scholarly journals A Third-Orderp-Laplacian Boundary Value Problem Solved by an SL(3,ℝ)Lie-Group Shooting Method

2013 ◽  
Vol 2013 ◽  
pp. 1-13
Author(s):  
Chein-Shan Liu
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Lie Group ◽  
Shooting Method ◽  
Boundary Value ◽  
Boundary Values ◽  
Third Order ◽  
Left Boundary ◽  
Boundary Layer Problem ◽  
The Third

The boundary layer problem for power-law fluid can be recast to a third-orderp-Laplacian boundary value problem (BVP). In this paper, we transform the third-orderp-Laplacian into a new system which exhibits a Lie-symmetry SL(3,ℝ). Then, the closure property of the Lie-group is used to derive a linear transformation between the boundary values at two ends of a spatial interval. Hence, we can iteratively solve the missing left boundary conditions, which are determined by matching the right boundary conditions through a finer tuning ofr∈[0,1]. The present SL(3,ℝ)Lie-group shooting method is easily implemented and is efficient to tackle the multiple solutions of the third-orderp-Laplacian. When the missing left boundary values can be determined accurately, we can apply the fourth-order Runge-Kutta (RK4) method to obtain a quite accurate numerical solution of thep-Laplacian.

scholarly journals NONLOCAL THIRD ORDER BOUNDARY VALUE PROBLEMS WITH SOLUTIONS THAT CHANGE SIGN

2014 ◽  
Vol 19 (2) ◽  
pp. 145-154
Author(s):  
Sergey Smirnov
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Shooting Method ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Oscillatory Behavior ◽  
Third Order ◽  
Scaling Method ◽  
Number Of Solutions

We investigate the existence and the number of solutions for a third order boundary value problem with nonlocal boundary conditions in connection with the oscillatory behavior of solutions. The combination of the shooting method and scaling method is used in the proofs of our main results. Examples are included to illustrate the results.

An Numeric Method on the Third Order Term of KDV Equation

2011 ◽  
Vol 135-136 ◽  
pp. 253-255
Author(s):  
Yi Min Tian
Keyword(s):  
Difference Scheme ◽  
Kdv Equation ◽  
Order Term ◽  
Boundary Value ◽  
Difference Schemes ◽  
Boundary Values ◽  
Implicit Difference Scheme ◽  
Third Order ◽  
The Third ◽  
Numeric Experiment

Numeric scheme and numeric result was in this paper. First, We proposes a kind of explicit - implicit difference scheme to solve the initial and boundary value questions of the third order term of KDV equation here,and so we can solve the problem that the additional boundary values must be given first for present difference schemes when we try to realize the calculation by then., second, numeric experiment results was given ay the end of this article.

scholarly journals Boundary value problem with integral conditions for a linear third-order equation

2003 ◽  
Vol 2003 (11) ◽  
pp. 553-567 ◽  
Author(s):  
M. Denche ◽  
A. Memou
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Strong Solution ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Integral Boundary Conditions ◽  
Order Equation ◽  
Integral Conditions ◽  
Third Order ◽  
Integral Boundary

We prove the existence and uniqueness of a strong solution for a linear third-order equation with integral boundary conditions. The proof uses energy inequalities and the density of the range of the generated operator.

scholarly journals Positive Solutions for Third-Order Boundary-Value Problems with the Integral Boundary Conditions and Dependence on the First-Order Derivatives

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yanping Guo ◽  
Fei Yang
Keyword(s):  
Fixed Point ◽  
Green Function ◽  
Boundary Conditions ◽  
Boundary Value ◽  
Integral Boundary Conditions ◽  
Third Order ◽  
Integral Boundary ◽  
First Order ◽  
The Third ◽  
Nonnegative Continuous Function

By using a fixed point theorem in a cone and the nonlocal third-order BVP's Green function, the existence of at least one positive solution for the third-order boundary-value problem with the integral boundary conditionsx′′′(t)+f(t,x(t),x′(t))=0,t∈J,x(0)=0,x′′(0)=0, andx(1)=∫01g(t)x(t)dtis considered, wherefis a nonnegative continuous function,J=[0,1], andg∈L[0,1].The emphasis here is thatfdepends on the first-order derivatives.

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