Understanding Linear Homogeneous PDE with Constant Coefficients
Explore the fundamentals of linear homogeneous partial differential equations with constant coefficients, including the complementary function and particular integral. Learn to solve such equations through detailed explanations and examples for distinct cases of roots in the auxiliary equation.
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Presentation Transcript
Partial differential equations with constant coefficients 1- homogenous linear equation with constant coefficients : A PDE which is linear with respect to the dependent variables ,and its derivative ,linear homogenous PDE is of the from all the derivatives are of the some order such as
??? ???+ ?1 ??? ??? ??? ??? 1??+ ?2 ??? ???= ? ?,? (1) ??? 2??2+ ?3 ??? 3??3 + .+?? ? ??,? = ? Now denote ? = ?? Then equation (1) can be written as the from (??+ ?1?? 1? + ?2?? 2? ?+ + ??? ?)? = ?(?,?)
Or ? ?,? ? = ? ?,? Solution of linear PDE Complete solution of (1) will consist of two parts a- the complementary function (C.F) b- the particular integral (P.I) the complementary function is a solution of ? ?,? ? = 0 (2)
To find the complementary function Let ? = ?(? + ??) be a the complementary function of (2) Now ??=?? ??= ?? ? + ?? ?2? ??2= ?2? (? + ??) ??? ???= ???(?)(? + ??) And ??2= Then ???=
Also ?? ??= ? ? + ?? ? ?= ?2? ??2= ? (? + ??) 2= ? ? Then ??? ???= ?(?)(? + ??) ?= ? ?
so ??? 1?? = ???(?)(? + ??) Substitution in equation (2) we get (??+ ?? 1?? 1+ .+??) ??( ) + ?? = 0 Which is satisfied if ??+ ?? 1?? 1+ .+??=0 (3) ?
the equation (4) is known as auxiliary equation The auxiliary equation obtained by putting ? = ? And ? = 1
Case1 : If ?1 ?2 ?3 .??are the distant real roots of the auxiliary equation the C.F of (1)is C.F=?1? + ?1? + ?2? + ?2? + .. ..+??? + ???
Case2 : If ?1= ?2= ?3= .= ??are the equal real roots of the auxiliary equation the C.F of (1)is C.F=?1? + ?1? + ??2? + ?2? + .. ..+????? + ???
Case 3 If ? = ? ?? are complex roots of auxiliary then C,F of equation (1) is C.F=?1? + ?1? + ?2? + ?2? + ? (?1? + ?1? ?2? + ?2? )
