and Applied Analysis - Year 2024

  In this article, we formulate and solve the representation problem for diffusion equations: giving a discretization of the Laplace transform of a diffusion equation under a space discretization over a space scale determined by an increment h > 0, can we construct a continuous in h family of Cauer ladder networks whose constitutive equations match for all h > 0 the discretization. It is proved that for a finite differences discretization...

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Year 2019

• About the Noether’s theorem for fractional Lagrangian systems and a generalization of the classical Jost method of proof

  Publication

  • J. Cresson
  • A. Szafrańska

  - Fractional Calculus and Applied Analysis - Year 2019

  Recently, the fractional Noether's theorem derived by G. Frederico and D.F.M. Torres in [10] was proved to be wrong by R.A.C. Ferreira and A.B. Malinowska in (see [7]) using a counterexample and doubts are stated about the validity of other Noether's type Theorem, in particular ([9],Theorem 32). However, the counterexample does not explain why and where the proof given in [10] does not work. In this paper, we make a detailed analysis...

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Year 2016

• Functional delay fractional equations

  Publication

  • T. Jankowski

  - Fractional Calculus and Applied Analysis - Year 2016

  In this paper, we discuss functional delay fractional equations. A Banach fixed point theorem is applied to obtain the existence (uniqueness) theorem. We also discuss such problems when a delay argument has a form α(t) = αt, 0 < α < 1, by Rusing the method of successive approximations. Some existence results are also formulated in this case. An example illustrates the main result.

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Year 2015

• Systems of Nonlinear Fractional Differential Equations

  Publication

  • T. Jankowski

  - Fractional Calculus and Applied Analysis - Year 2015

  Using the iterative method, this paper investigates the existence of a unique solution to systems of nonlinear fractional differential equations, which involve the right-handed Riemann-Liouville fractional derivatives D(T)(q)x and D(T)(q)y. Systems of linear fractional differential equations are also discussed. Two examples are added to illustrate the results.

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