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Legendre's Differential Equations
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Computational methods for Legendre's equation
Numerical solutions such as finite difference methods or spectral methods can be used when analytical solutions are intractable.
General form of Legendre's differential equation
Solve by converting it to a power series and finding the coefficients recursively; the general solution is a combination of Legendre polynomials and Legendre functions of the second kind.
Orthogonality of Legendre polynomials
Apply the orthogonality property to simplify the calculation of coefficients in series solutions, especially for boundary value problems.
Boundary conditions for physical applications
In physical applications, enforce boundary conditions that often lead to a requirement that the series solution must terminate.
Legendre's differential equation with non-integer orders
Non-integer orders lead to Legendre functions of the first and second kind, which are important in many applications including geophysics and electromagnetic theory.
Associated Legendre's differential equation
The associated equation involves an additional term and is solved by methods similar to the ordinary Legendre's equation, resulting in associated Legendre functions.
Recurrence relations of Legendre polynomials
Leverage recurrence relations for efficient computation of Legendre polynomials, especially for high orders.
Solving Legendre's equation for integer orders
Use Rodrigues' formula to obtain explicit expressions for integer order solutions, which are known as Legendre polynomials.
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