scholarly journals Effect of Shear on Ultrasonic Flow Measurement Using Nonaxisymmetric Wave Modes

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
Yong Chen ◽  
Yiyong Huang ◽  
Xiaoqian Chen ◽  
Dengpeng Hu
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Wave Attenuation ◽  
Inviscid Fluid ◽  
General Boundary Condition ◽  
Algebraic Equations ◽  
Nonaxisymmetric Wave ◽  
Laminar And Turbulent Flow ◽  
Ultrasonic Flow Meter ◽  
Semianalytical Method

Nonaxisymmetric wave propagation in an inviscid fluid with a pipeline shear flow is investigated. Mathematical equation is deduced from the conservations of mass and momentum, leading to a second-order differential equation in terms of the acoustic pressure. Meanwhile a general boundary condition is formulated to cover different types of wall configurations. A semianalytical method based on the Fourier-Bessel theory is provided to transform the differential equation to algebraic equations. Numerical analysis of phase velocity and wave attenuation in water is addressed in the laminar and turbulent flow. Meanwhile comparison among different kinds of boundary condition is given. In the end, the measurement performance of an ultrasonic flow meter is demonstrated.

scholarly journals AN ALGORITHM FOR SOLVING A CONTROL PROBLEM FOR A DIFFERENTIAL EQUATION WITH A PARAMETER

2018 ◽  
pp. 25-32
Author(s):  
Dzhumabaev D.S. ◽  
Bakirova E.A. ◽  
Kadirbayeva Zh.M.
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Control Problem ◽  
Ordinary Differential Equations ◽  
Boundary Value ◽  
Cauchy Problems ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

On a finite interval, a control problem for a linear ordinary differential equations with a parameter is considered. By partitioning the interval and introducing additional parameters, considered problem is reduced to the equivalent multipoint boundary value problem with parameters. To find the parameters introduced, the continuity conditions of the solution at the interior points of partition and boundary condition are used. For the fixed values of the parameters, the Cauchy problems for ordinary differential equations are solved. By substituting the Cauchy problem’s solutions into the boundary condition and the continuity conditions of the solution, a system of linear algebraic equations with respect to parameters is constructed. The solvability of this system ensures the existence of a solution to the original control problem. The system of linear algebraic equations is composed by the solutions of the matrix and vector Cauchy problems for ordinary differential equations on the subintervals. A numerical method for solving the origin control problem is offered based on the Runge-Kutta method of the 4-th order for solving the Cauchy problem for ordinary differential equations. Key words: boundary value problem with parameter, differential equation, solvability, algorithm.

Problem for an ordinary differential equation with a general boundary condition

2021 ◽  
Vol 21 (2) ◽  
pp. 9-14
Author(s):  
L.Kh. Gadzova ◽  
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Ordinary Differential Equation ◽  
Fractional Order ◽  
Unique Solvability ◽  
Explicit Representation ◽  
General Boundary ◽  
General Boundary Condition ◽  
General Representation

For an ordinary differential equation of fractional order with a general boundary condition, a general representation of the solution of the equation is found, a condition of unique solvability is found, and an explicit representation of the solution is constructed.

scholarly journals Solutions for impulsive fractional pantograph differential equation via generalized anti-periodic boundary condition

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Idris Ahmed ◽  
Poom Kumam ◽  
Jamilu Abubakar ◽  
Piyachat Borisut ◽  
Kanokwan Sitthithakerngkiet
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Fixed Point ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Fixed Point Theorems ◽  
Uniqueness Of The Solution ◽  
Fractional Differential

Abstract This study investigates the solutions of an impulsive fractional differential equation incorporated with a pantograph. This work extends and improves some results of the impulsive fractional differential equation. A differential equation of an impulsive fractional pantograph with a more general anti-periodic boundary condition is proposed. By employing the well-known fixed point theorems of Banach and Krasnoselskii, the existence and uniqueness of the solution of the proposed problem are established. Furthermore, two examples are presented to support our theoretical analysis.

scholarly journals Existence of Positive Solutions for Fractional Differential Equation with Nonlocal Boundary Condition

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Hongliang Gao ◽  
Xiaoling Han
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Fixed Point ◽  
Fractional Differential Equation ◽  
Positive Solutions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Existence Of Positive Solutions ◽  
Fractional Differential ◽  
The Fixed Point Theorem

By using the fixed point theorem, existence of positive solutions for fractional differential equation with nonlocal boundary conditionD0+αu(t)+a(t)f(t,u(t))=0,0<t<1,u(0)=0,u(1)=∑i=1∞αiu(ξi)is considered, where1<α≤2is a real number,D0+αis the standard Riemann-Liouville differentiation, andξi∈(0,1),  αi∈[0,∞)with∑i=1∞αiξiα-1<1,a(t)∈C([0,1],[0,∞)),  f(t,u)∈C([0,1]×[0,∞),[0,∞)).

scholarly journals New Properties of Modified Second Kind Chebyshev Polynomials

2020 ◽  
Vol 55 (3) ◽  
Author(s):  
Semaa Hassan Aziz ◽  
Mohammed Rasheed ◽  
Suha Shihab
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Chebyshev Polynomial ◽  
Chebyshev Polynomials ◽  
Digital Computer ◽  
Higher Order ◽  
Algebraic Equations ◽  
Equation Problem ◽  
Higher Order Differential Equations

Modified second kind Chebyshev polynomials for solving higher order differential equations are presented in this paper. This technique, along with some new properties of such polynomials, will reduce the original differential equation problem to the solution of algebraic equations with a straightforward and computational digital computer. Some illustrative examples are included. The modified second kind Chebyshev polynomial is calculated using only a small number of the modified second kind Chebyshev polynomials, which leads to attractive results.

scholarly journals Wellposedness of Neumann boundary-value problems of space-fractional differential equations

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Hong Wang ◽  
Danping Yang
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Neumann Boundary Condition ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Fractional Differential ◽  
Neumann Boundary Value Problems ◽  
Well Posed

AbstractFractional differential equation (FDE) provides an accurate description of transport processes that exhibit anomalous diffusion but introduces new mathematical difficulties that have not been encountered in the context of integer-order differential equation. For example, the wellposedness of the Dirichlet boundary-value problem of one-dimensional variable-coefficient FDE is not fully resolved yet. In addition, Neumann boundary-value problem of FDE poses significant challenges, partly due to the fact that different forms of FDE and different types of Neumann boundary condition have been proposed in the literature depending on different applications.We conduct preliminary mathematical analysis of the wellposedness of different Neumann boundary-value problems of the FDEs. We prove that five out of the nine combinations of three different forms of FDEs that are closed by three types of Neumann boundary conditions are well posed and the remaining four do not admit a solution. In particular, for each form of the FDE there is at least one type of Neumann boundary condition such that the corresponding boundary-value problem is well posed, but there is also at least one type of Neumann boundary condition such that the corresponding boundary-value problem is ill posed. This fully demonstrates the subtlety of the study of FDE, and, in particular, the crucial mathematical modeling question: which combination of FDE and fractional Neumann boundary condition, rather than which form of FDE or fractional Neumann boundary condition, should be used and studied in applications.

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