非规范伯格斯方程

非规范伯格斯方程 (Unnormalized Burgers equation)是一个非线性偏微分方程： u t − α ∗ u x x − β ∗ u ∗ u x = 0 {\displaystyle u_{t}-\alpha *u_{xx}-\beta *u*u_{x}=0} u ( x , t ) = C 3 / ( β ∗ C 2 ) + 2 ∗ α ∗ C 2 ∗ c o t ( C 1 + C 2 ∗ x + C 3 ∗ t ) / β {\displaystyle u(x,t)=_{C}3/(\beta *_{C}2)+2*\alpha *_{C}2*cot(_{C}1+_{C}2*x+_{C}3*t)/\beta } u ( x , t ) = C 3 / ( β ∗ C 2 ) + 2 ∗ α ∗ C 2 ∗ c o t h ( C 1 + C 2 ∗ x + C 3 ∗ t ) / β {\displaystyle {u(x,t)=_{C}3/(\beta *_{C}2)+2*\alpha *_{C}2*coth(_{C}1+_{C}2*x+_{C}3*t)/\beta }} u ( x , t ) = C 3 / ( β ∗ C 2 ) − 2 ∗ α ∗ C 2 ∗ t a n ( C 1 + C 2 ∗ x + C 3 ∗ t ) / β {\displaystyle {u(x,t)=_{C}3/(\beta *_{C}2)-2*\alpha *_{C}2*tan(_{C}1+_{C}2*x+_{C}3*t)/\beta }} u ( x , t ) = C 3 / ( β ∗ C 2 ) + 2 ∗ α ∗ C 2 ∗ t a n h ( C 1 + C 2 ∗ x + C 3 ∗ t ) / β {\displaystyle {u(x,t)=_{C}3/(\beta *_{C}2)+2*\alpha *_{C}2*tanh(_{C}1+_{C}2*x+_{C}3*t)/\beta }} u ( x , t ) = ( C 5 + t a n h ( ( 1 / 2 ) ∗ C 1 ∗ ( C 3 + C 4 ∗ x + C 5 ∗ t − C 2 ) / ( α ∗ C 4 2 ) ) ∗ C 1 ) / ( β ∗ C 4 ) {\displaystyle {u(x,t)=(_{C}5+tanh((1/2)*_{C}1*(_{C}3+_{C}4*x+_{C}5*t-_{C}2)/(\alpha *_{C}4^{2}))*_{C}1)/(\beta *_{C}4)}} u ( x , t ) = ( C 5 + t a n h ( ( 1 / 2 ) ∗ C 1 ∗ ( ( c s c ( C 3 + C 4 ∗ x + C 5 ∗ t ) − 1 ) ∗ ( c s c ( C 3 + C 4 ∗ x + C 5 ∗ t ) + 1 ) ∗ a r c t a n ( 1 / ( c s c ( C 3 + C 4 ∗ x + C 5 ∗ t ) 2 − 1 ) ) + C 2 ∗ ( c s c ( C 3 + C 4 ∗ x + C 5 ∗ t ) 2 − 1 ) ) / ( α ∗ C 4 2 ∗ ( c s c ( C 3 + C 4 ∗ x + C 5 ∗ t ) 2 − 1 ) ) ) ∗ C 1 ) / ( β ∗ C 4 ) {\displaystyle {u(x,t)=(_{C}5+tanh((1/2)*_{C}1*({\sqrt {(}}csc(_{C}3+_{C}4*x+_{C}5*t)-1)*{\sqrt {(}}csc(_{C}3+_{C}4*x+_{C}5*t)+1)*arctan(1/{\sqrt {(}}csc(_{C}3+_{C}4*x+_{C}5*t)^{2}-1))+_{C}2*{\sqrt {(}}csc(_{C}3+_{C}4*x+_{C}5*t)^{2}-1))/(\alpha *_{C}4^{2}*{\sqrt {(}}csc(_{C}3+_{C}4*x+_{C}5*t)^{2}-1)))*_{C}1)/(\beta *_{C}4)}} u ( x , t ) = ( C 5 − t a n ( ( 1 / 2 ) ∗ ( 2 ∗ C 1 ∗ α ∗ C 4 3 ∗ β − C 5 2 ) ∗ ( l n ( c o s h ( C 3 + C 4 ∗ x + C 5 ∗ t ) + ( c o s h ( C 3 + C 4 ∗ x + C 5 ∗ t ) 2 − 1 ) ) + C 2 ) / ( α ∗ C 4 2 ) ) ∗ ( 2 ∗ C 1 ∗ α ∗ C 4 3 ∗ β − C 5 2 ) ) / ( β ∗ C 4 ) {\displaystyle {u(x,t)=(_{C}5-tan((1/2)*{\sqrt {(}}2*_{C}1*\alpha *_{C}4^{3}*\beta -_{C}5^{2})*(ln(cosh(_{C}3+_{C}4*x+_{C}5*t)+{\sqrt {(}}cosh(_{C}3+_{C}4*x+_{C}5*t)^{2}-1))+_{C}2)/(\alpha *_{C}4^{2}))*{\sqrt {(}}2*_{C}1*\alpha *_{C}4^{3}*\beta -_{C}5^{2}))/(\beta *_{C}4)}} u ( x , t ) = ( C 5 − t a n ( ( 1 / 2 ) ∗ ( 2 ∗ C 1 ∗ α ∗ C 4 3 ∗ β − C 5 2 ) ∗ ( C 3 + C 4 ∗ x + C 5 ∗ t + C 2 ) / ( α ∗ C 4 2 ) ) ∗ ( 2 ∗ C 1 ∗ α ∗ C 4 3 ∗ β − C 5 2 ) ) / ( β ∗ C 4 ) {\displaystyle {u(x,t)=(_{C}5-tan((1/2)*{\sqrt {(}}2*_{C}1*\alpha *_{C}4^{3}*\beta -_{C}5^{2})*(_{C}3+_{C}4*x+_{C}5*t+_{C}2)/(\alpha *_{C}4^{2}))*{\sqrt {(}}2*_{C}1*\alpha *_{C}4^{3}*\beta -_{C}5^{2}))/(\beta *_{C}4)}} u ( x , t ) = ( C 5 + t a n h ( ( 1 / 2 ) ∗ ( 2 ∗ C 1 ∗ α ∗ C 4 3 ∗ β + C 5 2 ) ∗ ( a r c t a n h ( 1 / ( 1 + c s c h ( C 3 + C 4 ∗ x + C 5 ∗ t ) 2 ) ) − C 2 ) / ( α ∗ C 4 2 ) ) ∗ ( 2 ∗ C 1 ∗ α ∗ C 4 3 ∗ β + C 5 2 ) ) / ( β ∗ C 4 ) {\displaystyle {u(x,t)=(_{C}5+tanh((1/2)*{\sqrt {(}}2*_{C}1*\alpha *_{C}4^{3}*\beta +_{C}5^{2})*(arctanh(1/{\sqrt {(}}1+csch(_{C}3+_{C}4*x+_{C}5*t)^{2}))-_{C}2)/(\alpha *_{C}4^{2}))*{\sqrt {(}}2*_{C}1*\alpha *_{C}4^{3}*\beta +_{C}5^{2}))/(\beta *_{C}4)}} u ( x , t ) = ( C 5 − t a n h ( ( 1 / 2 ) ∗ ( 2 ∗ C 1 ∗ α ∗ C 4 3 ∗ β + C 5 2 ) ∗ ( ( 1 / 2 ) ∗ P i − C 3 − C 4 ∗ x − C 5 ∗ t + C 2 ) / ( α ∗ C 4 2 ) ) ∗ ( 2 ∗ C 1 ∗ α ∗ C 4 3 ∗ β + C 5 2 ) ) / ( β ∗ C 4 ) {\displaystyle {u(x,t)=(_{C}5-tanh((1/2)*{\sqrt {(}}2*_{C}1*\alpha *_{C}4^{3}*\beta +_{C}5^{2})*((1/2)*Pi-_{C}3-_{C}4*x-_{C}5*t+_{C}2)/(\alpha *_{C}4^{2}))*{\sqrt {(}}2*_{C}1*\alpha *_{C}4^{3}*\beta +_{C}5^{2}))/(\beta *_{C}4)}} u ( x , t ) = − ( − C 5 + t a n h ( ( 1 / 2 ) ∗ C 1 ∗ ( ( s e c ( C 3 + C 4 ∗ x + C 5 ∗ t ) − 1 ) ∗ ( s e c ( C 3 + C 4 ∗ x + C 5 ∗ t ) + 1 ) ∗ a r c t a n ( 1 / ( s e c ( C 3 + C 4 ∗ x + C 5 ∗ t ) 2 − 1 ) ) + C 2 ∗ ( s e c ( C 3 + C 4 ∗ x + C 5 ∗ t ) 2 − 1 ) ) / ( α ∗ C 4 2 ∗ ( s e c ( C 3 + C 4 ∗ x + C 5 ∗ t ) 2 − 1 ) ) ) ∗ C 1 ) / ( β ∗ C 4 ) {\displaystyle {u(x,t)=-(-_{C}5+tanh((1/2)*_{C}1*({\sqrt {(}}sec(_{C}3+_{C}4*x+_{C}5*t)-1)*{\sqrt {(}}sec(_{C}3+_{C}4*x+_{C}5*t)+1)*arctan(1/{\sqrt {(}}sec(_{C}3+_{C}4*x+_{C}5*t)^{2}-1))+_{C}2*{\sqrt {(}}sec(_{C}3+_{C}4*x+_{C}5*t)^{2}-1))/(\alpha *_{C}4^{2}*{\sqrt {(}}sec(_{C}3+_{C}4*x+_{C}5*t)^{2}-1)))*_{C}1)/(\beta *_{C}4)}} u ( x , t ) = ( 1 / 2 ) ∗ ( ( 2 ) ∗ C 5 + 2 ∗ t a n h ( ( 1 / 2 ) ∗ ( β ∗ C 1 ∗ C 4 ∗ α ) ∗ ( C 3 + C 4 ∗ x + C 5 ∗ t + C 2 ) ∗ ( 2 ) / ( C 4 ∗ α ) ) ∗ C 4 ∗ ( β ∗ C 1 ∗ C 4 ∗ α ) ) ∗ ( 2 ) / ( β ∗ C 4 ) {\displaystyle {u(x,t)=(1/2)*({\sqrt {(}}2)*_{C}5+2*tanh((1/2)*{\sqrt {(}}\beta *_{C}1*_{C}4*\alpha )*(_{C}3+_{C}4*x+_{C}5*t+_{C}2)*{\sqrt {(}}2)/(_{C}4*\alpha ))*_{C}4*{\sqrt {(}}\beta *_{C}1*_{C}4*\alpha ))*{\sqrt {(}}2)/(\beta *_{C}4)}} Andrei D. 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