Functional Solutions for a Plane Problem in Magnetohydrodynamics

Functional Solutions for a Plane Problem in Magnetohydrodynamics

GIOVANNICIMATTI

ABSTRACT - We study the flow in a channel of a fluid obeying the equations of magnetohydrodynamics under the hypotheses that viscosity, resistivity and thermal conductivity depend on the temperature. The special class of solutions for which two functional dependences between temperature and magnetic field and velocity and magnetic field exist is considered.

1. Introduction

We study in this paper a class of steady, incompressible and recti- linear flows of viscous, electrically and thermally conducting fluids along cylindrical channels of arbitrary cross-section in the framework of the equations of magnetohydrodynamics. The open and bounded subset of R², representing the cross-section of the pipe is referred to the orthogonal frame Ox₁x₂x₃.Ox₃ is the axis of the channel with unit vector i3. The magnetic fieldH and the velocityvare assu med to be of the form

Hˆh(x1;x2)i3

(1:1)

vˆv(x₁;x₂)i₃: (1:2)

Moreover, we suppose

pˆp(x1;x2):

(1:3)

The electrical resistivityr, the viscositynand the thermal conductivitykare assumed to be given functions of the temperature uand ofv andh. The

(*) Indirizzo dell'A.: Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy.

E-mail: cimatti@dm.unipi.it

domainVis multiply connected with its boundaryGconsisting of two dis- joint closed curvesG1 andG2 of classC¹. From the general equations of magnetohydrodynamics and from the energy equation [6] we obtain, in view of (1.1) and (1.2) the following boundary value problem (P) [4]. For more details we refer to the Appendix.

r (r(u;v;h)rh)ˆ0 inV (1:4)

r (n(u;v;h)rv)ˆ0 inV (1:5)

r

k(u;v;h)ru‡n(u;v;h)vrv‡r(u;v;h)hrh

ˆ0 inV (1:6)

hˆ0 onG1; hˆh onG2

vˆ0 onG₁; vˆvonG₂ uˆ0 onG₁; uˆu onG₂;

whereh,vanduare given positive constants. We prove in Section 2 that if the functionsk,nandrhave the special form (2.9) below, problem (P) has one and only one solution. Section 3 deals with the functional solutions of problem (P) i.e. with solutions for which there exist functional de- pendences betweenuandhand betweenvandh. We prove the existence of a one-to-one correspondence between the functional solutions and the solutions of a two-point problem for a system of first order differential equations depending on two parameters. First order ordinary differ- ential equations depending on parameters whose solutions must satisfy two boundary conditions have been considered, among others, in [2], [3], [7] and [8]. In Section 4 a theorem of existence and uniqueness is proved when resistivity, viscosity and thermal conductivity depend only on the temperature.

2. A result of existence and uniqueness

To prove the next theorem we need the following elementary LEMMA1. Letk(u)^ 2C¹([0;1)); ^n(v)2C¹([0;v]); 2C¹([0;h])

^

k(u)k₀ >0 (2:1)

and

^n(u)n0>0:

(2:2)

The transformation

uˆK(u)‡M(v)‡h² (2:3) 2

cˆN(v);

(2:4)

defined for u2[0;1), v2[0;h]and depending on the parameter h2[0;h], where

K(u)ˆ Z^u

0

^

k(t)dt; M(v)ˆ Z^v

0

t^n(t)dt; N(v)ˆ Z^v

0

^ n(t)dt

is globally invertible for every h2[0;h].

PROOF. By (2.2), (2.4) has an inverse vˆN ¹(c) (2:5)

defined in [0;N(v)]. Substituting (2.5) into (2.3) we obtain uˆK(u)‡M(N ¹(c))‡h²

2 : (2:6)

Since by (2.1)K(u) is strictly increasing and

u!1lim K(u)ˆ 1 (2:7)

(2.6) has, with respect tou, a unique solution uˆK ¹

u‡M(N ¹(c))‡h² 2

: (2:8)

The inverse of (2.3), (2.4) is therefore (2.8), (2.5). p THEOREM1. Ifkandnare given by

k(u;v;h)ˆr(u;v;h)^k(u); n(u;v;h)ˆr(u;v;h)^n(v);

(2:9)

wherer(u;v;h),^k(u)and^n(v)are C¹functions such that r(u;v;h)r₀>0; ^k(u)k0>0; ^n(v)n0>0 (2:10)

when u0,vv0,hh0, then problem(P)has one and only one solution.

PROOF. By the maximum principle the solutions of problem (P) satisfy u(x)0; vv(x)0; hh(x)0 inV; xˆ(x1;x2):

DefineK(u),M(v) andN(v) as in Lemma 1. The transformation uˆK(u)‡M(v)‡h²

2 cˆN(v) by Lemma 1 is globally invertible with inverse

uˆU(u;c;h)ˆK ¹

u‡M(N ¹(c))‡h² 2 (2:11)

vˆV(c)ˆN ¹(c):

(2:12) Define

a(u;c;h)ˆr(U(u;c;h);V(c);h):

(2:13)

Let (u(x);v(x);h(x)) be any classical solution of problem (P) and set u(x)ˆK(u(x))‡M(v(x))‡h(x)²

2 c(x)ˆN(v(x)):

Problem (P) can be reformulated in terms ofu,candhas problem (P) r (a(u;c;h)ru)ˆ0 inV

(2:14)

r (a(u;c;h)rc)ˆ0 inV (2:15)

r (a(u;c;h)rh)ˆ0 inV (2:16)

uˆ0 onG₁; uˆuˆK(u)‡M(v)‡h² 2 onG₂ cˆ0 onG1; cˆc ˆN(v) on G2

hˆ0 onG₁; hˆh onG₂:

We claim that problem (P) has one and only one solution. For, let (u(x);c(x);h(x)) be any solution of (P). Define

z(x)ˆu(x) u hh(x) j(x)ˆc(x) c

hh(x):

We havezˆ0 andjˆ0 onG. On the other hand, by (2.14) and (2.16) r (a(u;c;h)rz)ˆr (a(u;c;h)ru) u

hr (a(u;c;h)rh)ˆ0 inV (2:17)

and, by (2.15) and (2.16),

r(a(u;c;h)rj)ˆ r(a(u;c;h)rc) c

hr(a(u;c;h)rh)ˆ0 inV:

(2:18)

Multiplying (2.17) byzand integrating by parts overVwe have Z

V

a(u;c;h)jrzj²dx₁dx₂ˆ0:

Similarly from (2.18) we obtain Z

V

a(u;c;h)jrjj²dx₁dx₂ˆ0:

Hence

z(x)ˆ0; j(x)ˆ0 inV:

Thusu(x),c(x) andh(x) are related by the functional relations u(x)ˆu

hh(x) (2:19)

c(x)ˆc hh(x):

(2:20)

Hence the equation (2.16) can be rewritten r a

u hh;c

hh;h

rh

!

ˆ0 (2:21)

hˆ0 onG₁; hˆhonG₂: (2:22)

This problem has one and only one solution. For, let wˆ F(h)ˆ

Z^h

0

a u

ht;c ht;t

dt:

Equation (2.21) under this transformation becomes

Dwˆ0 inV; wˆ0 onG₁; wˆ F(h) onG₂:

Hence

h(x)ˆ F ¹(w(x)); u(x)ˆu

hF ¹(w(x)); c(x)ˆc

hF ¹(w(x)):

By (2.11) and (2.12) we obtain as the only solution of problem (P) u(x)ˆK ¹

u

hF ¹(w(x))‡M

N ¹ c

hF ¹(w(x))

‡1 2

F ¹(w(x)) ₂

v(x)ˆN ¹ c

hF ¹(w(x))

h(x)ˆ F ¹(w(x)):

p

3. The functional solutions

The assumptions (2.9) are quite special. To treat more general situa- tions we use a different definition of solutions suggested by the proof of Theorem 1 and in particular by the functional relations (2.19) and (2.20) betweenu,candh.

DEFINITION. We say that a classical solution (u(x);v(x);h(x)) of problem (P) is a functional solution if two functions U(h)2C²([0;h]), V(h)2C²([0;h]) exist such that

u(x)ˆ U(h(x)); v(x)ˆ V(h(x)):

THEOREM2. If

r(u;v;h)r₀>0 (3:1)

then there exists a one-to-one correspondence between the setfFSgof the functional solutions of problem (P) and the set fTPPg of the solutions (U(h);V(h);g;m)of the two-point problem(TPP)

n(U;V;h)dV

dh ˆgr(U;V;h) (3:2)

k(U;V;h)dU

dh‡n(U;V;h)VdV

dh‡hr(U;V;h)ˆmr(U;V;h) (3:3)

U(0)ˆ0; U(h)ˆu (3:4)

V(0)ˆ0; V(h)ˆv:

(3:5)

PROOF. We define the map I :fTPPg ! fPg as follows. Suppose (U(h);V(h);g;m)2 fTPPgand consider the nonlinear boundary value pro- blem

r (r(U(h);V(h);h)rh)ˆ0 inV (3:6)

hˆ0 onG₁; hˆhonG₂: (3:7)

It is easily seen that (3.6), (3.7) has one and only one solution. For, let wˆR(h)ˆ

Z^h

0

r(U(t);V(t);t)dt:

(3:8)

By (3.1) R maps one-to-one [0;h] onto [0; R(h)]. Moreover, under (3.8) problem (3.6), (3.7) becomes

Dwˆ0 inV (3:9)

wˆ0 onG1; wˆR(h) on G2: (3:10)

Problem (3.9), (3.10) has one and only one solution. Therefore, the only solution to problem (3.6), (3.7) is given byh(x)ˆR ¹(w(x)). We claim that

(u(x);v(x);h(x))ˆ(U(h(x));V(h(x));h(x))2 fPg:

(3:11)

We have by (3.2) and (3.6)

r (n(u(x);v(x);h(x))rv)ˆ r (n(U(h(x));V(h(x));h(x))dV dhrh)ˆ (3:12)

ˆgr (r(U(h(x));V(h(x));h(x))rh)ˆ0 inV:

Moreover, by (3.3) and (3.6) r

k(u;v;h)ru‡vn(u;v;h)rv‡hr(u;v;h)rh

ˆ (3:13)

ˆ r

(k(U(h);V(h);h)dU

dh‡ V(h)n(U(h);V(h);h)dV

dh‡hr(U(h);V(h);h)rh

ˆ

ˆmr (r(U(h);V(h);h)rh)ˆ0 inV:

All the boundary conditions of problem (P) are also satisfied in view of (3.4) and (3.5). Hence I is well-defined. On the other hand,I is one-to-one. For, let (U(h);V(h);g;m)2 fTPPg; (U(h);~ V(h);~ ~g;m)~ 2 fTPPg and (u(x);v(x);h(x))ˆ I(U(h);V(h);g;m), (~u(x);~v(x);~h(x))ˆ I(U(h);~ V(h);~ ~g;m). If~ I(U(h);V(h);g;m)ˆ I(U~(h);V(h);~ ~g;~m) we have u(x)ˆu(x),~ v(x)ˆ~v(x) and h(x)ˆ~h(x), but u(x)ˆ U(h(x)), u(x)~ ˆU(h(x)) and~ v(x)ˆ V(h(x)), ~v(x)ˆV(h(x)). Hence~

U(h(x))ˆU(~~h(x))ˆU(h(x)) and~ V(h(x))ˆV(~h(x))~ ˆV(h(x)). Thus~ U(h)ˆU~(h); V(h)ˆV(h):~

(3:14)

This implies also gˆ~g and mˆ~m. Finally I is surjective. For, let (u(x);v(x);h(x))ˆ(U(h(x));V(h(x));h(x))2 fFSg. Define

uˆ Z^h

0

n(U(t);V(t);t)dV dt(t)dt (3:15)

cˆ Z^h

0

k(U(t);V(t);t)dU

dt(t)‡ V(t)n(U(t);V(t);t)dV

dt(t)‡tr(U(t);V(t);t)

dt

zˆ Z^h

0

r(U(t);V(t);t)dt (3:16)

and

U(x)ˆu(h(x)); C(x)ˆc(h(x)); Z(x)ˆz(h(x)):

(3:17)

By (1.4), (1.5) and (1.6) we have

DUˆ0; DCˆ0; DZˆ0 inV with

Uˆ0 onG₁; UˆAonG₂ (3:18)

Cˆ0 onG1; CˆBonG2

(3:19)

Zˆ0 onG₁; ZˆConG₂ where

Aˆ Z^h

0

n(U(t);V(t);t)dV dt(t)dt

Bˆ Z^h

0

k(U(t);V(t);t)dU

dt (t)‡ V(t)n(U(t);V(t);t)dV

dt (t)‡tr(U(t);V(t);t)

dt

Cˆ Z^h

0

r(U(t);V(t);t)dt

andC6ˆ0 by (3.1). Letz(x) be given by the solution of the problem Dzˆ0 inV; zˆ0 onG1; zˆ1 onG2:

(3:20) We have

U(x)ˆAz(x); C(x)ˆBz(x); Z(x)ˆCz(x):

(3:21) Setting

gˆA

C; mˆB (3:22) C

we obtain

U(x)ˆgZ(x); C(x)ˆmZ(x) (3:23)

and by (3.15), (3.16), and (3.17) we haveuˆgzandcˆmzi.e.

Z^h

0

n(U(t);V(t);t)dV

dt(t)dtˆg Z^h

0

r(U(t);V(t);t)dt (3:24)

Z^h

0

k(U(t);V(t);t)dU

dt(t)‡V(t)n(U(t);V(t);t)dV

dt(t)‡tr(U(t);V(t);t)

dtˆ (3:25)

ˆm Z^h

0

r(U(t);V(t);t)dt:

Taking the derivative with respect to h in (3.24) and (3.25) we obtain (3.2) and (3.3). Hence, with g and m given by (3.22) we have

I(U(h);V(h);g;m)ˆ(u(x);v(x);h(x)). p

As an immediate consequence of Theorem 2 problem (P) has one and only one functional solution if the corresponding problem (TPP) has one and only one solution. Moreover, the search of the functional solution of problem (P) is broken into two steps: (i) to find the solutions (U(h);V(h);g;m) of problem (TPP) and (ii) to solve problem (3.20). Step (ii) contains, so-to-speak, the geometry of problem (P) and step (i) the nonlinear part.

To give a theorem of existence for the two-point problem (TPP) we note that (3.2), (3.3), (3.4) and (3.5) can be rewritten as follows

dV

dh ˆg r(U;V;h) n(U;V;h) (3:26)

dU

dh ˆ( gV h‡m)r(U;V;h) k(U;V;h) (3:27)

U(0)ˆ0; U(h)ˆu (3:28)

V(0)ˆ0; V(h)ˆv:

(3:29)

THEOREM3. If

k(u;v;h)>0; r(u;v;h)>0; n(u;v;h)>0 (3:30)

K2r(u;v;h)

k(u;v;h))K1>0 (3:31)

and

N2r(u;v;h)

n(u;v;h))N1>0 (3:32)

the two-point problem(3.26)-(3.29)has at least one solution.

PROOF. We transform problem (3.26)-(3.29) in a system of two Vol- terra-Fredholm integral equations. From (3.26) and (3.29) we have

gˆ~gˆ v R^h

0

_r

n]dt (3:33)

where

r n

ˆr(U(t);V(t);h(t)) n(U(t);V(t);h(t)): Moreover, from (3.27) and (3.28) we obtain

mˆm~ˆ 1 R^h

0

r k

dt

u‡~g Z^h

0

V(t) r

k

dt‡ Z^h

0

t r

k

dt (3:34)

where

r k

ˆr(U(t);V(t);h(t)) k(U(t);V(t);h(t)):

Substituting (3.33) and (3.34) in (3.26) and (3.27) and integrating with re-

spect tohwe have the system of integral equations U(h)ˆ~m

Z^h

0

r k

dt ~g Z^h

0

V(t) r

k

dt Z^h

0

t r

k

dt (3:35)

V(h)ˆ~g Z^h

0

r n

dt:

(3:36)

From (3.33) we have

0~gG2ˆ: v N₁h (3:37)

and from (3.36) and (3.26)

0 V(h)V2ˆ:N2

N₁vh² (3:38)

dV

dh(h)N2

N₁vh:

(3:39)

Moreover, by (3.34) and (3.37) 0~mM₂ˆ: 1

K1h

u‡vh³N₂K₂ N₁² ‡h²K₂

2

: (3:40)

Hence, by (3.35) and (3.27)

jU(h)j U2ˆ:M2K2h‡v²h⁴K2N2

N²₁ ‡K²h² (3:41)

dU dh

D2ˆ:(V2G2‡h‡M2)K2: (3:42)

Using (3.38), (3.39), (3.41) and (3.42) we can apply the Schauder fixed point theorem. For, define

Bˆ f(U(h);V(h))2C⁰([0;h]) C⁰([0;h]); jU(h)j U₂; jV(h)j V₂g and letT :B!C⁰([0;h])C⁰([0;h]),T ˆ(T1;T2)

T₁(U;V)ˆ~g Z^h

0

r n

dt:

T₂(U;V)ˆ~m Z^h

0

r k

dt ~g Z^h

0

V(t) r

k

dt Z^h

0

t r

k

dt:

We haveT(B)B. Moreover, proceeding as in the proof of (3.39) and (3.42) we find thatT(B) is bounded inC¹([0;h])C¹([0;h]) and thus compact in C⁰([0;h])C⁰([0;h]) by Arzela's theorem. We conclude that the two-point problem (3.26)-(3.29) has at least one solution. p As a consequence of the theorem just proved and of Theorem 1 problem (P) has at least one functional solution if (3.1), (3.30), (3.31) and (3.32) are satisfied.

4. A result of uniqueness

In general [5] the two-point problem (TPP) has more than one solution.

It is therefore interesting to give conditions under which the solution is unique. For example, if condition (2.9) holds we have as corresponding two- point problem

^n(V)dV dhˆg

^k(U)dU

dh‡ V^n(V)dV

dh‡h^r(h)ˆm U(0)ˆ0; U(h)ˆu

V(0)ˆ0; V(h)ˆv

and it is easily seen that in this case the solution is unique. In the next theorem a result of uniqueness is given whenk,nandrdepend only on the temperatureu. In practical cases the main dependence ofk,nandr is precisely from the temperature. In the proof we use the following Lemma [1] .

LEMMA2. Ifb(u)2C¹([0;1)),b(u)0andb⁰(u)0then the problem d²u

dh²ˆ1‡g²b(u); u(0)ˆ0; u(h)ˆu>0 (4:1)

has, for everyg2R¹, one and only one solution.

PROOF. Simply note thatu(h)0 by the maximum principle and that d

du(1‡g²b(u))ˆg²b⁰(u)0. p

THEOREM4. Ifk(u),n(u)andr(u)2C¹([0;1))and satisfy k(u)>0; r(u)>0; n(u)>0

(4:2)

Z¹

0

k(t) r(t)dtˆ 1 (4:3)

r⁰(u)n(u)r(u)n⁰(u) (4:4)

and

r(u)

n(u)r0>0 (4:5)

then there exists one and only one functional solution of problem(P).

PROOF. According to Theorem 2 the two-point problem corresponding to problem (P) is, in this case:

n(U)dV

dhˆgr(U) (4:6)

k(U) r(U)

dU

dh‡gV ‡hˆm (4:7)

U(0)ˆ0; U(h)ˆu (4:8)

V(0)ˆ0; V(h) ˆv:

(4:9)

We note that

g>0 (4:10)

since g0 is incompatible with (4.2) and (4.8). By (4.3) the function uˆF(U)

F(U)ˆ Z^U

0

k(t) r(t)dt (4:11)

maps diffeomorphically [0;1) onto [0;1). The system (4.6)-(4.9) becomes, in terms ofuandV,

dV

dh ˆgb(u); V(0)ˆ0; V(h)ˆv (4:12)

du

dh‡gV ‡hˆm; u(0)ˆ0; u(h)ˆuˆF(u);

(4:13)

where

b(u)ˆr(F ¹(u)) n(F ¹(u)): (4:14)

From (4.13) we have

du² dh²‡gdV

dh‡1ˆ0:

(4:15)

Substituting (4.12) into (4.15) we obtain du²

dh²ˆ1‡g²b(u) (4:16)

u(0)ˆ0; u(h) ˆu:

(4:17)

If (4.4) holds we haveb⁰(u)0. Thus, by Lemma 2 problem (4.16), (4.17) has for everygone and only one solutionu(h;g) and

U(h;g)ˆF ¹(u(h;g)):

(4:18)

Moreover, by (4.12)

V(h)ˆg Z^h

0

b(u(t;g))dt:

(4:19)

The parameterg, still at our disposal, is now used to satisfy the boundary conditionV(h) ˆvwhich, by (4.19), reads

vˆg Z^h

0

b(u(t;g))dt:

(4:20)

To prove that (4.20), as an equation ing, has one and only one solution we define

G(g)ˆg Z^h

0

b(u(t;g))dt:

We haveG(0)ˆ0 andG(g)>for allg>0. Moreover,

G⁰(g)ˆ Z^h

0

b(u(t;g))dt‡g Z^h

0

b⁰(u(t;g))@u

@g(t;g)dt:

(4:21)

We claim that @u

@g(t;g)0. By the continuous differentiability with re- spect to the parameterg and by (4.13) we have

dug

dh ˆ V(h) (4:22)

ug(0)ˆ0:

(4:23)

On the other hand, by (4.12) and (4.10) we haveV(h;g)>0. Thus, from (4.22) and (4.23) we obtain

ug(h;g)50:

(4:24)

Hence, by (4.21) and (4.5)

G⁰(g) Z^h

0

b(u(t;g;g))dtr0h>0:

Therefore (4.20) has, for everyv, one and only one solution~g. We conclude that the only solution of problem (4.6)-(4.9) is given by

U(h)ˆF ¹(u(h;~g)) (4:25)

V(h)ˆ~g Z^h

0

b(u(t;~g))dt:

(4:26)

By Theorem 2 there is only one functional solution of problem (P) which we obtain in the usual way: we solve the problem

r (r(U(h))rh)ˆ0 inV; hˆ0 onG₁; hˆhonG₂ (4:27)

defining

wˆ H(h)ˆ Z^h

0

r(U(t))dt:

By (4.2)Hmaps ono-to-one [0;h] onto [0;w] where w ˆ H(h). In term ofw problem (4.27) reads

Dwˆ0 inV; wˆ0 onG₁; wˆw onG₂: (4:28)

Letw(x) be the solution of (4.28). We have h(x)ˆ H ¹(w(x)) (4:29)

and

u(x)ˆ U(h(x))ˆ U(H ¹(w(x))) (4:30)

v(x)ˆ V(h(x))ˆ V(H ¹(w(x))):

(4:31)

The only functional solution of problem (P) is given by (4.29), (4.30) and

(4.31). p

We stress the difference between Theorem 1, where we found un- iqueness in the general class of classical solutionsfCSg, and Theorem 4 where uniqueness is proved in the smaller class of functional solutions fFSg. It would be interesting to show that the solution, under the hy- potheses of Theorem 4, is unique also infCSg.

5. Appendix

We derive here the equations (1.4), (1.5) and (1.6) entering in pro- blem (P). We assume the magnetic permeability m₀ to be a constant.

Moreover, the pressure p, the magnetic field H and the velocity v are supposed to have the special form (1.1), (1.2) and (1.3). Using the Maxwell equation

Jˆ c 4pr H (5:1)

the magnetohydrodynamic body forces f ˆm₀

c JH by (1.1) take the form

f ˆ m₀ 2ph @h

@x₁i1 m₀ 2ph @h

@x₂i2: (5:2)

Thus, by (5.2), (1.2) and (1.3) the Navier-Stokes equations reduce to r (nrv)ˆ0

(5:3)

@p

@x₁‡h @h

@x₁ˆ0 (5:4)

@p

@x₂‡h @h

@x₂ˆ0:

(5:5)

In view of (1.1) and (1.2), the Ohm's lawrJˆE‡vm₀Hgives rJˆE:

(5:6)

Moreover, sinceEis irrotational, we have, taking thecurlof (5.1), r (rr H)ˆ0

(5:7) and by (1.1)

r (rrh)ˆ0:

(5:8)

The energy equation reads

r (kru)ˆEJ‡nX³

i;kˆ1

@v_i

@xk‡@v_k

@xi

@v_i

@xk

(5:9)

where in the right hand side the first term represents the Joule heating and the second the viscous heating. Specializing (5.9) in our case we obtain, in view of (1.1), (1.2) and (5.6),

r (kru)ˆrjrhj²‡njrvj²: (5:10)

This equation, by (5.3) and (5.8), can also be written as follows r

kru‡rhrh‡vnrv]ˆ0:

Once h(x₁;x₂) is known as part of the solution of problem (P), we can computep(x₁;x₂) from (5.4) and (5.5).

REFERENCES

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[2] F. CAFIERO, Su un problema ai limiti relativo all'equazione y⁰ˆf(x;y;l), Giornale di Matematiche di Battaglini,77(1947), pp. 28-35.

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Padova,18(1949), pp. 239-257.

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gence form with constant boundary conditions, to appear in Proc. (A) Roy.

Soc. Edin.

[6] A. JEFFREY,Magnetohydrodynamics, Oliver and Boyd LTD, John Wiley and Sons, Inc., New York 1966.

[7] K. ZAWISCHA,UÈ ber die Differentialgleichung y⁰ˆkf(x;y)deren LoÈsungkurve durch zwei gegebene Punkte hindurchgehen soll, Monatsh. fuÈr Math. und Phys.37(1930), pp. 103-124.

[8] G. ZWIRNER,Un criterio d'esistenza e unicitaÁ per gli integrali dell'equazione y⁰ˆlf(x;y)passanti per due punti assegnati, Boll. Un. Mat. Ital.,3(1948), pp. 15-18.

Manoscritto pervenuto in redazione il 25 gennaio 2011.