• Title/Summary/Keyword: Caputo fractional derivative

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THE SPACE-TIME FRACTIONAL DIFFUSION EQUATION WITH CAPUTO DERIVATIVES

  • HUANG F.;LIU F.
    • Journal of applied mathematics & informatics
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    • v.19 no.1_2
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    • pp.179-190
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    • 2005
  • We deal with the Cauchy problem for the space-time fractional diffusion equation, which is obtained from standard diffusion equation by replacing the second-order space derivative with a Caputo (or Riemann-Liouville) derivative of order ${\beta}{\in}$ (0, 2] and the first-order time derivative with Caputo derivative of order ${\beta}{\in}$ (0, 1]. The fundamental solution (Green function) for the Cauchy problem is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. We derive explicit expression of the Green function. The Green function also can be interpreted as a spatial probability density function evolving in time. We further explain the similarity property by discussing the scale-invariance of the space-time fractional diffusion equation.

A NOTE ON EXPLICIT SOLUTIONS OF CERTAIN IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS

  • Koo, Namjip
    • Journal of the Chungcheong Mathematical Society
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    • v.30 no.1
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    • pp.159-164
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    • 2017
  • This paper deals with linear impulsive differential equations involving the Caputo fractional derivative. We provide exact solutions of nonhomogeneous linear impulsive fractional differential equations with constant coefficients by means of the Mittag-Leffler functions.

THE FUNDAMENTAL SOLUTION OF THE SPACE-TIME FRACTIONAL ADVECTION-DISPERSION EQUATION

  • HUANG F.;LIU F.
    • Journal of applied mathematics & informatics
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    • v.18 no.1_2
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    • pp.339-350
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    • 2005
  • A space-time fractional advection-dispersion equation (ADE) is a generalization of the classical ADE in which the first-order time derivative is replaced with Caputo derivative of order $\alpha{\in}(0,1]$, and the second-order space derivative is replaced with a Riesz-Feller derivative of order $\beta{\in}0,2]$. We derive the solution of its Cauchy problem in terms of the Green functions and the representations of the Green function by applying its Fourier-Laplace transforms. The Green function also can be interpreted as a spatial probability density function (pdf) evolving in time. We do the same on another kind of space-time fractional advection-dispersion equation whose space and time derivatives both replacing with Caputo derivatives.

A NOTE ON LINEAR IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS

  • Choi, Sung Kyu;Koo, Namjip
    • Journal of the Chungcheong Mathematical Society
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    • v.28 no.4
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    • pp.583-590
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    • 2015
  • This paper deals with linear impulsive fractional differential equations involving the Caputo derivative with non-integer order q. We provide exact solutions of linear impulsive fractional differential equations with constant coefficient by mean of the Mittag-Leffler functions. Then we apply the exact solutions to improve impulsive integral inequalities with singularity.

A GENERALIZED APPROACH OF FRACTIONAL FOURIER TRANSFORM TO STABILITY OF FRACTIONAL DIFFERENTIAL EQUATION

  • Mohanapriya, Arusamy;Sivakumar, Varudaraj;Prakash, Periasamy
    • Korean Journal of Mathematics
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    • v.29 no.4
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    • pp.749-763
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    • 2021
  • This research article deals with the Mittag-Leffler-Hyers-Ulam stability of linear and impulsive fractional order differential equation which involves the Caputo derivative. The application of the generalized fractional Fourier transform method and fixed point theorem, evaluates the existence, uniqueness and stability of solution that are acquired for the proposed non-linear problems on Lizorkin space. Finally, examples are introduced to validate the outcomes of main result.

ANALYSIS OF SOLUTIONS FOR THE BOUNDARY VALUE PROBLEMS OF NONLINEAR FRACTIONAL INTEGRODIFFERENTIAL EQUATIONS INVOLVING GRONWALL'S INEQUALITY IN BANACH SPACES

  • KARTHIKEYAN, K.;RAJA, D. SENTHIL;SUNDARARAJAN, P.
    • Journal of applied mathematics & informatics
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    • v.40 no.1_2
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    • pp.305-316
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    • 2022
  • We study the existence and uniqueness of solutions for a class of boundary value problems of nonlinear fractional order differential equations involving the Caputo fractional derivative by employing the Banach's contraction principle and the Schauder's fixed point theorem. In addition, an example is given to demonstrate the application of our main results.

A New Analytical Series Solution with Convergence for Nonlinear Fractional Lienard's Equations with Caputo Fractional Derivative

  • Khalouta, Ali
    • Kyungpook Mathematical Journal
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    • v.62 no.3
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    • pp.583-593
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    • 2022
  • Lienard's equations are important nonlinear differential equations with application in many areas of applied mathematics. In the present article, a new approach known as the modified fractional Taylor series method (MFTSM) is proposed to solve the nonlinear fractional Lienard equations with Caputo fractional derivatives, and the convergence of this method is established. Numerical examples are given to verify our theoretical results and to illustrate the accuracy and effectiveness of the method. The results obtained show the reliability and efficiency of the MFTSM, suggesting that it can be used to solve other types of nonlinear fractional differential equations that arise in modeling different physical problems.

ANALYSIS OF MALARIA DYNAMICS USING ITS FRACTIONAL ORDER MATHEMATICAL MODEL

  • PAWAR, D.D.;PATIL, W.D.;RAUT, D.K.
    • Journal of applied mathematics & informatics
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    • v.39 no.1_2
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    • pp.197-214
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    • 2021
  • In this paper, we have studied dynamics of fractional order mathematical model of malaria transmission for two groups of human population say semi-immune and non-immune along with growing stages of mosquito vector. The present fractional order mathematical model is the extension of integer order mathematical model proposed by Ousmane Koutou et al. For this study, Atangana-Baleanu fractional order derivative in Caputo sense has been implemented. In the view of memory effect of fractional derivative, this model has been found more realistic than integer order model of malaria and helps to understand dynamical behaviour of malaria epidemic in depth. We have analysed the proposed model for two precisely defined set of parameters and initial value conditions. The uniqueness and existence of present model has been proved by Lipschitz conditions and fixed point theorem. Generalised Euler method is used to analyse numerical results. It is observed that this model is more dynamic as we have considered all classes of human population and mosquito vector to analyse the dynamics of malaria.

FRACTIONAL GREEN FUNCTION FOR LINEAR TIME-FRACTIONAL INHOMOGENEOUS PARTIAL DIFFERENTIAL EQUATIONS IN FLUID MECHANICS

  • Momani, Shaher;Odibat, Zaid M.
    • Journal of applied mathematics & informatics
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    • v.24 no.1_2
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    • pp.167-178
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    • 2007
  • This paper deals with the solutions of linear inhomogeneous time-fractional partial differential equations in applied mathematics and fluid mechanics. The fractional derivatives are described in the Caputo sense. The fractional Green function method is used to obtain solutions for time-fractional wave equation, linearized time-fractional Burgers equation, and linear time-fractional KdV equation. The new approach introduces a promising tool for solving fractional partial differential equations.