scholarly journals Two scale defect measure and linear equations with oscillating coefficients

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2867-2873
Author(s):  
Jelena Aleksic ◽  
Stevan Pilipovic
Keyword(s):  
Linear Equations ◽  
Convergent Sequence ◽  
Dimensional Torus ◽  
Strong Limit ◽  
Absolutely Continuous ◽  
Order Linear ◽  
First Order ◽  
Linear Pde ◽  
Defect Measure ◽  
Oscillating Coefficients

Microlocal measure ? is associated to a two-scale convergent sequence un over Rd with the limit u ? L2(Rd x Td), Td is a torus, to analyze possible strong limit. ? is an operator valued measure absolutely continuous with respect to the product of scalar microlocal defect measure and a measure on the d-dimensional torus. The result is applied to the first order linear PDE with the oscillating coefficients.

First Order Linear Equations

2001 ◽  
pp. 51-79
Author(s):  
Martin Bohner ◽  
Allan Peterson
Keyword(s):  
Linear Equations ◽  
Order Linear ◽  
First Order

A numerical method of solving second-order linear differential equations with two-point boundary conditions

1957 ◽  
Vol 53 (2) ◽  
pp. 442-447 ◽  
Author(s):  
E. Cicely Ridley
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Riccati Equation ◽  
Direct Method ◽  
Linear Equations ◽  
Factorization Method ◽  
Order Linear ◽  
First Order ◽  
The Matrix ◽  
Linear Differential

ABSTRACTA direct method of integrating the equation y″ + g(x) y = h(x), with the two-point linear boundary conditions y′(a) + αy(a) = A, y′(b) + βy(b) = B, is based on the factorization of the equation into two first-order linear equations v′ − sv = h and y′ + sy = v, where s is a solution of the Riccati equation s′ − s2 = g. The first-order equations for v and y are integrated in succession, one in the direction of x increasing, and one in the direction of x decreasing, one boundary condition being used in each of these integrations. The appropriate solution of the Riccati equation is determined by the boundary condition at the end of the range from which the integration of the equation for v is started. The process is compared with the matrix factorization method of Thomas and Fox, and its stability discussed.

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