Orthogonal polynomials in two variables and second-order partial differential equations
نویسندگان
چکیده
منابع مشابه
Sobolev Orthogonal Polynomials in Two Variables and Second Order Partial Differential Equations
We consider polynomials in two variables which satisfy an admissible second order partial differential equation of the form (*) Auxx + 2Buxy + Cuyy +Dux + Euy = u; and are orthogonal relative to a symmetric bilinear form de ned by '(p; q) = h ; pqi+ h ; pxqxi ; where A; ; E are polynomials in x and y; is an eigenvalue parameter, and are linear functionals on polynomials. We nd a condition for ...
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The second order partial difference equation of two variables Du := A1,1(x)∆1∇1u+A1,2(x)∆1∇2u+ A2,1(x)∆2∇1u+ A2,2(x)∆2∇2u + B1(x)∆1u+ B2(x)∆2u = λu, is studied to determine when it has orthogonal polynomials as solutions. We derive conditions on D so that a weight function W exists for which WDu is self-adjoint and the difference equation has polynomial solutions which are orthogonal with respe...
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Orthogonal polynomials of degree n with respect to the weight function Wμ(x) = (1 − ‖x‖2)μ on the unit ball in R are known to satisfy the partial differential equation [ ∆− 〈x,∇〉 − (2μ+ d)〈x,∇〉 ] P = −n(n+ 2μ+ d)P for μ > −1. The singular case of μ = −1,−2, . . . is studied in this paper. Explicit polynomial solutions are constructed and the equation for ν = −2,−3, . . . is shown to have comple...
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ژورنال
عنوان ژورنال: Journal of Computational and Applied Mathematics
سال: 1997
ISSN: 0377-0427
DOI: 10.1016/s0377-0427(97)00082-4