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Costruction of classic exact solutions for Tricomi equation
Gianluca Argentini
To cite this version:
Gianluca Argentini. Costruction of classic exact solutions for Tricomi equation. 2015. �hal-01196780�
arXiv:1010.1681v1 [math.AP] 8 Oct 2010
Costruction of classic exact solutions for Tricomi equation
Gianluca Argentini
Research & Development Dept., Riello Burners - Italy gianluca.argentini@rielloburners.com
gianluca.argentini@gmail.com
2010, October
Abstract
A formula to construct classic exact solutions to Tricomi partial differential equation. The steps to obtain this formula require only elementary resolution of a simple system of first order PDEs.
Keywords
second order PDE, fluid dynamics, system of first order partial differential equa- tions.
The (generalized) Tricomi equation is the second order partial differential equation
∂xxf+a(x)∂yyf = 0 (1)
where f(x, y) and a(x) are regular functions. In the case a(x) =x, (1) is the classic Tricomi equation, as described by Prof. Francesco in his work dated 1923 [5]. The equation is an abstraction of the Euler equation on a 2D fluid mo- tion in the case of a flow speed near the sonic condition. For Tricomi equation some exact solutions formulas exist ([4] or [2]), and solutions to some partic- ular boundary values problems are known [1]. Also, some results about weak solutions are found [3].
Let beu=∂xf =fx,v=∂yf =fy. Then Tricomi equation can be written as a system of two first order equations:
ux(x, y) +a(x)vy(x, y) = 0
uy(x, y)−vx(x, y) = 0 (2)
We define, for convenience reasons, a function t=t(x, y) =v(x, y). From the first equation we can write
u(x, y) =− Z x
[a(s)t (s, y)]ds+g(y) (3)
where a∈ R and g is an arbitrary function of real variable. Now we have to find the condition thatgandtmust satisfy to verify the second equation of the system (2). This equation is now, from the usual rule of derivation of integrals depending on parameters,
− Z x
a
[a(s)tyy(s, y)]ds+g′(y) =tx (4) and hence
g(y) = Z y
b
tx(x, r) + Z x
a
[a(s)tyy(s, r)]ds
dr (5)
withb∈R. Butg must depend only ony: using the Fundamental Theorem of the Integral Calculus we have
0 =∂xg(y) = Z y
b
[txx(x, r) +a(x)tyy(x, r)]dr ∀y (6) If t is a solution of Tricomi equation (1), previous condition is verified. Note that, integrating with respect toying(y) formula, the expression forubecomes
u(x, y) = Z y
b
tx(x, r)dr− Z x
a
[a(s)ty(s, b)]ds (7) So, we have found that ift is a solution of Tricomi equation, then a functionf such thatfx=uandfy =t, withugiven by (7), is a solution too. Hence, from fy=t we can write
f(x, y) = Z y
b
t(x, r)dr+h(x) (8)
wherehis an arbitrary function of real variable. From fx=ufollows Z y
b
tx(x, r)dr+h′(x) = Z y
b
tx(x, r)dr− Z x
a
[a(s)ty(s, b)]ds (9) so that
h(x) =− Z x
a
Z q
a
[a(s)ty(s, b)]ds dq (10) We have so proven thatif t=t(x, y)is a generic solution of Tricomi equation, then the following formula gives a solution of the same equation too:
f(x, y) = Z y
b
t(x, r)dr− Z x
a
Z q
a
[a(s)ty(s, b)]ds dq (11)
Fora-posterioriverification, let be tsuch thattxx+a(x)tyy= 0. We have
fxx= Z y
b
txx(x, r)dr−a(x)ty(x, b) =
=− Z y
b
a(x)tyy(x, r)dr−a(x)ty(x, b) =
=−a(x)ty(x, y) +a(x)ty(x, b)−a(x)ty(x, b) =
=−a(x)ty(x, y) =
=−a(x)fyy (12)
so the solving formula (11) is verified.
Example1. Let be fxx+xfyy= 0 the original Tricomi equation. Note that t(x, y) =yis a (trivial) solution. Then, from (11) witha=b= 0, the function
f(x, y) = Z y
0
y dr− Z x
0
Z q
0
s ds dq= 1 2y2−1
6x3 (13)
is a solution too.
Example2. Let befxx+ cos(x)fyy= 0 a generalized Tricomi equation. Note that t(x, y) = y is a (trivial) solution. Then, from (11) with a = b = 0, the function
f(x, y) = Z y
0
y dr− Z x
0
Z q
0
cos(s)ds dq =−1 + 1
2y2+ cos(x) (14) is a solution too.
Note that formula (11) can be used as a solutions machine: starting from the most simple not null solution, that is fromt(x, y) = 1, it can be used for itera- tive construction of solutions.
Also, formula (11) can be used to find solutions to some particular boundary values problems for Tricomi equation. For example, in the casea(x) =xand b= 0, if the functiont(x, y) is such thatty(x,0) = 0∀x(condition satisfied e.g.
byt=−16x3+12y2), then from (11) follows f(x, y) =
Z y
0
t(x, r)dr (15)
so thatf(x,0) = 0∀x∈R. Hence, ifty(x,0) = 0∀x, thenf(x, y) =Ry
0 t(x, r)dr is a solution of the boundary values differential problem
∂xxf +a(x)∂yyf = 0, f(x,0) = 0 x∈R (16) Iftis a solution of (1) satisfying the boundary conditionst(x,0) =g(x), withg a function of real variable, then from (11) follows thatfy(x,0) =g(x), that is
∂xxf+a(x)∂yyf = 0, ∂yf(x,0) =g(x) x∈R (17)
References
[1] M. Cinquini Cibrario, Equazioni a derivate parziali di tipo misto, Milan Journal of Mathematics, vol. 25,1, Springer, 1955
[2] http://eqworld.ipmnet.ru/en/solutions/lpde/lpde402.pdf, 2010
[3] K. Yagdjian, Global Existence for the n-Dimensional Semilinear Tricomi- Type Equations,Comm. in partial differential equations, Vol. 31,6, 2006 [4] A. D. Polyanin,Handbook of Linear Partial Differential Equations for En-
gineers and Scientists, Chapman & Hall/CRC, 2002
[5] F. Tricomi, Sulle equazioni differenziali alle derivate parziali di secondo or- dine di tipo misto, Memorie della Reale Accademia dei Lincei, S. 5a, Vol.
XIV, fasc. 7, 1923
