• 제목/요약/키워드: Fractional order partial differential equation

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FOURIER'S TRANSFORM OF FRACTIONAL ORDER VIA MITTAG-LEFFLER FUNCTION AND MODIFIED RIEMANN-LIOUVILLE DERIVATIVE

  • Jumarie, Guy
    • Journal of applied mathematics & informatics
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    • 제26권5_6호
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    • pp.1101-1121
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    • 2008
  • One proposes an approach to fractional Fourier's transform, or Fourier's transform of fractional order, which applies to functions which are fractional differentiable but are not necessarily differentiable, in such a manner that they cannot be analyzed by using the so-called Caputo-Djrbashian fractional derivative. Firstly, as a preliminary, one defines fractional sine and cosine functions, therefore one obtains Fourier's series of fractional order. Then one defines the fractional Fourier's transform. The main properties of this fractal transformation are exhibited, the Parseval equation is obtained as well as the fractional Fourier inversion theorem. The prospect of application for this new tool is the spectral density analysis of signals, in signal processing, and the analysis of some partial differential equations of fractional order.

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FRACTIONAL HAMILTON-JACOBI EQUATION FOR THE OPTIMAL CONTROL OF NONRANDOM FRACTIONAL DYNAMICS WITH FRACTIONAL COST FUNCTION

  • Jumarie, Gyu
    • Journal of applied mathematics & informatics
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    • 제23권1_2호
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    • pp.215-228
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    • 2007
  • By using the variational calculus of fractional order, one derives a Hamilton-Jacobi equation and a Lagrangian variational approach to the optimal control of one-dimensional fractional dynamics with fractional cost function. It is shown that these two methods are equivalent, as a result of the Lagrange's characteristics method (a new approach) for solving non linear fractional partial differential equations. The key of this results is the fractional Taylor's series $f(x+h)=E_{\alpha}(h^{\alpha}D^{\alpha})f(x)$ where $E_{\alpha}(.)$ is the Mittag-Leffler function.

UPPER AND LOWER SOLUTION METHOD FOR FRACTIONAL EVOLUTION EQUATIONS WITH ORDER 1 < α < 2

  • Shu, Xiao-Bao;Xu, Fei
    • 대한수학회지
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    • 제51권6호
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    • pp.1123-1139
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    • 2014
  • In this work, we investigate the existence of the extremal solutions for a class of fractional partial differential equations with order 1 < ${\alpha}$ < 2 by upper and lower solution method. Using the theory of Hausdorff measure of noncompactness, a series of results about the solutions to such differential equations is obtained.

ANALYTIC TRAVELLING WAVE SOLUTIONS OF NONLINEAR COUPLED EQUATIONS OF FRACTIONAL ORDER

  • AN, JEONG HYANG;LEE, YOUHO
    • 호남수학학술지
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    • 제37권4호
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    • pp.411-421
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    • 2015
  • This paper investigates the issue of analytic travelling wave solutions for some important coupled models of fractional order. Analytic travelling wave solutions of the considered model are found by means of the Q-function method. The results give us that the Q-function method is very simple, reliable and effective for searching analytic exact solutions of complex nonlinear partial differential equations.

Image Denoising Based on Adaptive Fractional Order Anisotropic Diffusion

  • Yu, Jimin;Tan, Lijian;Zhou, Shangbo;Wang, Liping;Wang, Chaomei
    • KSII Transactions on Internet and Information Systems (TIIS)
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    • 제11권1호
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    • pp.436-450
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    • 2017
  • Recently, the method based on fractional order partial differential equation has been used in image processing. Usually, the optional order of fractional differentiation is determined by a lot of experiments. In this paper, a denoising model is proposed based on adaptive fractional order anisotropic diffusion. In the proposed model, the complexity of the local image texture is reflected by the local variance, and the order of the fractional differentiation is determined adaptively. In the process of the adaptive fractional order model, the discrete Fourier transform is applied to compute the fractional order difference as well as the dynamic evolution process. Experimental results show that the peak signal-to-noise ratio (PSNR) and structural similarity index measurement (SSIM) of the proposed image denoising algorithm is better than that of other some algorithms. The proposed algorithm not only can keep the detailed image information and edge information, but also obtain a good visual effect.

GENERALISED COMMON FIXED POINT THEOREM FOR WEAKLY COMPATIBLE MAPPINGS VIA IMPLICIT CONTRACTIVE RELATION IN QUASI-PARTIAL Sb-METRIC SPACE WITH SOME APPLICATIONS

  • Lucas Wangwe;Santosh Kumar
    • 호남수학학술지
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    • 제45권1호
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    • pp.1-24
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    • 2023
  • In the present paper, we prove common fixed point theorems for a pair of weakly compatible mappings under implicit contractive relation in quasi-partial Sb-metric spaces. We also provide an illustrative example to support our results. Furthermore, we will use the results obtained for application to two boundary value problems for the second-order differential equation. Also, we prove a common solution for the nonlinear fractional differential equation.

A STUDY OF A WEAK SOLUTION OF A DIFFUSION PROBLEM FOR A TEMPORAL FRACTIONAL DIFFERENTIAL EQUATION

  • Anakira, Nidal;Chebana, Zinouba;Oussaeif, Taki-Eddine;Batiha, Iqbal M.;Ouannas, Adel
    • Nonlinear Functional Analysis and Applications
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    • 제27권3호
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    • pp.679-689
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    • 2022
  • In this paper, we establish sufficient conditions for the existence and uniqueness of solution for a class of initial boundary value problems with Dirichlet condition in regard to a category of fractional-order partial differential equations. The results are established by a method based on the theorem of Lax Milligram.

BIFURCATION PROBLEM FOR A CLASS OF QUASILINEAR FRACTIONAL SCHRÖDINGER EQUATIONS

  • Abid, Imed
    • 대한수학회지
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    • 제57권6호
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    • pp.1347-1372
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    • 2020
  • We study bifurcation for the following fractional Schrödinger equation $$\{\left.\begin{eqnarray}(-{\Delta})^su+V(x)u&=&{\lambda}f(u)&&{\text{in}}\;{\Omega}\\u&>&0&&{\text{in}}\;{\Omega}\\u&=&0&&{\hspace{32}}{\text{in}}\;{\mathbb{R}}^n{\backslash}{\Omega}\end{eqnarray}\right$$ where 0 < s < 1, n > 2s, Ω is a bounded smooth domain of ℝn, (-∆)s is the fractional Laplacian of order s, V is the potential energy satisfying suitable assumptions and λ is a positive real parameter. The nonlinear term f is a positive nondecreasing convex function, asymptotically linear that is $\lim_{t{\rightarrow}+{\infty}}\;{\frac{f(t)}{t}}=a{\in}(0,+{\infty})$. We discuss the existence, uniqueness and stability of a positive solution and we also prove the existence of critical value and the uniqueness of extremal solutions. We take into account the types of Bifurcation problem for a class of quasilinear fractional Schrödinger equations, we also establish the asymptotic behavior of the solution around the bifurcation point.

BARRIER OPTIONS UNDER THE MFBM WITH JUMPS : APPLICATION OF THE BDF2 METHOD

  • Choi, Heungsu;Lee, Younhee
    • 충청수학회지
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    • 제33권1호
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    • pp.165-171
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    • 2020
  • In this paper we consider a mixed fractional Brownian motion (mfBm) with jumps. The prices of European barrier options can be evaluated by solving a partial integro-differential equation (PIDE) with variable coefficients, which is derived from the mfBm with jumps. The 2-step backward differentiation formula (BDF2 method) proposed in [6] is applied with the second-order convergence rate in the time and spatial variables. Numerical simulations are carried out to observe the convergence behaviors of the BDF2 method under the mfBm with the Kou model.