The Burgers equation on the semiline with general boundary conditions at the origin

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Abstract
A technique is given to solve the initial/boundary value problem for the Burgers equation ut(x,t)=uxx(x,t)+2 ux(x,t) u(x,t) on the semiline 0≤x<∞, with the general boundary condition at the origin H[u(0,t),ux(0,t);t]=0. Here ‘‘to solve’’ means ‘‘to reduce to an equation in one variable only.’’ This equation is generally nonlinear and integrodifferent ial; it comes in several (equivalent) avatars, which contain nontrivially a free parameter, whose value can be assigned arbitrarily since the solution of the equation is independent of it. In the special case when H(y,z;t)=a(t)y+b(t)(z+y2) −F(t), which is the case relevant for most applications, the equations reduce to linear integral equations of Volterra type, which can in fact be solved by quadratures if a(t)/F(t)=c1 and b(t)/F(t)=c2 are time‐independent.

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