scholarly journals On Saulyev's Methods

2017 ◽  
Vol 43 (599) ◽  
Author(s):  
Ole Østerby
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Parabolic Equations ◽  
Order Derivative ◽  
First Order ◽  
Two Space Dimensions ◽  
Stability And Consistency

In 1957 V. K. Saulyev proposed two so-called asymmetric methods for solving parabolic equations. We study these methods w.r.t. their stability and consistency, how to include first order derivative terms, how to apply boundary conditions with a derivative, and how to extend the methods to two space dimensions. We also prove that the various modifications proposed by Saulyev, Barakat and Clark, and Larkin also (as was to be expected) require k = o(h) in order to be consistent. As a curiosity we show that the two original Saulyev methods in fact solve two different differential equations.

A Note on the Disadvantages Associated with the Treatment of an Eigenvalue as an Additional Dependent Variable in Differential Eigenvalue Problems

1968 ◽  
Vol 72 (696) ◽  
pp. 1068
Author(s):  
B. Dawson ◽  
M. Davies
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Eigenvalue Problems ◽  
The Other ◽  
Unknown Boundary ◽  
Boundary Values ◽  
First Order ◽  
Basic Set ◽  
Non Linear ◽  
First Order Differential Equations

A novel technique of dealing with differential eigenvalue problems has recently been introduced by Wadsworth and Wilde . The differential equation is expressed as a set of simultaneous first-order differential equations, the eigenvalueλbeing regarded as an additional variable by adding the equationto the basic set. The differential eigenvalue problem is thus reduced to a set of non-linear first-order differential equations with two-point boundary conditions. This treatment of the problem, although novel, suffers from two serious disadvantages. First, it introduces non-linearity into an otherwise linear set of equations. Thus, the solution can no longer be obtained by linear combinations of independent particular solutions. One method of solving the non-linear systems is by assigning arbitrary starting values at one boundary and performing a step-by-step integration to the other boundary where in general the boundary conditions are not satisfied. The problem can be solved by adjustment of the initial assigned arbitrary values until the given conditions at the other boundary are satisfied. A second method and the one used by Wadsworth and Wilde is to estimate the unknown boundary values at both boundaries and integrate inwards to a meeting point. Changes can then be made to the unknown boundary values to make the two branches of the curve fit together.

Exact Equations for the Large Inextensional Motion of Elastic Plates

1981 ◽  
Vol 48 (1) ◽  
pp. 109-112 ◽  
Author(s):  
J. G. Simmonds
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Space Variable ◽  
Second Order ◽  
Straight Edge ◽  
Elastic Plates ◽  
Natural Coordinates ◽  
Governing Equations ◽  
First Order ◽  
Second Order In Time

The governing equations for plates that twist as they deform are reduced to 14 differential equations, first-order in a single space variable and second-order in time. Many of the equations are the same as for statics. Nevertheless, the extension to dynamics is nontrivial because the natural coordinates to use to describe the deformed, developable midsurface are not Lagrangian. The plate is assumed to have two curved, stress-free edges, one built-in straight edge, and one free straight edge acted upon by a force and a couple. There are 7 boundary conditions at the built-in end and 7 at the free end.

Decay of Turbulent Axisymmetrical Free Flows With Rotation

1967 ◽  
Vol 34 (4) ◽  
pp. 806-812 ◽  
Author(s):  
A. Chervinsky ◽  
D. Lorenz
Keyword(s):  
Boundary Layer ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Algebraic Equations ◽  
Free Jets ◽  
Swirling Jets ◽  
First Order ◽  
Free Jet ◽  
Order Algebraic ◽  
First Order Differential Equations

Previous studies on turbulent swirling jets are extended to cover general axisymmetrical turbulent free flows with rotation, with particular considerations of free jets and wakes. The equations governing the flow are integrated subject to the pertaining boundary conditions making use of the usual boundary-layer approximations. The axial and tangential components of velocity are assumed to retain similar forms of radial distributions and a set of two ordinary first-order differential equations is derived, the solution of which describes the axial decay of the maximal axial and tangential velocities. Integration of the differential equations in the particular case of uniform external velocity subject to conditions at the orifice results in a set of two second-order algebraic equations which are readily solved. The theoretical solutions derived for a free jet are compared with available experimental results.

scholarly journals Existence and uniqueness of solutions for the system ofintegro-differential equations with three-point and nonlinear integral boundary conditions

2020 ◽  
Vol 99 (3) ◽  
pp. 23-37
Author(s):  
M.J. Mardanov ◽  
◽  
Y.A. Sharifov ◽  
K.E. Ismayilova ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Original Problem ◽  
Integral Boundary Conditions ◽  
Uniqueness Of Solutions ◽  
Integral Boundary ◽  
First Order ◽  
Fixed Point Principle ◽  
Uniqueness Of A Solution

The paper examines a system of nonlinear integro-differential equations with three-point and nonlinear integral boundary conditions. The original problem demonstrated to be equivalent to integral equations by using Green function. Theorems on the existence and uniqueness of a solution to the boundary value problems for the first order nonlinear system of integro- differential equations with three-point and nonlinear integral boundary conditions are proved. A proof of uniqueness theorem of the solution is obtained by Banach fixed point principle, and the existence theorem then follows from Schaefer’s theorem.

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