scholarly journals Approximation and application of the Riesz-Caputo fractional derivative of variable order with fixed memory

Meccanica ◽  
2021 ◽  
Author(s):  
Tomasz Blaszczyk ◽  
Krzysztof Bekus ◽  
Krzysztof Szajek ◽  
Wojciech Sumelka
Keyword(s):  
Differential Operator ◽  
Fractional Derivative ◽  
Rate Of Convergence ◽  
Continuum Mechanics ◽  
Caputo Fractional Derivative ◽  
Polynomial Interpolation ◽  
Piecewise Linear ◽  
Variable Order ◽  
Piecewise Constant ◽  
Quadratic Interpolation

AbstractIn this paper, the Riesz-Caputo fractional derivative of variable order with fixed memory is considered. The studied non-integer differential operator is approximated by means of modified basic rules of numerical integration. The three proposed methods are based on polynomial interpolation: piecewise constant, piecewise linear, and piecewise quadratic interpolation. The errors generated by the described methods and the experimental rate of convergence are reported. Finally, an application of the Riesz-Caputo fractional derivative of space-dependent order in continuum mechanics is depicted.

scholarly journals Boundary Layers in a Two-Point Boundary Value Problem with a Caputo Fractional Derivative

2015 ◽  
Vol 15 (1) ◽  
pp. 79-95 ◽  
Author(s):  
Martin Stynes ◽  
José Luis Gracia
Keyword(s):  
Boundary Layer ◽  
Differential Operator ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Order Term ◽  
Boundary Value ◽  
Point Boundary ◽  
Leading Term ◽  
Special Case

AbstractA two-point boundary value problem is considered on the interval $[0,1]$, where the leading term in the differential operator is a Caputo fractional derivative of order δ with $1<\delta <2$. Writing u for the solution of the problem, it is known that typically $u^{\prime \prime }(x)$ blows up as $x\rightarrow 0$. A numerical example demonstrates the possibility of a further phenomenon that imposes difficulties on numerical methods: u may exhibit a boundary layer at x = 1 when δ is near 1. The conditions on the data of the problem under which this layer appears are investigated by first solving the constant-coefficient case using Laplace transforms, determining precisely when a layer is present in this special case, then using this information to enlighten our examination of the general variable-coefficient case (in particular, in the construction of a barrier function for u). This analysis proves that usually no boundary layer can occur in the solution u at x = 0, and that the quantity $M = \max _{x\in [0,1]}b(x)$, where b is the coefficient of the first-order term in the differential operator, is critical: when $M<1$, no boundary layer is present when δ is near 1, but when M ≥ 1 then a boundary layer at x = 1 is possible. Numerical results illustrate the sharpness of most of our results.

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

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Karel Van Bockstal
Keyword(s):  
Weak Solution ◽  
Diffusion Equation ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Differential Operators ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Variable Order ◽  
Unique Weak Solution ◽  
Bounded Lipschitz Domain

AbstractIn 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 $u\in \operatorname{L}^{\infty } ((0,T),\operatorname{H}^{1}_{0}( \Omega ) )$ u ∈ L ∞ ( ( 0 , T ) , H 0 1 ( Ω ) ) to the problem if the initial data belongs to $\operatorname{H}^{1}_{0}(\Omega )$ H 0 1 ( Ω ) . We show that the solution belongs to $\operatorname{C} ([0,T],{\operatorname{H}^{1}_{0}(\Omega )}^{*} )$ C ( [ 0 , T ] , H 0 1 ( Ω ) ∗ ) in the case of a Caputo fractional derivative of constant order. We generalise a fundamental identity for integro-differential operators of the form $\frac{\mathrm{d}}{\mathrm{d}t} (k\ast v)(t)$ d d t ( 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.

scholarly journals Numerical evaluation of the interpolation accuracy of simple elementary functions

2018 ◽  
Vol 9 (4) ◽  
pp. 93-116
Author(s):  
Sergej Vital'evich Znamenskij
Keyword(s):  
Polynomial Interpolation ◽  
Piecewise Linear ◽  
Rational Functions ◽  
Linear Interpolation ◽  
Graphical Form ◽  
Elementary Functions ◽  
Advantages And Disadvantages ◽  
Piecewise Linear Interpolation ◽  
Interpolation Accuracy ◽  
Quadratic Interpolation

The paper contains a comparison of the accuracy of the restoration of elementary functions by the values in the nodes for algorithms of low-degree piecewise-polynomial interpolation. The test results demonstrate in graphical form the advantages and disadvantages of the widely used cubic interpolation splines. The comparison revealed that, contrary to popular belief, the smoothness of the interpolant is not directly related to the accuracy of the approximation. In the 20 different examples considered, the piecewise quadratic interpolation is rarely and only slightly inferior in the form of the used classical cubic splines, often by orders of magnitude better than many of them. In several examples, the high interpolation error of simple functions on a fixed grid appears to be almost independent of the degree of the algorithm and the smoothness of the interpolant. The piecewise-linear interpolation unexpectedly appeared the most accurate in one of the examples. A new problem arises: to find a local interpolation algorithm, accurately restoring any rational functions of the second order.

scholarly journals Numerical approximation of stochastic time-fractional diffusion

2019 ◽  
Vol 53 (4) ◽  
pp. 1245-1268 ◽  
Author(s):  
Bangti Jin ◽  
Yubin Yan ◽  
Zhi Zhou
Keyword(s):  
Fractional Derivative ◽  
Gaussian Noise ◽  
Caputo Fractional Derivative ◽  
Fractional Integral ◽  
Convergence Rates ◽  
Piecewise Linear ◽  
Fractional Diffusion ◽  
Strong And Weak Convergence ◽  
Standard Galerkin Method ◽  
Stochastic Time

We develop and analyze a numerical method for stochastic time-fractional diffusion driven by additive fractionally integrated Gaussian noise. The model involves two nonlocal terms in time, i.e., a Caputo fractional derivative of order α ∈ (0,1), and fractionally integrated Gaussian noise (with a Riemann-Liouville fractional integral of order γ ∈ [0,1] in the front). The numerical scheme approximates the model in space by the standard Galerkin method with continuous piecewise linear finite elements and in time by the classical Grünwald-Letnikov method (for both Caputo fractional derivative and Riemann-Liouville fractional integral), and the noise by the L2-projection. Sharp strong and weak convergence rates are established, using suitable nonsmooth data error estimates for the discrete solution operators for the deterministic inhomogeneous problem. One- and two-dimensional numerical results are presented to support the theoretical findings.

scholarly journals Comparison of Dynamical Behavior Between Fractional Order Delayed and Discrete Conformable Fractional Order Tumor-Immune System

2020 ◽  
Author(s):  
ERCAN BALCI ◽  
Senol KARTAL ◽  
İlhan ÖZTÜRK
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Bifurcation Analysis ◽  
Caputo Fractional Derivative ◽  
Discrete System ◽  
Nonlinear Differential Equations ◽  
Dynamical Behavior ◽  
Tumor Model ◽  
System Stability ◽  
Piecewise Constant

In this paper, we analyze the dynamical behavior of the delayed fractional-order tumor model with Caputo sense and discretized conformable fractional-order tumor model. The model is constituted with the group of nonlinear differential equations having effector and tumor cells. First of all, stability and bifurcation analysis of the delayed fractional-order tumor model in the sense of Caputo fractional derivative is studied, and the existence of Hopf bifurcation depending on the time delay parameter is proved by using center manifold and bifurcation theory. Applying the discretization process based on using the piecewise constant arguments to the conformable version of the model gives a two-dimensional discrete system. Stability and Neimark-Sacker bifurcation analysis of the discrete system are demonstrated using the Schur-Cohn criterion and projection method. This study reveals that the delay parameter $ \tau $ in the model with Caputo fractional derivative and the discretization parameter $ h $ in the discrete-time conformable fractional-order model have similar effects on the dynamical behavior of corresponding systems. Moreover, the effect of the order of fractional derivative on the dynamical behavior of the systems is discussed. Finally, all results obtained are interpreted biologically, and numerical simulations are presented to illustrate and support theoretical results.

scholarly journals A subinterval-based method for circuits with fractional order elements

2014 ◽  
Vol 62 (3) ◽  
pp. 449-454 ◽  
Author(s):  
M. Sowa
Keyword(s):  
Fractional Derivative ◽  
Numerical Method ◽  
Fractional Order ◽  
Caputo Fractional Derivative ◽  
Polynomial Interpolation ◽  
Time Interval ◽  
Traditional Methods ◽  
Linear Circuit ◽  
Entire Time ◽  
Analytical Formulae

Abstract The paper deals with the solution of problems that concern fractional time derivatives. Specifically the author’s interest lies in solving circuit problems with so called fractional capacitors and fractional inductors. A numerical method is proposed that involves polynomial interpolation and the division of the entire time interval (for which computations are performed) into subintervals. Analytical formulae are derived for the integro-differentiation according to the Caputo fractional derivative. The rules that concern the subinterval dynamics throughout the computation are also presented in the paper. For exemplary linear circuit problems (AC and transient) involving fractional order elements the solutions have been obtained. These solutions are compared with ones obtained by means of traditional methods

scholarly journals Variable-Order Fractional Models for Wall-Bounded Turbulent Flows

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 782
Author(s):  
Fangying Song ◽  
George Em Karniadakis
Keyword(s):  
Fractional Derivative ◽  
Turbulent Flows ◽  
Fractional Order ◽  
Variable Order ◽  
Universal Curve ◽  
Reynolds Stresses ◽  
Continuous Change ◽  
Mean Velocity ◽  
Symmetric Variable ◽  
Fractional Models

Modeling of wall-bounded turbulent flows is still an open problem in classical physics, with relatively slow progress in the last few decades beyond the log law, which only describes the intermediate region in wall-bounded turbulence, i.e., 30–50 y+ to 0.1–0.2 R+ in a pipe of radius R. Here, we propose a fundamentally new approach based on fractional calculus to model the entire mean velocity profile from the wall to the centerline of the pipe. Specifically, we represent the Reynolds stresses with a non-local fractional derivative of variable-order that decays with the distance from the wall. Surprisingly, we find that this variable fractional order has a universal form for all Reynolds numbers and for three different flow types, i.e., channel flow, Couette flow, and pipe flow. We first use existing databases from direct numerical simulations (DNSs) to lean the variable-order function and subsequently we test it against other DNS data and experimental measurements, including the Princeton superpipe experiments. Taken together, our findings reveal the continuous change in rate of turbulent diffusion from the wall as well as the strong nonlocality of turbulent interactions that intensify away from the wall. Moreover, we propose alternative formulations, including a divergence variable fractional (two-sided) model for turbulent flows. The total shear stress is represented by a two-sided symmetric variable fractional derivative. The numerical results show that this formulation can lead to smooth fractional-order profiles in the whole domain. This new model improves the one-sided model, which is considered in the half domain (wall to centerline) only. We use a finite difference method for solving the inverse problem, but we also introduce the fractional physics-informed neural network (fPINN) for solving the inverse and forward problems much more efficiently. In addition to the aforementioned fully-developed flows, we model turbulent boundary layers and discuss how the streamwise variation affects the universal curve.

scholarly journals On the sub–diffusion fractional initial value problem with time variable order

2021 ◽  
Vol 10 (1) ◽  
pp. 1301-1315
Author(s):  
Eduardo Cuesta ◽  
Mokhtar Kirane ◽  
Ahmed Alsaedi ◽  
Bashir Ahmad
Keyword(s):  
Asymptotic Behavior ◽  
Fractional Derivative ◽  
Initial Value Problem ◽  
Regularity Result ◽  
Time Variable ◽  
Variable Order ◽  
Initial Value ◽  
Integration By Parts ◽  
Integration By Parts Formula ◽  
Variable Order Fractional Derivative

Abstract We consider a fractional derivative with order varying in time. Then, we derive for it a Leibniz' inequality and an integration by parts formula. We also study an initial value problem with our time variable order fractional derivative and present a regularity result for it, and a study on the asymptotic behavior.

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