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Euler-Darboux Equation

The Euler-Darboux equation is the partial differential equation

u_(xy)+(alphau_x-betau_y)/(x-y)=0
(1)

for constants alpha and beta, away from x=y.

Let x,xi in I and y,eta in J, where I and J are real intervals with disjoint closures. On the rectangle I×J, its Riemann function is

R(x,y;xi,eta)=((y-xi)/(y-x))^beta((eta-x)/(y-x))^alpha_2F_1(-alpha,-beta;1;z),
(2)

where z=(x-xi)(y-eta)/[(x-eta)(y-xi)] and _2F_1 is the hypergeometric function. The positive bases specify the powers, and R(xi,eta;xi,eta)=1. With (xi,eta) fixed, R satisfies

R_(xy)-alpha/(x-y)R_x+beta/(x-y)R_y+(alpha+beta)/((x-y)^2)R=0,
(3)

and its characteristic boundary values are

R(xi,y;xi,eta)=((eta-xi)/(y-xi))^alpha
(4)
R(x,eta;xi,eta)=((eta-xi)/(eta-x))^beta
(5)

(O. Marichev, pers. comm., July 7, 2008).

See also

Euler-Poisson-Darboux Equation

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References

Miller, W. Jr. Symmetry and Separation of Variables. Reading, MA: Addison-Wesley, 1977.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 129, 1997.

Referenced on Wolfram|Alpha

Euler-Darboux Equation

Cite this as:

Weisstein, Eric W. "Euler-Darboux Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Euler-DarbouxEquation.html

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