scholarly journals 14-point difference operator for the approximation of the first derivatives of a solution of Laplace’s equation in a rectangular parallelepiped

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 791-800 ◽  
Author(s):  
Adiguzel Dosiyev ◽  
Hediye Sarikaya
Keyword(s):  
Exact Solution ◽  
Difference Operator ◽  
Grid Point ◽  
Rectangular Parallelepiped ◽  
Constant Independent ◽  
Compatibility Conditions ◽  
First Order ◽  
First Derivative ◽  
Boundary Functions ◽  
Derivatives Of

A 14-point difference operator is used to construct finite difference problems for the approximation of the solution, and the first order derivatives of the Dirichlet problem for Laplace?s equations in a rectangular parallelepiped. The boundary functions ?j on the faces ?j, j = 1,2,...,6 of the parallelepiped are supposed to have pth order derivatives satisfying the H?lder condition, i.e., ?j ? Cp,?(?j), 0 < ? < 1, where p = {4,5}. On the edges, the boundary functions as a whole are continuous, and their second and fourth order derivatives satisfy the compatibility conditions which result from the Laplace equation. For the error uh - u of the approximate solution uh at each grid point (x1,x2,x3), ?uh-u?? c?p-4(x1,x2,x3)h4 is obtained, where u is the exact solution, ? = ? (x1, x2,x3) is the distance from the current grid point to the boundary of the parallelepiped, h is the grid step, and c is a constant independent of ? and h. It is proved that when ?j ? Cp,?, 0 < ? < 1, the proposed difference scheme for the approximation of the first derivative converges uniformly with order O(hp-1), p ? {4,5}.

scholarly journals On the difference method for approximation of second order derivatives of a solution of Laplace’s equation in a rectangular parallelepiped

Filomat ◽  
2019 ◽  
Vol 33 (2) ◽  
pp. 633-643
Author(s):  
Adiguzel Dosiyev ◽  
Hediye Sarikaya
Keyword(s):  
Difference Schemes ◽  
Rectangular Parallelepiped ◽  
Finite Difference Schemes ◽  
Compatibility Conditions ◽  
Averaging Operator ◽  
Second Derivatives ◽  
Boundary Functions ◽  
The Difference ◽  
Derivatives Of ◽  
Theoretical Results

We present and justify finite difference schemes with the 14-point averaging operator for the second derivatives of the solution of the Dirichlet problem for Laplace?s equations on a rectangular parallelepiped. The boundary functions ?j on the faces ?j,j = 1,2,..., 6 of the parallelepiped are supposed to have fifth derivatives belonging to the H?lder classes C5?, 0 < ? < 1. On the edges, the boundary functions as a whole are continuous, and their second and fourth order derivatives satisfy the compatibility conditions which result from the Laplace equation. It is proved that the proposed difference schemes for the approximation of the pure and mixed second derivatives converge uniformly with order O(h3+?), 0 < ? < 1 and O(h3), respectively. Numerical experiments are illustrated to support the theoretical results.

Modeling Extreme-Event Precursors with the Fractional Diffusion Equation

2015 ◽  
Vol 18 (1) ◽  
Author(s):  
Michele Caputo ◽  
José M. Carcione ◽  
Marco A. B. Botelho
Keyword(s):  
Diffusion Equation ◽  
Difference Operator ◽  
Extreme Event ◽  
Inverse Fourier Transform ◽  
Time Derivative ◽  
First Order ◽  
The Past ◽  
Moderate Earthquake ◽  
The Green Function ◽  
Derivatives Of

AbstractExtreme catastrophic events such as earthquakes, terrorism and economic collapses are difficult to predict. We propose a tentative mathematical model for the precursors of these events based on a memory formalism and apply it to earthquakes suggesting a physical interpretation. In this case, a precursor can be the anomalous increasing rate of events (aftershocks) following a moderate earthquake, contrary to Omori's law. This trend constitute foreshocks of the main event and can be modelled with fractional time derivatives. A fractional derivative of order 0 < v < 2 replaces the first-order time derivative in the classical diffusion equation.We obtain the frequency-domain Green's function and the corresponding time-domain solution by performing an inverse Fourier transform. Alternatively, we propose a numerical algorithm, where the time derivative is computed with the Grünwald-Letnikov expansion, which is a finitedifference generalization of the standard finite-difference operator to derivatives of fractional order. The results match the analytical solution obtained from the Green function. The calculation requires to store the whole field in the computer memory since anomalous diffusion “remembers the past”.

scholarly journals On the high order convergence of the difference solution of Laplace’s equation in a rectangular parallelepiped

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 893-901 ◽  
Author(s):  
Adiguzel Dosiyev ◽  
Ahlam Abdussalam
Keyword(s):  
Approximate Solution ◽  
Dirichlet Problem ◽  
Laplace Equation ◽  
Grid Point ◽  
Order Convergence ◽  
Compatibility Conditions ◽  
Second Derivatives ◽  
Boundary Functions ◽  
High Order Convergence ◽  
The Difference

The boundary functions ?j of the Dirichlet problem, on the faces ?j, j = 1,2,..., 6 of the parallelepiped R are supposed to have seventh derivatives satisfying the H?lder condition and on the edges their second, fourth and sixth order derivatives satisfy the compatibility conditions which result from the Laplace equation. For the error uh-u of the approximate solution uh at each grid point (x1,x2,x3), a pointwise estmation O(?h6) is obtained, where ?= ?(x1,x2,x3) is the distance from the current grid point to the boundary of R; h is the grid step. The solution of difference problems constructed for the approximate values of the first and pure second derivatives converge with orders O(h6 ?ln h?) and O(h5+?), 0 < ? < 1, respectivly.

Complete monotonicity of entropy production in chemical reaction

1981 ◽  
Vol 46 (2) ◽  
pp. 452-456
Author(s):  
Milan Šolc
Keyword(s):  
Entropy Production ◽  
Equilibrium Constant ◽  
Chemical Reaction ◽  
Relative Entropy ◽  
Order Reaction ◽  
Complete Monotonicity ◽  
First Order Reaction ◽  
First Order ◽  
Completely Monotonic ◽  
Derivatives Of

The successive time derivatives of relative entropy and entropy production for a system with a reversible first-order reaction alternate in sign. It is proved that the relative entropy for reactions with an equilibrium constant smaller than or equal to one is completely monotonic in the whole definition interval, and for reactions with an equilibrium constant larger than one this function is completely monotonic at the beginning of the reaction and near to equilibrium.

scholarly journals Existence results for first derivative dependent ϕ-Laplacian boundary value problems

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Imran Talib ◽  
Thabet Abdeljawad
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Continuous Functions ◽  
Existence Of Solution ◽  
Existence Results ◽  
Main Concern ◽  
First Derivative ◽  
Boundary Functions ◽  
Carathéodory Function

Abstract Our main concern in this article is to investigate the existence of solution for the boundary-value problem $$\begin{aligned}& (\phi \bigl(x'(t)\bigr)'=g_{1} \bigl(t,x(t),x'(t)\bigr),\quad \forall t\in [0,1], \\& \Upsilon _{1}\bigl(x(0),x(1),x'(0)\bigr)=0, \\& \Upsilon _{2}\bigl(x(0),x(1),x'(1)\bigr)=0, \end{aligned}$$ ( ϕ ( x ′ ( t ) ) ′ = g 1 ( t , x ( t ) , x ′ ( t ) ) , ∀ t ∈ [ 0 , 1 ] , ϒ 1 ( x ( 0 ) , x ( 1 ) , x ′ ( 0 ) ) = 0 , ϒ 2 ( x ( 0 ) , x ( 1 ) , x ′ ( 1 ) ) = 0 , where $g_{1}:[0,1]\times \mathbb{R}^{2}\rightarrow \mathbb{R}$ g 1 : [ 0 , 1 ] × R 2 → R is an $L^{1}$ L 1 -Carathéodory function, $\Upsilon _{i}:\mathbb{R}^{3}\rightarrow \mathbb{R} $ ϒ i : R 3 → R are continuous functions, $i=1,2$ i = 1 , 2 , and $\phi :(-a,a)\rightarrow \mathbb{R}$ ϕ : ( − a , a ) → R is an increasing homeomorphism such that $\phi (0)=0$ ϕ ( 0 ) = 0 , for $0< a< \infty $ 0 < a < ∞ . We obtain the solvability results by imposing some new conditions on the boundary functions. The new conditions allow us to ensure the existence of at least one solution in the sector defined by well ordered functions. These ordered functions do not require one to check the definitions of lower and upper solutions. Moreover, the monotonicity assumptions on the arguments of boundary functions are not required in our case. An application is considered to ensure the applicability of our results.

Hodograph Method for Solving the Problem of Shallow Water Under a Solid Lid

2021 ◽  
pp. 15-24
Author(s):  
Tatiana F. Dolgikh
Keyword(s):  
Cauchy Problem ◽  
Green Function ◽  
Differential Equations ◽  
Shallow Water ◽  
Ordinary Differential Equations ◽  
Riemann Invariants ◽  
Hodograph Method ◽  
First Order ◽  
The Cauchy Problem ◽  
Derivatives Of

One of the mathematical models describing the behavior of two horizontally infinite adjoining layers of an ideal incompressible liquid under a solid cover moving at different speeds is investigated. At a large difference in the layer velocities, the Kelvin-Helmholtz instability occurs, which leads to a distortion of the interface. At the initial point in time, the interface is not necessarily flat. From a mathematical point of view, the behavior of the liquid layers is described by a system of four quasilinear equations, either hyperbolic or elliptic, in partial derivatives of the first order. Some type shallow water equations are used to construct the model. In the simple version of the model considered in this paper, in the spatially one-dimensional case, the unknowns are the boundary between the liquid layers h(x,t) and the difference in their velocities γ(x,t). The main attention is paid to the case of elliptic equations when |h|&lt;1 and γ&gt;1. An evolutionary Cauchy problem with arbitrary sufficiently smooth initial data is set for the system of equations. The explicit dependence of the Riemann invariants on the initial variables of the problem is indicated. To solve the Cauchy problem formulated in terms of Riemann invariants, a variant of the hodograph method based on a certain conservation law is used. This method allows us to convert a system of two quasilinear partial differential equations of the first order to a single linear partial differential equation of the second order with variable coefficients. For a linear equation, the Riemann-Green function is specified, which is used to construct a two-parameter implicit solution to the original problem. The explicit solution of the problem is constructed on the level lines (isochrons) of the implicit solution by solving a certain Cauchy problem for a system of ordinary differential equations. As a result, the original Cauchy problem in partial derivatives of the first order is transformed to the Cauchy problem for a system of ordinary differential equations, which is solved by numerical methods. Due to the bulkiness of the expression for the Riemann-Green function, some asymptotic approximation of the problem is considered, and the results of calculations, and their analysis are presented.

scholarly journals Hexagonal Grid Computation of the Derivatives of the Solution to the Heat Equation by Using Fourth-Order Accurate Two-Stage Implicit Methods

2021 ◽  
Vol 5 (4) ◽  
pp. 203
Author(s):  
Suzan Cival Buranay ◽  
Nouman Arshad ◽  
Ahmed Hersi Matan
Keyword(s):  
Heat Equation ◽  
Fourth Order ◽  
Boundary Value ◽  
Time Variable ◽  
Implicit Methods ◽  
First Order ◽  
Mixed Derivatives ◽  
Hexagonal Grids ◽  
Spatial Derivatives ◽  
Derivatives Of

We give fourth-order accurate implicit methods for the computation of the first-order spatial derivatives and second-order mixed derivatives involving the time derivative of the solution of first type boundary value problem of two dimensional heat equation. The methods are constructed based on two stages: At the first stage of the methods, the solution and its derivative with respect to time variable are approximated by using the implicit scheme in Buranay and Arshad in 2020. Therefore, Oh4+τ of convergence on constructed hexagonal grids is obtained that the step sizes in the space variables x1, x2 and in time variable are indicated by h, 32h and τ, respectively. Special difference boundary value problems on hexagonal grids are constructed at the second stages to approximate the first order spatial derivatives and the second order mixed derivatives of the solution. Further, Oh4+τ order of uniform convergence of these schemes are shown for r=ωτh2≥116,ω>0. Additionally, the methods are applied on two sample problems.

scholarly journals An Euler-Lagrange Equation only Depending on Derivatives of Caputo for Fractional Variational Problems with Classical Derivatives

2020 ◽  
Vol 8 (2) ◽  
pp. 590-601
Author(s):  
Melani Barrios ◽  
Gabriela Reyero
Keyword(s):  
Exact Solution ◽  
Variational Problem ◽  
Lagrange Equation ◽  
Integral Form ◽  
Variational Problems ◽  
The Other ◽  
Isoperimetric Problems ◽  
Caputo Derivatives ◽  
Fractional Variational Problems ◽  
Derivatives Of

In this paper we present advances in fractional variational problems with a Lagrangian depending on Caputofractional and classical derivatives. New formulations of the fractional Euler-Lagrange equation are shown for the basic and isoperimetric problems, one in an integral form, and the other that depends only on the Caputo derivatives. The advantage is that Caputo derivatives are more appropriate for modeling problems than the Riemann-Liouville derivatives and makes the calculations easier to solve because, in some cases, its behavior is similar to the behavior of classical derivatives. Finally, anew exact solution for a particular variational problem is obtained.

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