scholarly journals AN EFFICIENT APPROACH FOR SOLVING FRACTIONAL VARIABLE ORDER REACTION SUB-DIFFUSION BASED ON HERMITE FORMULA

Fractals ◽  
2021 ◽  
pp. 2240020
Author(s):  
MOHAMED ADEL ◽  
MOHAMED ELSAID
Keyword(s):  
Nonlinear Models ◽  
Finite Difference Methods ◽  
Order Reaction ◽  
Variable Order ◽  
Von Neumann ◽  
Discretization Schemes ◽  
Anomalous Reaction ◽  
Linear And Nonlinear ◽  
The Stability ◽  
Variable Order Derivative

Anomalous Reaction-Sub-diffusion equations play an important role transferred in a lot of our daily applications in our life, especially in applied chemistry. In the presented work, a modified type of these models is considered which is the Reaction-Sub-diffusion equation of variable order, the linear and nonlinear models and we will refer to it by VORSDE. An accurate technique depends on a mix of the finite difference methods (FDM) together with Hermite formula is introduced to study these important types of anomalous equations. Regarding the analysis of the stability for the mentioned, it is done using the variable Von-Neumann technique; also the convergent analysis is introduced. As a result of the previous steps, we derived a stability condition which will be held for many discretization schemes of the variable order derivative and some other parameters and we checked it numerically.

scholarly journals Modified Variational Iteration Algorithm-II: Convergence and Applications to Diffusion Models

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-14 ◽  
Author(s):  
Hijaz Ahmad ◽  
Tufail A. Khan ◽  
Predrag S. Stanimirović ◽  
Yu-Ming Chu ◽  
Imtiaz Ahmad
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Nonlinear Models ◽  
Nonlinear Differential Equations ◽  
Variational Iteration Method ◽  
Iteration Algorithm ◽  
Compact Finite Difference Method ◽  
Variational Iteration ◽  
Linear And Nonlinear ◽  
The Stability

Variational iteration method has been extensively employed to deal with linear and nonlinear differential equations of integer and fractional order. The key property of the technique is its ability and flexibility to investigate linear and nonlinear models conveniently and accurately. The current study presents an improved algorithm to the variational iteration algorithm-II (VIA-II) for the numerical treatment of diffusion as well as convection-diffusion equations. This newly introduced modification is termed as the modified variational iteration algorithm-II (MVIA-II). The convergence of the MVIA-II is studied in the case of solving nonlinear equations. The main advantage of the MVIA-II improvement is an auxiliary parameter which makes sure a fast convergence of the standard VIA-II iteration algorithm. In order to verify the stability, accuracy, and computational speed of the method, the obtained solutions are compared numerically and graphically with the exact ones as well as with the results obtained by the previously proposed compact finite difference method and second kind Chebyshev wavelets. The comparison revealed that the modified version yields accurate results, converges rapidly, and offers better robustness in comparison with other methods used in the literature. Moreover, the basic idea depicted in this study is relied upon the possibility of the MVIA-II being utilized to handle nonlinear differential equations that arise in different fields of physical and biological sciences. A strong motivation for such applications is the fact that any discretization, transformation, or any assumptions are not required for this proposed algorithm in finding appropriate numerical solutions.

scholarly journals Stability and Convergence of a Time-Fractional Variable Order Hantush Equation for a Deformable Aquifer

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Abdon Atangana ◽  
S. C. Oukouomi Noutchie
Keyword(s):  
Partial Derivative ◽  
Mathematical Formulation ◽  
Stability And Convergence ◽  
Order Derivative ◽  
Variable Order ◽  
Leaky Aquifer ◽  
The Stability ◽  
Nicolson Scheme ◽  
Variable Order Derivative ◽  
Modified Equation

The medium through which the groundwater moves varies in time and space. The Hantush equation describes the movement of groundwater through a leaky aquifer. To include explicitly the deformation of the leaky aquifer into the mathematical formulation, we modify the equation by replacing the partial derivative with respect to time by the time-fractional variable order derivative. The modified equation is solved numerically via the Crank-Nicolson scheme. The stability and the convergence in this case are presented in details.

scholarly journals NUMERICAL SIMULATIONS FOR THE VARIABLE ORDER TWO-DIMENSIONAL REACTION SUB-DIFFUSION EQUATION: LINEAR AND NONLINEAR

Fractals ◽  
2021 ◽  
pp. 2240019
Author(s):  
MOHAMED ADEL
Keyword(s):  
Diffusion Equation ◽  
Numerical Technique ◽  
Weighted Average ◽  
Transport Processes ◽  
Diffusion Equations ◽  
Variable Order ◽  
High Concentration ◽  
Fractal Media ◽  
Linear And Nonlinear ◽  
The Stability

The applications and the fields that use the anomalous sub-diffusion equations cannot be easily listed due to their wide area. Sure, one of the main physical reasons for using and researching fractional order diffusion equations is to explain anomalous diffusion that occurs in transport processes through complex and/or disordered structures, such as fractal media. One of the important applications is their use in chemical reactions, where a single material continues to shift from a high concentration area to a low concentration area until the concentration across the space is equal. The mathematical model that describes these physical-chemical phenomena is called the reaction sub-diffusion equation. In our study, we try to solve the 2D variable order version of these equations (2DVORSE) (linear and nonlinear) by using an accurate numerical technique which is the variable weighted average finite difference method (WAFDM). We will analyze the stability of the resulting scheme by using a modified suitable version of the John Von Neumann procedure. Specific stability conditions that occur or some parameters in the resulting schemes are derived and checked. At the end of the study, numerical examples are simulated to check the stability and the accuracy of the proposed technique.

scholarly journals The Discrete Ordinate /Pseudo-spectral Method: Review and Application from a Physicist's Perspective

1993 ◽  
Vol 46 (4) ◽  
pp. 465 ◽  
Author(s):  
RE Robson ◽  
Arnstein Prytz
Keyword(s):  
Boundary Layer ◽  
Nonlinear Models ◽  
Finite Difference Methods ◽  
Comprehensive Analysis ◽  
Discrete Ordinate Method ◽  
Discrete Ordinate ◽  
Linear And Nonlinear ◽  
Difference Methods ◽  
Pseudo Spectral Method ◽  
Pseudo Spectral

A discussion of the discrete ordinate method for solving differential equations is presented along with a number of examples that have application in various fields of physics. In particular, diffusion cooling, boundary layer meteorology and the diffusion of water in soils are studied. It is shown that the discrete ordinate method is considerably more accurate than finite difference methods of the same order. Results are presented for linear and nonlinear models, with a comprehensive analysis of the results and accuracies.

Stability of radical anions derived from substituted 5-nitrofurans investigated by ESR spectroscopy

1985 ◽  
Vol 50 (7) ◽  
pp. 1594-1601 ◽  
Author(s):  
Jiří Klíma ◽  
Larisa Baumane ◽  
Janis Stradinš ◽  
Jiří Volke ◽  
Romualds Gavars
Keyword(s):  
Radical Anion ◽  
High Stability ◽  
Esr Spectroscopy ◽  
Reaction Path ◽  
Order Reaction ◽  
Second Order ◽  
Radical Anions ◽  
Second Order Reaction ◽  
Unpaired Electron Density ◽  
The Stability

It has been found that the decay in dimethylformamide and dimethylformamide-water mixtures of radical anions in five of the investigated 5-nitrofurans is governed by a second-order reaction. Only the decay of the radical anion generated from 5-nitro-2-furfural III may be described by an equation including parallel first- and second-order reactions; this behaviour is evidently caused by the relatively high stability of the corresponding dianion, this being an intermediate in the reaction path. The presence of a larger conjugated system in the substituent in position 2 results in a decrease of the unpaired electron density in the nitro group and, consequently, an increase in the stability of the corresponding radical anions.

Unbounded Linear Operators

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Hilbert Spaces ◽  
Linear Operators ◽  
Von Neumann ◽  
Limit Circle ◽  
Sectorial Operators ◽  
Closed Operators ◽  
The Stability ◽  
Symmetric Operators ◽  
Unbounded Linear Operators ◽  
Special Case

This chapter is concerned with closable and closed operators in Hilbert spaces, especially with the special classes of symmetric, J-symmetric, accretive and sectorial operators. The Stone–von Neumann theory of extensions of symmetric operators is treated as a special case of results for compatible adjoint pairs of closed operators. Also discussed in detail is the stability of closedness and self-adjointness under perturbations. The abstract results are applied to operators defined by second-order differential expressions, and Sims’ generalization of the Weyl limit-point, limit-circle characterization for symmetric expressions to J-symmetric expressions is proved.

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.

Comparison of linear and nonlinear models to estimate body weight of Pelibuey ewes from body measurements

2021 ◽  
Vol 53 (1) ◽  
Author(s):  
Rafael Macedo-Barragán ◽  
Victalina Arredondo-Ruiz ◽  
Carlos Haubi-Segura ◽  
Paola Castillo-Zamora
Keyword(s):  
Body Weight ◽  
Nonlinear Models ◽  
Body Measurements ◽  
Linear And Nonlinear

OPTIMAL VELOCITY MODEL WITH RELATIVE VELOCITY

2006 ◽  
Vol 17 (01) ◽  
pp. 65-73 ◽  
Author(s):  
SHIRO SAWADA
Keyword(s):  
Nonlinear Analysis ◽  
Relative Velocity ◽  
Velocity Model ◽  
Fundamental Diagram ◽  
Optimal Velocity ◽  
Traffic Jam ◽  
Optimal Velocity Model ◽  
Linear And Nonlinear ◽  
The Stability ◽  
Good Agreement

The optimal velocity model which depends not only on the headway but also on the relative velocity is analyzed in detail. We investigate the effect of considering the relative velocity based on the linear and nonlinear analysis of the model. The linear stability analysis shows that the improvement in the stability of the traffic flow is obtained by taking into account the relative velocity. From the nonlinear analysis, the relative velocity dependence of the propagating kink solution for traffic jam is obtained. The relation between the headway and the velocity and the fundamental diagram are examined by numerical simulation. We find that the results by the linear and nonlinear analysis of the model are in good agreement with the numerical results.

scholarly journals On the exact and numerical solutions to a nonlinear model arising in mathematical biology

2018 ◽  
Vol 22 ◽  
pp. 01061 ◽  
Author(s):  
Asif Yokus ◽  
Tukur Abdulkadir Sulaiman ◽  
Haci Mehmet Baskonus ◽  
Sibel Pasali Atmaca
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Soliton Solutions ◽  
Hyperbolic Function ◽  
Von Neumann ◽  
Convection Diffusion Equation ◽  
Wolfram Mathematica ◽  
Forward Difference ◽  
Von Neumann Analysis ◽  
The Stability

This study acquires the exact and numerical approximations of a reaction-convection-diffusion equation arising in mathematical bi- ology namely; Murry equation through its analytical solutions obtained by using a mathematical approach; the modified exp(-Ψ(η))-expansion function method. We successfully obtained the kink-type and singular soliton solutions with the hyperbolic function structure to this equa- tion. We performed the numerical simulations (3D and 2D) of the obtained analytical solutions under suitable values of parameters. We obtained the approximate numerical and exact solutions to this equa- tion by utilizing the finite forward difference scheme by taking one of the obtained analytical solutions into consideration. We investigate the stability of the finite forward difference method with the equation through the Fourier-Von Neumann analysis. We present the L2 and L∞ error norms of the approximations. The numerical and exact approx- imations are compared and the comparison is supported by a graphic plot. All the computations and the graphics plots in this study are car- ried out with help of the Matlab and Wolfram Mathematica softwares. Finally, we submit a comprehensive conclusion to this study.

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