scholarly journals Coupling kinetic models and advection-diffusion equations. 1. Framework development and application to sucrose translocation and metabolism in sugarcane

2021 ◽  
Author(s):  
Lafras Uys ◽  
Jan-Hendrik S Hofmeyr ◽  
Johann M Rohwer
Keyword(s):  
Differential Equations ◽  
Kinetic Models ◽  
Numerical Solutions ◽  
Spatial Information ◽  
Initial Conditions ◽  
Reaction Rates ◽  
Diffusion Reaction ◽  
Sucrose Accumulation ◽  
Fluid Medium ◽  
Advection Diffusion

Abstract The sugarcane stalk, besides being the main structural component of the plant, is also the major storage organ for carbohydrates. Previous studies have modelled the sucrose accumulation pathway in the internodal storage parenchyma of sugarcane using kinetic models cast as systems of ordinary differential equations. To address the shortcomings of these models, which did not include subcellular compartmentation or spatial information, the present study extends the original models within an advection-diffusion-reaction framework, requiring the use of partial differential equations to model sucrose metabolism coupled to phloem translocation.We propose a kinetic model of a coupled reaction network where species can be involved in chemical reactions and/or be transported over long distances in a fluid medium by advection or diffusion. Darcy’s law is used to model fluid flow and allows a simplified, phenomenological approach to be applied to translocation in the phloem. Similarly, generic reversible Hill equations are used to model biochemical reaction rates. Numerical solutions to this formulation are demonstrated with time-course analysis of a simplified model of sucrose accumulation. The model shows sucrose accumulation in the vacuoles of stalk parenchyma cells, and is moreover able to demonstrate the up-regulation of photosynthesis in response to a change in sink demand. The model presented is able to capture the spatio-temporal evolution of the system from a set of initial conditions by combining phloem flow, diffusion, transport of metabolites between compartments and biochemical enzyme-catalysed reactions in a rigorous, quantitative framework that can form the basis for future modelling and experimental design.

An efficient Bi-cubic B-spline ADI method for numerical solutions of two-dimensional unsteady advection diffusion equations

2018 ◽  
Vol 28 (11) ◽  
pp. 2620-2649 ◽  
Author(s):  
Rajni Rohila ◽  
R.C. Mittal
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Diffusion Equations ◽  
Spline Functions ◽  
Two Dimensional ◽  
Partial Differential ◽  
Content Type ◽  
B Spline ◽  
Advection Diffusion

Purpose This paper aims to develop a novel numerical method based on bi-cubic B-spline functions and alternating direction (ADI) scheme to study numerical solutions of advection diffusion equation. The method captures important properties in the advection of fluids very efficiently. C.P.U. time has been shown to be very less as compared with other numerical schemes. Problems of great practical importance have been simulated through the proposed numerical scheme to test the efficiency and applicability of method. Design/methodology/approach A bi-cubic B-spline ADI method has been proposed to capture many complex properties in the advection of fluids. Findings Bi-cubic B-spline ADI technique to investigate numerical solutions of partial differential equations has been studied. Presented numerical procedure has been applied to important two-dimensional advection diffusion equations. Computed results are efficient and reliable, have been depicted by graphs and several contour forms and confirm the accuracy of the applied technique. Stability analysis has been performed by von Neumann method and the proposed method is shown to satisfy stability criteria unconditionally. In future, the authors aim to extend this study by applying more complex partial differential equations. Though the structure of the method seems to be little complex, the method has the advantage of using small processing time. Consequently, the method may be used to find solutions at higher time levels also. Originality/value ADI technique has never been applied with bi-cubic B-spline functions for numerical solutions of partial differential equations.

Key words: Nonlinear Differential-Difference Equations; Exp-Function Method; N-Soliton Solutions

2010 ◽  
Vol 65 (11) ◽  
pp. 935-949 ◽  
Author(s):  
Mehdi Dehghan ◽  
Jalil Manafian ◽  
Abbas Saadatmandi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Analysis Method ◽  
Partial Differential ◽  
Integer Order ◽  
Homotopy Analysis

In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the homotopy analysis method for partial differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.

scholarly journals Numerical Solutions of Fractional Differential Equations Arising in Engineering Sciences

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 215 ◽  
Author(s):  
Alessandra Jannelli
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Diffusion Reaction ◽  
Unconditionally Stable ◽  
Advection Diffusion ◽  
Engineering Sciences ◽  
Hydrodynamic Processes ◽  
Fractional Differential

This paper deals with the numerical solutions of a class of fractional mathematical models arising in engineering sciences governed by time-fractional advection-diffusion-reaction (TF–ADR) equations, involving the Caputo derivative. In particular, we are interested in the models that link chemical and hydrodynamic processes. The aim of this paper is to propose a simple and robust implicit unconditionally stable finite difference method for solving the TF–ADR equations. The numerical results show that the proposed method is efficient, reliable and easy to implement from a computational viewpoint and can be employed for engineering sciences problems.

scholarly journals Controllability for Fuzzy Fractional Evolution Equations in Credibility Space

2021 ◽  
Vol 5 (3) ◽  
pp. 112
Author(s):  
Azmat Ullah Khan Niazi ◽  
Naveed Iqbal ◽  
Rasool Shah ◽  
Fongchan Wannalookkhee ◽  
Kamsing Nonlaopon
Keyword(s):  
Differential Equations ◽  
Evolution Equations ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Exact Controllability ◽  
Fractional Evolution Equations ◽  
Liu Process ◽  
Bounded Space ◽  
Fractional Differential ◽  
Derivatives Of

This article addresses exact controllability for Caputo fuzzy fractional evolution equations in the credibility space from the perspective of the Liu process. The class or problems considered here are Caputo fuzzy differential equations with Caputo derivatives of order β∈(1,2), 0CDtβu(t,ζ)=Au(t,ζ)+f(t,u(t,ζ))dCt+Bx(t)Cx(t)dt with initial conditions u(0)=u0,u′(0)=u1, where u(t,ζ) takes values from U(⊂EN),V(⊂EN) is the other bounded space, and EN represents the set of all upper semi-continuously convex fuzzy numbers on R. In addition, several numerical solutions have been provided to verify the correctness and effectiveness of the main result. Finally, an example is given, which expresses the fuzzy fractional differential equations.

ON THE ISSUE OF PREDICTING THE ADVANCEMENT OF UNIVERSITIES IN GLOBAL UNIVERSITY RANKINGS

2021 ◽  
Vol 1 (2) ◽  
pp. 15-28
Author(s):  
S. V. Blazhevich ◽  
◽  
V. M. Moskovkin ◽  
He Zhang ◽  
◽  
...  
Keyword(s):  
Population Dynamics ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Original System ◽  
Analytic Solutions ◽  
Volterra Equations ◽  
University Ranking ◽  
Simplified Approach ◽  
Integral Indicator

A simplified approach to solving the equations of population dynamics (Lotka–Volterra equations), which is a nonlinear multidimensional system of ordinary differential equations of the first order, describing the competitive interaction of universities included in some world university ranking, is proposed. The phase variables in these equations are the values of the integral indicator of the university ranking, which is called Overall or Total Score. The simplification consists in reducing this system to a system of independent Verhulst equations with analytic solutions in exponents of time and passing from them to stationary solutions when time tends to infinity. It is shown that with this approach and a given growth rate Overall (Total) Score, it is possible to find symmetric coefficients of interuniversity competition for no more than three competing universities. When finding such coefficients for the first three universities in the THE ranking, numerical solutions of the original system of population dynamics equations were obtained using the Runge–Kutta method in MatLab. It is shown that the development of this approach, based on the equations of population dynamics, can consist in turning to the concept of competitive – cooperative university interactions. The system of differential equations describes the process of changing the integral indicator during the period between two ratings. Using the found values of the coefficients of interuniversity competition, the system is solved sequentially for all stages of the ranking, and the decisions at the previous stage are used as the initial conditions for the next one.

scholarly journals Coupling kinetic models and advection-diffusion equations. 2. Sensitivity analysis of an advection-diffusion-reaction model

2021 ◽  
Author(s):  
Lafras Uys ◽  
Jan-Hendrik S Hofmeyr ◽  
Johann M Rohwer
Keyword(s):  
Sensitivity Analysis ◽  
Initial Conditions ◽  
Phloem Loading ◽  
Reaction Model ◽  
Diffusion Reaction ◽  
Phloem Unloading ◽  
Advection Diffusion ◽  
Steady State Model ◽  
Sensitivity Analysis Method ◽  
Spatio Temporal

Abstract The accompanying paper (Uys et al., in silico Plants, XXXX) presented a core model of sucrose accumulation within the advection-diffusion-reaction framework, which is able to capture the spatio-temporal evolution of the system from a set of initial conditions. This paper presents a sensitivity analysis of this model. Because this is a non-steady-state model based on partial differential equations, we performed the sensitivity analysis using two approaches from engineering. The Morris method is based on a one-at-a-time design, perturbing parameters individually and calculating the influence on model output in terms of elementary effects. FAST is a global sensitivity analysis method, where all parameters are perturbed simultaneously, oscillating at different frequencies, enabling the calculation of the contribution of each parameter through Fourier analysis. Overall, both methods gave similar results. Perturbations in reactions tended to have a large influence on their own rate, as well as on directly connected metabolites. Sensitivities varied both with the time of the simulation and the position along the sugarcane stalk. Our results suggest that vacuolar sucrose concentrations are most sensitive to vacuolar invertase in the centre of the stalk, but that phloem unloading and vacuolar sucrose uptake also contribute, especially towards the stalk edges. Sucrose in the phloem was most sensitive to phloem loading at the nodes, but most sensitive to phloem unloading in the middle of the internodes. Sink concentrations of sucrose in the symplast were most sensitive to phloem unloading in the middle of the internodes, but at the nodes cytosolic invertase had the greatest effect.

scholarly journals Construction a distributed order smoking model and its nonstandard finite difference discretization

2021 ◽  
Vol 7 (3) ◽  
pp. 4636-4654
Author(s):  
Mehmet Kocabiyik ◽  
◽  
Mevlüde Yakit Ongun ◽  
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Health Problems ◽  
Potential Health ◽  
Fractional Differential ◽  
Distributed Order ◽  
Nonstandard Finite Difference

<abstract><p>Smoking is currently one of the most important health problems in the world and increases the risk of developing diseases. For these reasons, it is important to determine the effects of smoking on humans. In this paper, we discuss a new system of distributed order fractional differential equations of the smoking model. With the use of distributed order fractional differential equations, it is possible to solve both ordinary and fractional-order equations. We can make these solutions with the density function included in the definition of the distributed order fractional differential equation. We construct the Nonstandard Finite Difference (NSFD) schemes to obtain numerical solutions of this model. Positivity solutions are preserved under positive initial conditions with this discretization method. Also, since NSFD schemes can preserve all the properties of the continuous models for any discretization parameter, the method is successful in dynamical consistency. We use the Schur-Cohn criteria for stability analysis of the discretized model. With the solutions obtained, we can understand the effects of smoking on people in a short time, even in different situations. Thus, by knowing these effects in advance, potential health problems can be predicted, and life risks can be minimized according to these predictions.</p></abstract>

scholarly journals Finite element based hybrid techniques for advection-diffusion-reaction processes

2018 ◽  
Vol 8 (2) ◽  
pp. 127-136
Author(s):  
Murat Sari ◽  
Huseyin Tunc
Keyword(s):  
Exact Solution ◽  
Finite Element ◽  
Numerical Solutions ◽  
Diffusion Reaction ◽  
Optimization Approach ◽  
B Spline ◽  
Hybrid Techniques ◽  
Advection Diffusion ◽  
Pure Diffusion ◽  
Spline Basis

In this paper, numerical solutions of the advection-diffusion-reaction (ADR) equation are investigated using the Galerkin, collocation and Taylor-Galerkin cubic B-spline finite element method in strong form of spatial elements using an ?-family optimization approach for time variation. The main objective of this article is to capture effective results of the finite element techniques with B-spline basis functions under the consideration of the ADR processes. All produced results are compared with the exact solution and the literature for various versions of problems including pure advection, pure diffusion, advection-diffusion, and advection-diffusion-reaction equations. It is proved that the present methods have good agreement with the exact solution and the literature.

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