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.

A conditional Lipschitz stability for determining a space-dependent source coefficient in the 2D/3D advection-dispersion equation

2017 ◽  
Vol 25 (2) ◽  
Author(s):  
Gongsheng Li ◽  
Xianzheng Jia ◽  
Chunlong Sun
Keyword(s):  
Inverse Problem ◽  
Dispersion Equation ◽  
Adjoint Method ◽  
Noisy Data ◽  
Adjoint Problem ◽  
Admissible Set ◽  
Lipschitz Stability ◽  
Optimal Perturbation ◽  
Advection Dispersion ◽  
Bilinear Functional

Abstract This paper deals with an inverse problem of determining a space-dependent source coefficient in the 2D/3D advection-dispersion equation with final observations using the variational adjoint method. Data compatibility for the inverse problem is analyzed by which an admissible set for the unknowns is induced. With the aid of an adjoint problem, a bilinear functional based on the variational identity is set forth with which a norm for the unknown is well-defined under suitable conditions, and then a conditional Lipschitz stability for the inverse problem is established. Furthermore, numerical inversions with random noisy data are performed using the optimal perturbation algorithm, and the inversion solutions give good approximations to the exact solution as the noise level goes to small.

Direct and inverse source problems for a space fractional advection dispersion equation

2017 ◽  
Vol 25 (2) ◽  
Author(s):  
Abeer Aldoghaither ◽  
Taous-Meriem Laleg-Kirati ◽  
Da-Yan Liu
Keyword(s):  
Inverse Problem ◽  
Dispersion Equation ◽  
Direct Problem ◽  
Source Term ◽  
Analytic Solution ◽  
Finite Domain ◽  
Inverse Source Problem ◽  
Direct And Inverse Problems ◽  
Advection Dispersion ◽  
Inverse Source Problems

Abstract In this paper, direct and inverse problems for a space fractional advection dispersion equation on a finite domain are studied. The inverse problem consists in determining the source term from final observations. We first derive the analytic solution to the direct problem which we use to prove the uniqueness and the unstability of the inverse source problem using final measurements. Finally, we illustrate the results with a numerical example.

scholarly journals Analytical Solution of 1D Advection-Dispersion Equation In Finite Groundwater Reservoir With Spatially Dependent Dispersivity And Two Inputs Sources With Source-Sink Term Impact

2021 ◽  
Author(s):  
Thomas TJOCK-MBAGA ◽  
Patrice Ele Abiama ◽  
Jean Marie Ema'a Ema'a ◽  
Germain Hubert Ben-Bolie
Keyword(s):  
Analytical Solution ◽  
Linear Function ◽  
Dispersion Equation ◽  
Source Term ◽  
Numerical Solutions ◽  
Integral Transform ◽  
Liouville Problem ◽  
Finite Domain ◽  
Advection Dispersion ◽  
Sink Term

Abstract This study derives an analytical solution of a one-dimensional (1D) advection-dispersion equation (ADE) for solute transport with two contaminant sources that takes into account the source term. For a heterogeneous medium, groundwater velocity is considered as a linear function while the dispersion as a nth-power of linear function of space and analytical solutions are obtained for and . The solution in a heterogeneous finite domain with unsteady coefficients is obtained using the Generalized Integral Transform Technique (GITT) with a new regular Sturm-Liouville Problem (SLP). The solutions are validated with the numerical solutions obtained using MATLAB pedpe solver and the existing solution from the proposed solutions. We exanimated the influence of the source term, the heterogeneity parameters and the unsteady coefficient on the solute concentration distribution. The results show that the source term produces a solute build-up while the heterogeneity level decreases the concentration level in the medium. As an illustration, model predictions are used to estimate the time histories of the radiological doses of uranium at different distances from the sources boundary in order to understand the potential radiological impact on the general public.

scholarly journals Improved reproducing kernel method to solve space-time fractional advection-dispersion equation

2021 ◽  
Author(s):  
Tofigh Allahviranloo ◽  
Hussein Sahihi ◽  
Soheil Salahshour ◽  
D. Baleanu
Keyword(s):  
Dispersion Equation ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
Space Time ◽  
Finite Domain ◽  
Reproducing Kernel Method ◽  
Advection Dispersion ◽  
Schmidt Orthogonalization

In this paper, we consider the Space-Time Fractional Advection-Dispersion equation on a finite domain with variable coefficients. Fractional Advection- Dispersion equation as a model for transporting heterogeneous subsurface media as one approach to the modeling of the generally non-Fickian behavior of transport. We use a semi-analytical method as Reproducing kernel Method to solve the Space-Time Fractional Advection-Dispersion equation so that we can get better approximate solutions than the methods with which this problem has been solved. The main obstacle to solve this problem is the existence of a Gram-Schmidt orthogonalization process in the general form of the reproducing kernel method, which is very time-consuming. So, we introduce the Improved Reproducing Kernel Method, which is a different implementation for the general form of the reproducing kernel method. In this method, the Gram-Schmidt orthogonalization process is eliminated to significantly reduce the CPU-time. Also, the present method increases the accuracy of approximate solutions.

Numerical simulation and parameters inversion for non-symmetric two-sided fractional advection-dispersion equations

2016 ◽  
Vol 24 (1) ◽  
Author(s):  
Xianzheng Jia ◽  
Gongsheng Li
Keyword(s):  
Fractional Order ◽  
Neumann Boundary ◽  
Coefficient Matrix ◽  
Stability And Convergence ◽  
Finite Domain ◽  
Dispersion Coefficients ◽  
Optimal Perturbation ◽  
Advection Dispersion ◽  
Definition Of ◽  
Parameters Inversion

AbstractThis paper deals with numerical solution and parameters inversion for a one-dimensional non-symmetric two-sided fractional advection-dispersion equation (FADE) with zero Neumann boundary condition in a finite domain. A fully discretized finite difference scheme is set forth based on Grünwald–Letnikov's definition of the fractional derivative, and its stability and convergence are proved by estimating the spectral radius of the coefficient matrix of the scheme. Furthermore, an inverse problem of simultaneously determining the fractional order and the dispersion coefficients is investigated, and numerical inversions are carried out by using the optimal perturbation regularization algorithm. The inversion results show that the fractional order and the dispersion coefficients in the FADE can be determined successfully by the final observations.

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 Conditional Stability for an Inverse Problem of Determining a Space-Dependent Source Coefficient in the Advection-Dispersion Equation with Robin’s Boundary Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Shunqin Wang ◽  
Chunlong Sun ◽  
Gongsheng Li
Keyword(s):  
Boundary Condition ◽  
Inverse Problem ◽  
Dispersion Equation ◽  
Integral Identity ◽  
Adjoint Problem ◽  
Conditional Stability ◽  
Admissible Set ◽  
Lipschitz Stability ◽  
One Dimensional ◽  
Advection Dispersion

This paper deals with an inverse problem of determining the space-dependent source coefficient in one-dimensional advection-dispersion equation with Robin’s boundary condition. Data compatibility for the inverse problem is analyzed by which an admissible set for the unknown is set forth. Furthermore, with the help of an integral identity, a conditional Lipschitz stability is established by suitably controlling the solution of an adjoint problem.

Sign in / Sign up

Export Citation Format

Share Document