scholarly journals Analytical solution of the time fractional diffusion equation and fractional convection-diffusion equation

2018 ◽  
Vol 65 (1) ◽  
pp. 82 ◽  
Author(s):  
Francisco Gomez ◽  
Victor Morales ◽  
Marco Taneco
Keyword(s):  
Diffusion Equation ◽  
Analytical Solutions ◽  
Fourier Transforms ◽  
Time Derivative ◽  
Fractional Diffusion ◽  
Transport Processes ◽  
Diffusion Equations ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Diffusion And Convection

In this paper, we obtain analytical solutions for the time-fractional diffusion and time-fractional convection-diffusion equations. These equations are obtained from the standard equations by replacing the time derivative with a fractional derivative of order $\alpha$. Fractional operators of type Liouville-Caputo, Atangana-Baleanu-Caputo, fractional conformable derivative in Liouville-Caputo sense and Atangana-Koca-Caputo were used to model diffusion and convection-diffusion equation. The Laplace and Fourier transforms were applied to obtain the analytical solutions for the fractional order diffusion and convection-diffusion equations. The solutions obtained can be useful to understand the modeling of anomalous diffusive, subdiffusive systems and super-diffusive systems, transport processes, random walk and wave propagation phenomenon.

Asymptotically Self-Similar Solution for the Convection-Diffusion Equation

2014 ◽  
Vol 623 ◽  
pp. 97-100 ◽  
Author(s):  
Xue Yu ◽  
Qiu Sheng Zhang
Keyword(s):  
Diffusion Equation ◽  
Global Solutions ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Self Similar Solution ◽  
The Cauchy Problem ◽  
Self Similar ◽  
Diffusion And Convection ◽  
Physical Phenomena ◽  
Physical Quantities

We study the Cauchy problem for the convection-diffusion equation, which describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to diffusion and convection processes. For, we show the continuous dependence upon the initial data. Moreover, asymptotically self–similar global solutions are investigated with nonhomogeneous initial date.

scholarly journals Flow regimes for fluid injection into a confined porous medium

2015 ◽  
Vol 767 ◽  
pp. 881-909 ◽  
Author(s):  
Zhong Zheng ◽  
Bo Guo ◽  
Ivan C. Christov ◽  
Michael A. Celia ◽  
Howard A. Stone
Keyword(s):  
Porous Medium ◽  
Diffusion Equation ◽  
Governing Equation ◽  
Analytical Solutions ◽  
Nonlinear Diffusion ◽  
Flow Behaviour ◽  
Late Time ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Time Period

AbstractWe report theoretical and numerical studies of the flow behaviour when a fluid is injected into a confined porous medium saturated with another fluid of different density and viscosity. For a two-dimensional configuration with point source injection, a nonlinear convection–diffusion equation is derived to describe the time evolution of the fluid–fluid interface. In the early time period, the fluid motion is mainly driven by the buoyancy force and the governing equation is reduced to a nonlinear diffusion equation with a well-known self-similar solution. In the late time period, the fluid flow is mainly driven by the injection, and the governing equation is approximated by a nonlinear hyperbolic equation that determines the global spreading rate; a shock solution is obtained when the injected fluid is more viscous than the displaced fluid, whereas a rarefaction wave solution is found when the injected fluid is less viscous. In the late time period, we also obtain analytical solutions including the diffusive term associated with the buoyancy effects (for an injected fluid with a viscosity higher than or equal to that of the displaced fluid), which provide the structure of the moving front. Numerical simulations of the convection–diffusion equation are performed; the various analytical solutions are verified as appropriate asymptotic limits, and the transition processes between the individual limits are demonstrated. The flow behaviour is summarized in a diagram with five distinct dynamical regimes: a nonlinear diffusion regime, a transition regime, a travelling wave regime, an equal-viscosity regime, and a rarefaction regime.

scholarly journals Space–Time Radial Basis Function–Based Meshless Approach for Solving Convection–Diffusion Equations

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1735
Author(s):  
Cheng-Yu Ku ◽  
Jing-En Xiao ◽  
Chih-Yu Liu
Keyword(s):  
Diffusion Equation ◽  
Radial Basis Function ◽  
Time Domain ◽  
Basis Function ◽  
Numerical Solutions ◽  
Space Time ◽  
Diffusion Equations ◽  
Numerical Examples ◽  
Convection Diffusion ◽  
Convection Diffusion Equation

This article proposes a space–time meshless approach based on the transient radial polynomial series function (TRPSF) for solving convection–diffusion equations. We adopted the TRPSF as the basis function for the spatial and temporal discretization of the convection–diffusion equation. The TRPSF is constructed in the space–time domain, which is a combination of n–dimensional Euclidean space and time into an n + 1–dimensional manifold. Because the initial and boundary conditions were applied on the space–time domain boundaries, we converted the transient problem into an inverse boundary value problem. Additionally, all partial derivatives of the proposed TRPSF are a series of continuous functions, which are nonsingular and smooth. Solutions were approximated by solving the system of simultaneous equations formulated from the boundary, source, and internal collocation points. Numerical examples including stationary and nonstationary convection–diffusion problems were employed. The numerical solutions revealed that the proposed space–time meshless approach may achieve more accurate numerical solutions than those obtained by using the conventional radial basis function (RBF) with the time-marching scheme. Furthermore, the numerical examples indicated that the TRPSF is more stable and accurate than other RBFs for solving the convection–diffusion equation.

scholarly journals New High-Order Compact ADI Algorithms for 3D Nonlinear Time-Fractional Convection-Diffusion Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Shuying Zhai ◽  
Xinlong Feng ◽  
Zhifeng Weng
Keyword(s):  
Diffusion Equation ◽  
Time Derivative ◽  
Three Dimensional ◽  
Cost Effective ◽  
Linearization Method ◽  
High Order ◽  
Thomas Algorithm ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Alternating Direction

Numerical approximations of the three-dimensional (3D) nonlinear time-fractional convection-diffusion equation is studied, which is firstly transformed to a time-fractional diffusion equation and then is solved by linearization method combined with alternating direction implicit (ADI) method. By using fourth-order Padé approximation for spatial derivatives and classical backward differentiation method for time derivative, two new high-order compact ADI algorithms with ordersO(τmin(1+α,2−α)+h4)andO(τ2−α+h4)are presented. The resulting schemes in each ADI solution step corresponding to a tridiagonal matrix equation can be solved by the Thomas algorithm which makes the computation cost effective. Numerical experiments are shown to demonstrate the high accuracy and robustness of two new schemes.

scholarly journals Novel Cubic Trigonometric B-Spline Approach Based on the Hermite Formula for Solving the Convection-Diffusion Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Aatika Yousaf ◽  
Thabet Abdeljawad ◽  
Muhammad Yaseen ◽  
Muhammad Abbas
Keyword(s):  
Approximate Solution ◽  
Diffusion Equation ◽  
Time Derivative ◽  
High Accuracy ◽  
Spline Method ◽  
B Spline ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Numerical Tests ◽  
Piecewise Continuous

This paper introduces a cubic trigonometric B-spline method (CuTBM) based on the Hermite formula for numerically handling the convection-diffusion equation (CDE). The method utilizes a merger of the CuTBM and the Hermite formula for the approximation of a space derivative, while the time derivative is discretized using a finite difference scheme. This combination has greatly enhanced the accuracy of the scheme. A stability analysis of the scheme is also presented to confirm that the errors do not magnify. The main advantage of the scheme is that the approximate solution is obtained as a smooth piecewise continuous function empowering us to approximate a solution at any location in the domain of interest with high accuracy. Numerical tests are performed, and the outcomes are compared with the ones presented previously to show the superiority of the presented scheme.

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