A note on comparison theorems for graphs

Solution of heat and mass transfer problems with non-linear coefficients

Tyumen State University Herald Physical and Mathematical Modeling Oil Gas Energy ◽

10.21684/2411-7978-2019-5-4-10-20 ◽

2019 ◽

Vol 5 (4) ◽

pp. 10-20

Author(s):

Boris G. Aksenov ◽

Yuri E. Karyakin ◽

Svetlana V. Karyakina

Keyword(s):

Mass Transfer ◽

Exact Solution ◽

Numerical Methods ◽

Heat And Mass Transfer ◽

Unknown Function ◽

Comparison Theorems ◽

Upper And Lower Bounds ◽

Maximum Deviation ◽

Approximate Methods ◽

Nonmonotonic Dependence

Equations, which have nonlinear nonmonotonic dependence of one of the coefficients on an unknown function, can describe processes of heat and mass transfer. As a rule, existing approximate methods do not provide solutions with acceptable accuracy. Numerical methods do not involve obtaining an analytical expression for the unknown function and require studying the convergence of the algorithm used. The value of absolute error is uncertain. The authors propose an approximate method for solving such problems based on Westphal comparison theorems. The comparison theorems allow finding upper and lower bounds of the unknown exact solution. A special procedure developed for the stepwise improvement of these bounds provide solutions with a given accuracy. There are only a few problems for equations with nonlinear nonmonotonic coefficients for which the exact solution has been obtained. One of such problems, presented in this article, shows the efficiency of the proposed method. The results prove that the proposed method for obtaining bounds of the solution of a nonlinear nonmonotonic equation of parabolic type can be considered as a new method of the approximate analytical solution having guaranteed accuracy. In addition, the proposed here method allows calculating the maximum deviation from the unknown exact solution of the results of other approximate and numerical methods.

Download Full-text

Comparison theorems for fourth order differential equations

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171286000133 ◽

1986 ◽

Vol 9 (1) ◽

pp. 105-109

Author(s):

Garret J. Etgen ◽

Willie E. Taylor

Keyword(s):

Continuous Function ◽

Differential Equations ◽

Fourth Order ◽

Linear Equations ◽

Oscillation Criterion ◽

Comparison Theorems ◽

Positive Continuous Function ◽

Order Linear ◽

All Solutions

This paper establishes an apparently overlooked relationship between the pair of fourth order linear equationsyiv−p(x)y=0andyiv+p(x)y=0, wherepis a positive, continuous function defined on[0,∞). It is shown that if all solutions of the first equation are nonoscillatory, then all solutions of the second equation must be nonoscillatory as well. An oscillation criterion for these equations is also given.

Download Full-text