Mathematical Physics Vol 1

7.4 Linear second order PDE

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In this case it is said that the solution us obtained from the solution u 1 – by shifting the argument . Property 4. Combining the last two properties, we obtain that the solutions are also u = Z C ( α 1 ) u 1 ( x 1 − α 1 , x 2 ) d α 1 , (7.91) u = x C ( α 1 , α 2 ) u 1 ( x 1 − α 1 , x 2 − α 2 ) d α 1 d α 2 . (7.92) For such solutions, it is said that they are obtained by convolution or as a resultant of functions C and u 1 . Property 5. If the equation (7.83), with real constant coefficients, has a complex solution of the form u = P ( x 1 , x 2 )+ i · Q ( x 1 , x 2 ) , (7.93) where P and Q are real functions, and i = √ − 1 is the imaginary unit, then the functions P and Q are also solutions of this equation.

Proof L ( u )= L ( P + iQ )= L ( P )+ iL ( Q )= 0 ⇒ L ( P )= 0 ∧ L ( Q )= 0 .

(7.94)

7.4.2 Classification of second order LDE with two variables

When studying partial equations, the question arises as to whether it is possible to simplify the initial equation by introducing appropriate transformations. In this chapter, we take, for simplicity, that n = 2, i.e. that the unknown function is of the form u = u ( x 1 , x 2 ) . Observe now the homogeneous partial differential equation

L ( u )= 0 ,

(7.95)

which will be transformed by introducing new variables ξ 1 and ξ 2

∂ξ 1 ∂ x 1 ∂ξ 2 ∂ x 1

∂ξ 1 ∂ x 2

ξ i = ξ i ( x 1 , x 2 ) , J

ξ 1 , ξ 2 x 1 , x 2 ≡

= 0 , i = 1 , 2 .

(7.96)

∂ξ 2 ∂ x 2 ̸

where J is the Jacobi 9 functional determinant or Jacobian. The relation between new and old derivatives is given by the following equations

n (= 2 ) ∑ i = 1

∂ u ∂ x j

∂ u ∂ξ i

∂ξ i ∂ x j

j = 1 , 2

(7.97)

=

,

9 Carl Gustav Jacobi 1804-1851, German mathematician. Author of important works in analysis, especially theory of elliptic functions.