scholarly journals A Simultaneous Inversion Problem for the Variable-Order Time Fractional Differential Equation with Variable Coefficient

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Shengnan Wang ◽  
Zhendong Wang ◽  
Gongsheng Li ◽  
Yingmei Wang
Keyword(s):  
Inverse Problem ◽  
Noisy Data ◽  
Fractional Diffusion ◽  
Basis Functions ◽  
Variable Order ◽  
Inversion Problem ◽  
Inversion Algorithm ◽  
Regularization Algorithm ◽  
Simultaneous Inversion ◽  
Fractional Differential

This paper deals with an inverse problem of simultaneously determining the space-dependent diffusion coefficient and the fractional order in the variable-order time fractional diffusion equation by the measurements at one interior point. Numerical solution to the forward problem is given by the finite difference scheme, and the homotopy regularization algorithm is applied to solve the inverse problem utilizing Legendre polynomials as the basis functions of the approximate space. The inversion solutions with noisy data which give good approximations to the exact solution demonstrate effectiveness of the inversion algorithm for the simultaneous inversion problem.

scholarly journals Numerical Inversion for the Multiple Fractional Orders in the Multiterm TFDE

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Chunlong Sun ◽  
Gongsheng Li ◽  
Xianzheng Jia
Keyword(s):  
Numerical Stability ◽  
Noisy Data ◽  
Fractional Diffusion ◽  
Exact Order ◽  
Inversion Problem ◽  
Inversion Algorithm ◽  
Regularization Algorithm ◽  
Fractional Orders ◽  
Diffusion Behaviors ◽  
Key Parameter

The fractional order in a fractional diffusion model is a key parameter which characterizes the anomalous diffusion behaviors. This paper deals with an inverse problem of determining the multiple fractional orders in the multiterm time-fractional diffusion equation (TFDE for short) from numerics. The homotopy regularization algorithm is applied to solve the inversion problem using the finite data at one interior point in the space domain. The inversion fractional orders with random noisy data give good approximations to the exact order demonstrating the efficiency of the inversion algorithm and numerical stability of the inversion problem.

Numerical Inversion for the Initial Distribution in the Multi-Term Time-Fractional Diffusion Equation Using Final Observations

2017 ◽  
Vol 9 (6) ◽  
pp. 1525-1546 ◽  
Author(s):  
Chunlong Sun ◽  
Gongsheng Li ◽  
Xianzheng Jia
Keyword(s):  
Diffusion Equation ◽  
Initial Distribution ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Numerical Inversion ◽  
Inversion Problem ◽  
Inversion Algorithm ◽  
Regularization Algorithm ◽  
Lower Semi ◽  
Time Fractional Diffusion Equation

AbstractThis article deals with numerical inversion for the initial distribution in the multi-term time-fractional diffusion equation using final observations. The inversion problem is of instability, but it is uniquely solvable based on the solution's expression for the forward problem and estimation to the multivariate Mittag-Leffler function. From view point of optimality, solving the inversion problem is transformed to minimizing a cost functional, and existence of a minimum is proved by the weakly lower semi-continuity of the functional. Furthermore, the homotopy regularization algorithm is introduced based on the minimization problem to perform numerical inversions, and the inversion solutions with noisy data give good approximations to the exact initial distribution demonstrating the efficiency of the inversion algorithm.

scholarly journals Numerical Solution of Variable-Order Fractional Differential Equations Using Bernoulli Polynomials

2021 ◽  
Vol 5 (4) ◽  
pp. 219
Author(s):  
Somayeh Nemati ◽  
Pedro M. Lima ◽  
Delfim F. M. Torres
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Unknown Function ◽  
Operational Matrix ◽  
Bernoulli Polynomials ◽  
High Accuracy ◽  
Basis Functions ◽  
Variable Order ◽  
Algebraic Equations ◽  
Fractional Differential

We introduce a new numerical method, based on Bernoulli polynomials, for solving multiterm variable-order fractional differential equations. The variable-order fractional derivative was considered in the Caputo sense, while the Riemann–Liouville integral operator was used to give approximations for the unknown function and its variable-order derivatives. An operational matrix of variable-order fractional integration was introduced for the Bernoulli functions. By assuming that the solution of the problem is sufficiently smooth, we approximated a given order of its derivative using Bernoulli polynomials. Then, we used the introduced operational matrix to find some approximations for the unknown function and its derivatives. Using these approximations and some collocation points, the problem was reduced to the solution of a system of nonlinear algebraic equations. An error estimate is given for the approximate solution obtained by the proposed method. Finally, five illustrative examples were considered to demonstrate the applicability and high accuracy of the proposed technique, comparing our results with the ones obtained by existing methods in the literature and making clear the novelty of the work. The numerical results showed that the new method is efficient, giving high-accuracy approximate solutions even with a small number of basis functions and when the solution to the problem is not infinitely differentiable, providing better results and a smaller number of basis functions when compared to state-of-the-art methods.

scholarly journals Variable-Order Fractional Diffusion Model-Based Medical Image Denoising

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
A. Abirami ◽  
P. Prakash ◽  
Y.-K. Ma
Keyword(s):  
Diffusion Model ◽  
Scattering Theory ◽  
Medical Image ◽  
Signal To Noise Ratio ◽  
Fractional Diffusion ◽  
Vital Role ◽  
Variable Order ◽  
Order Model ◽  
Proposed Model ◽  
Fractional Differential

Fractional differential models are playing a vital role in many applications such as diffusion, probability potential theory, and scattering theory. In this study, the variable-order space and time fractional diffusion model is employed for denoising the medical images. The finite difference approach is implemented to find the numerical solution of the proposed model. Convergence and stability of the numerical method are presented. The experimental outcomes of the variable-order model are analyzed with those of the fractional and integer-order diffusion models. It was noticed that the peak signal-to-noise ratio (PSNR) value is increased considerably for the proposed model.

scholarly journals Numerical Identification of Multiparameters in the Space Fractional Advection Dispersion Equation by Final Observations

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Dali Zhang ◽  
Gongsheng Li ◽  
Guangsheng Chi ◽  
Xianzheng Jia ◽  
Huiling Li
Keyword(s):  
Inverse Problem ◽  
Diffusion Coefficient ◽  
Dispersion Equation ◽  
Average Velocity ◽  
Noisy Data ◽  
Finite Domain ◽  
Accurate Data ◽  
Regularization Algorithm ◽  
Optimal Perturbation ◽  
Advection Dispersion

This paper deals with an inverse problem for identifying multiparameters in 1D space fractional advection dispersion equation (FADE) on a finite domain with final observations. The parameters to be identified are the fractional order, the diffusion coefficient, and the average velocity in the FADE. The forward problem is solved by a finite difference scheme, and then an optimal perturbation regularization algorithm is introduced to determine the three parameters simultaneously. Numerical inversions are performed both with the accurate data and noisy data, and several factors having influences on realization of the algorithm are discussed. The inversion solutions are in good approximations to the exact solutions demonstrating the efficiency of the proposed algorithm.

scholarly journals Implicit nonlinear fractional differential equations of variable order

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Amar Benkerrouche ◽  
Mohammed Said Souid ◽  
Kanokwan Sitthithakerngkiet ◽  
Ali Hakem
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Variable Order ◽  
Stability Of Solutions ◽  
Nonlinear Fractional Differential Equations ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
The Stability

AbstractIn this manuscript, we examine both the existence and the stability of solutions to the implicit boundary value problem of Caputo fractional differential equations of variable order. We construct an example to illustrate the validity of the observed results.

scholarly journals A predator–prey model involving variable-order fractional differential equations with Mittag-Leffler kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Aziz Khan ◽  
Hashim M. Alshehri ◽  
J. F. Gómez-Aguilar ◽  
Zareen A. Khan ◽  
G. Fernández-Anaya
Keyword(s):  
Fractional Order ◽  
Existence And Uniqueness ◽  
Variable Order ◽  
Fractional Order Systems ◽  
New Approach ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Fractional Differential ◽  
The Fixed Point Theorem

AbstractThis paper is about to formulate a design of predator–prey model with constant and time fractional variable order. The predator and prey act as agents in an ecosystem in this simulation. We focus on a time fractional order Atangana–Baleanu operator in the sense of Liouville–Caputo. Due to the nonlocality of the method, the predator–prey model is generated by using another FO derivative developed as a kernel based on the generalized Mittag-Leffler function. Two fractional-order systems are assumed, with and without delay. For the numerical solution of the models, we not only employ the Adams–Bashforth–Moulton method but also explore the existence and uniqueness of these schemes. We use the fixed point theorem which is useful in describing the existence of a new approach with a particular set of solutions. For the illustration, several numerical examples are added to the paper to show the effectiveness of the numerical method.

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