Differential Equations Lecture Work Solutions 246

Differential Equations Lecture Work Solutions 246 - as the...

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b. ∂u ∂t + t 2 u ∂u ∂x = u 6. Solve ∂u ∂t + t 2 u ∂u ∂x =5 subject to u ( x, 0) = x. 7. Using implicit diferentiation, veriFy that u ( x, t )= f ( x tu ) is a solution oF u t + uu x =0 8. Consider the damped quasilinear wave equation u t + uu x + cu =0 where c is a positive constant. (a) Using the method oF characteristics, construct a solution oF the initial value problem with u ( x, 0) = f ( x ), in implicit Form. Discuss the wave motion and the efect oF the damping. (b) Determine the breaking time oF the solution by ±nding the envelope oF the charac- teristic curves and by using implicit diferentiation. With
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Unformatted text preview: as the parameter on the initial line, show that unless f ( ) < c , no breaking occurs. 9. Consider the one-dimensional Form oF Eulers equations For isentropic ow and assume that the pressure p is a constant. The equations reduce to t + u x + u x = 0 u t + uu x = 0 Let u ( x, 0) = f ( x ) and ( x, 0) = g ( x ). By rst solving the equation For u and then the equation For , obtain the implicit solution u = f ( x ut ) = g ( x ut ) 1 + tf ( x ut ) 246...
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