scholarly journals Conditional Lie-Bäcklund Symmetries and Differential Constraints of Radially Symmetric Nonlinear Convection-Diffusion Equations with Source

Entropy ◽  
2020 ◽  
Vol 22 (8) ◽  
pp. 873
Author(s):  
Lina Ji ◽  
Rui Wang
Keyword(s):  
Dynamical Systems ◽  
Diffusion Equations ◽  
Two Dimensional ◽  
Nonlinear Convection ◽  
Differential Constraint ◽  
Invariant Surface ◽  
Differential Constraints ◽  
Convection Diffusion ◽  
Constraint Method ◽  
Symmetry Method

A conditional Lie-Bäcklund symmetry method and differential constraint method are developed to study the radially symmetric nonlinear convection-diffusion equations with source. The equations and the admitted conditional Lie-Bäcklund symmetries (differential constraints) are identified. As a consequence, symmetry reductions to two-dimensional dynamical systems of the resulting equations are derived due to the compatibility of the original equation and the additional differential constraint corresponding to the invariant surface equation of the admitted conditional Lie-Bäcklund symmetry.

scholarly journals Second-Order Conditional Lie-Bäcklund Symmetry and Differential Constraint of Radially Symmetric Diffusion System

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Jianping Wang ◽  
Huijing Ba ◽  
Yaru Liu ◽  
Longqi He ◽  
Lina Ji
Keyword(s):  
Dynamical Systems ◽  
Exact Solutions ◽  
Surface Condition ◽  
Original System ◽  
Second Order ◽  
Diffusion System ◽  
Differential Constraint ◽  
Invariant Surface ◽  
Symmetric Diffusion ◽  
Symmetry Method

The classifications and reductions of radially symmetric diffusion system are studied due to the conditional Lie-Bäcklund symmetry method. We obtain the invariant condition, which is the so-called determining system and under which the radially symmetric diffusion system admits second-order conditional Lie-Bäcklund symmetries. The governing systems and the admitted second-order conditional Lie-Bäcklund symmetries are identified by solving the nonlinear determining system. Exact solutions of the resulting systems are constructed due to the compatibility of the original system and the admitted differential constraint corresponding to the invariant surface condition. For most of the cases, they are reduced to solving four-dimensional dynamical systems.

scholarly journals Approximate analytical fractional view of convection–diffusion equations

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 897-905
Author(s):  
Hassan Khan ◽  
Saima Mustafa ◽  
Izaz Ali ◽  
Poom Kumam ◽  
Dumitru Baleanu ◽  
...  
Keyword(s):  
Fractional Order ◽  
Variational Iteration Method ◽  
Absolute Error ◽  
Form Solution ◽  
Diffusion Equations ◽  
Nonlinear Convection ◽  
Convection Diffusion ◽  
Suggested Technique ◽  
First Time ◽  
Modified Variational Iteration Method

Abstract In this article, a modified variational iteration method along with Laplace transformation is used for obtaining the solution of fractional-order nonlinear convection–diffusion equations (CDEs). The proposed technique is applied for the first time to solve fractional-order nonlinear CDEs and attain a series-form solution with the quick rate of convergence. Tabular and graphical representations are presented to confirm the reliability of the suggested technique. The solutions are calculated for fractional as well as for integer orders of the problems. The solution graphs of the solutions at various fractional derivatives are plotted. The accuracy is measured in terms of absolute error. The higher degree of accuracy is observed from the table and figures. It is further investigated that fractional solutions have the convergence behavior toward the solution at integer order. The applicability of the present technique is verified by illustrative examples. The simple and effective procedure of the current technique supports its implementation to solve other nonlinear fractional problems in different areas of applied science.

scholarly journals Two-dimensional numerical modeling of chemical transport–transformation in fluvial rivers: formulation of equations and physical interpretation

2009 ◽  
Vol 11 (2) ◽  
pp. 106-118 ◽  
Author(s):  
Sui Liang Huang
Keyword(s):  
Sediment Transport ◽  
Chemical Transport ◽  
Transformation Model ◽  
Diffusion Equations ◽  
Desorption Kinetics ◽  
Two Dimensional ◽  
Transformation Equation ◽  
Governing Equations ◽  
Convection Diffusion ◽  
Kinetics Of

Based on previous work on the transport–transformation model of heavy metal pollutants in fluvial rivers, this paper presents the formulation of a two-dimensional model to describe chemical transport–transformation in fluvial rivers by considering basic principles of environmental chemistry, hydraulics and mechanics of sediment transport and recent developments along with three very simplified test cases. The model consists of water flow governing equations, sediment transport governing equations, transport–transformation equation of chemicals and convection–diffusion equations of sorption–desorption kinetics of particulate chemical concentrations on suspended load, bed load and bed sediment. The chemical transport–transformation equation is basically a mass balance equation. It demonstrates how sediment transport affects transport–transformation of chemicals in fluvial rivers. The convection–diffusion equations of sorption–desorption kinetics of chemicals, being an extension of batch reactor experimental results, take both physical transport, i.e. convection and diffusion, and chemical reactions, i.e. sorption–desorption into account. The effects of sediment transport on chemical transport–transformation were clarified through three simple examples. Specifically, the transport–transformation of chemicals in a steady, uniform and equilibrium sediment-laden flow was calculated by applying this model, and results were shown to be rational. Both theoretical analysis and numerical simulation indicated that the transport–transformation of chemicals in sediment-laden flows with a clay-enriched riverbed possesses not only the generality of common tracer pollutants, but also characteristics of transport–transformation induced by sediment motion. Future work will be conducted to present the validation/application of the model with available data.

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