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Esen HANAC
LONG -TERM SOLUTIONS OF INVISCID BURGERS EQUATION WITH FRACTIONAL TERM
 
I analyse the main results concerning the existence and structure of large -time solutions of an initial-value problem for equation, namely u_t+1/x^2 uu_x=u_xx -∞0, (1) with step initial data u(x,0)={█(1 as x≥0 0 as x<0)┤ and u(x,t)→{█(1,x→∞ 0,x→-∞ )┤ where respectively x represents distance and t represents time. Development of the large-time solutions of the equation which is given above has been examined by using the method of matched asymptotic coordinate expansions. In particulary, asymptotic solution to the equation above is searched firstly when as t tends to 0, then asymptotic solution to the equation (1) as x tends to infinity and after then asymptotic solution to the equation (1) as t tends to infinity are found separately. For each asymptotic schemas of the equation (1) have own regions which are obtained after some calculations depending on the term as mentioned before . After all, solutions to initial value problem (1) are identfied according to results in the cases ( which are as t tends to 0, as x tends to infinity and the final one as t tends to infinity). The aim is to see in which circumstance travelling wave is occured or stationary state condition is, may be seen both cases too.

Anahtar Kelimeler: Nonlinear Equation, matched asymptotic coordinate expansions method,TW



 


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