Abstract operators and higher-order linear partial differential equations

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Abstract: For the linear partial differential equation

P (p a r t i a l_{x}, p a r t i a l_{t}) u = f (x, t)

, where

x i n m a t h b b R^{n},; t i n m a t h b b R^{1}

, with

P (p a r t i a l_{x}, p a r t i a l_{t})

is

p r o d_{i = 1}^{m} (f r a c p a r t i a l p a r t i a l t - a_{i} P (p a r t i a l_{x}))

or

p r o d_{i = 1}^{m} (f r a c p a r t i a l^{2} p a r t i a l t^{2} - a_{i}^{2} P (p a r t i a l_{x}))

, the authors give the analytic solution of the cauchy problem using the abstract operators

e^{t P (p a r t i a l_{x})}

and

f r a c s i n h (t P (p a r t i a l_{x})^{1 / 2}) P (p a r t i a l_{x})^{1 / 2}

. By representing the operators with integrals, explicit solutions are obtained with an integral form of a given function.

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