Generalized Frankl-Rassias Problem for a Class of Mixed Type Equations in an Infinite Domain

RUZIEV M.Kh.

Institute of Mathematics,National University of Uzbekistan,Durmon yuli str.29, 100125 Tashkent,Uzbekistan.

Generalized Frankl-Rassias Problem for a Class of Mixed Type Equations in an Infinite Domain

RUZIEV M.Kh.?

Institute of Mathematics,National University of Uzbekistan,Durmon yuli str.29, 100125 Tashkent,Uzbekistan.

Received 14 December 2013;Accepted 27 February 2014

.In this paper,we study the boundary-value problem for mixed type equation with singular coefficient.We prove the unique solvability of the mentioned problem with the help of the extremum principle.The proof of the existence is based on the theory of singular integral equations,Wiener-Hopf equations and Fredholm integral equations.

Extremum principle;infinite domain;non-local problem;mixed type equation; Cauchy problem;Darboux formula;integral equations.

1 Introduction

Boundary-valueproblemsformixedtypeequationsinvarious unboundeddomains were studied in detail in works[1-2].The existence and the uniqueness of the solution of Frankl’s problem for several mixed type equations were proved by A.V.Bitsadze[3],U.V. Devingtal[4],N.M.Fleisher[5]and by other scientists.A series of interesting results, devoted to studying boundary-value problems for partial differential equations were obtained in works[6-8].In the work[9]a non-local problem with Bitsadze-Samarskii condition on parallel characteristics,one of which lies inside of characteristic triangle, for mixed type equation was investigated.In[10]the general Tricomi-Rassias problem was investigated for the generalized Chaplygin equation.The Tricomi problem with two parabolic lines of degeneracy,was considered in[11].

The uniqueness of quasi-regular solutions for the exterior Tricomi and Frankl problems for quaterelliptic-quaterhyperbolic mixed type partial differential equations of second order with eight parabolic degenerate lines proved by J.M.Rassias[12].In the work[13]some boundary-value problems with nonlocal initial condition for model and degenerate parabolic equations with parameter were considered.The well-posedness of the extended Tricomi-Chaplygin-Frankl problem in a multidimensional region was established in[14].The Tricomi and Frankl problems for partial differential equations were investigated by G.C.Wen[15-17].Boundary value problems for the wave equation and equations of mixed type considered in[18].

The present work is devoted to the investigation of the problem with Frankl type condition on a segment of line of degeneration for mixed type equation with singular coefficient.

Formulation of the problem

Consider the equation

in the domain D=D+∪D-∪I,where D+is a first open quadrant of the plain,D-is a bounded domain in the fourth quadrant of the plane bounded by characteristics OC and BC of Eq.(1.1)issuing from points O(0,0)and B(1,0),and by segment OB of the straight line y=0,I={(x,y):0＜x＜1,y=0}.

Introduce the following denotations:

C0and C1are,correspondingly,points of intersection of characteristics OC and BC with characteristic issuing from the point E(c,0),where c∈I is an arbitrary fixed value.

Let p(x)∈C2[0,c]be a diffeomorphism from the set of points of the segment[0,c]to the set of points of the segment[c,1]such that p′(x)＜0,p(0)=1,and p(c)=c.As an example of such a function consider the linear function p(x)=1-kx,where k=(1-c)/c.

ProblemTF.Find a function u(x,y)with the following properties:and satisfies Eq.(1.1)in this domain;

3)u(x,y)is a generalized solution from the class R1(see[19])in the domain D-;

4)

5)u(x,y)satisfies the boundary conditions

and the conjugation condition

In addition,these limits with x=0,x=1,and x=c can have singularities of order less than 1-2β,where,μis a constant,τ1(x)∈C(ˉI1),and the function τ1(x)near the point x=1 is representable in the form,it satisfies Hoelder condition on any interval,and with sufficiently large x it satisfies the inequality,where M and δ are positive constants,?(y)∈ satisfies the Hoelder condition on any interval .

Note that condition(1.6)is an analog of the Frankl condition[3,4,20-22].

2 The uniqueness of the solution of Problem TF

Theorem 2.1.Let .Problem TF has only trivial solution.

Proof.By using a solution to the modified Cauchy problem for Eq.(1.1)in the domain D-with the boundary-value condition(1.5),we get

Γ(x)is Eulers gamma-function.

Relation(2.1)is the first functional relation between unknown functions τ(x)and ν(x),transferred to the interval I from the domain D-.Using the notation u(x,0)=τ(x), x∈,we can rewrite condition(1.6)as

It is well known that at the point of positive maximum(negative minimum)of the function τ(x)the fractional differential operator satisfy the inequality

Hence in view of(2.1)with Ψ(x)≡0,we get

Inequalities(2.4)and(2.5)contradict to gluing condition(1.7),hence we deduce that x0/∈(0,c).Nowassumethat x0∈(c,1).Let x1∈(0,c)be asolutiontotheequation p(x1)=x0. In this case from formula(2.2)withf(x)≡0 we obtain

Now,restoring the desired function u(x,y),taking into account the continuity of the solution to Problem TF and the conjugation condition(1.7)in the domain D-,as a solution to the modified Cauchy problem with homogeneousdata(2.7)we obtain u(x,y)≡0 in the domain D-.

3 The existence of the solution of Problem TF

Theorem 3.1.Let the following conditions be satisfied:k1/2-αμsinαπ＜1,p(x)=1-kx,where α=(1-2β)/4.Then the solution of the Problem TF exists.

Proof.One can easily prove that a solution to the Dirichlet problem for Eq.(1.1)in the domain D+that satisfies conditions(1.2)-(1.4)and u(x,0)=τ(x),x∈I,is representable in the form

where

Jν(z)is the Bessel function of the first kind[26].From formula(3.1)by some transformations we obtain

where

Equality(3.2)is a functional relation between unknown functions τ(x)and ν(x)induced on I from the elliptic part D+of the mixed domain D.

Further,in the integral with the limit(c,1)we change the variable of integration as t=p(s)=1-ks and in view of(2.2)from(3.2)we obtain

Performingsometransformationsin(3.4),weobtainthefollowing singularintegralequation:

where

We rewrite equality(3.5)in the form

where

is a regular operator and

The first integral operator in the right-hand side of(3.6)is not regular,because the integrand with x=cand s=c has an isolated singularity of the order and therefore this item in(3.6)is separated.Provisionally,assuming that the right-hand side of Eq.(3.6)is known function,we rewrite it as

where

Setting τ(x)=x2-2βρ(x)and g0(x)=x2-2βg1(x)and taking into account the identity

we write Eq.(3.7)in the form Changing variables as,we obtain the following singular integral equation:

We will seek for its solution in the class of functions satisfying the Hoelder condition on(0,c2)and bounded with ξ→c2such that with ξ→0 they can tend to infinity of order less thanone.Inthis class the indexof theindicated equationequals zero,and its solution can be written explicitly

Hence,returning to the previous functions,we obtain

where α=(1-2β)/4.Now,substituting(3.8)into(3.9)and after some transformations, we obtain

where

is a regular kernel,

In view of the equality p(x)=1-kx,Eq.(3.10)can be written as

Let us evaluate the inner integral in(3.11):

Let us evaluate the inner integral in(3.11):

where

F(a,b,c;z)is the hypergeometric Gauss function[19].

Here B0(x,s)is continuous differentiable function in the square[0,c]×[0,c].Substituting(3.12)into(3.11),after simple calculations,we obtain the following integral equation:

where

is a regular operator.

Since(1-ks-c)α=kα(c-s)α,we can write equality(3.13)as follows:

Further we transform Eq.(3.14)by extracting its characteristic part and get

where

Taking into account the identity 1-kc-c=0,we write Eq.(3.15)in the form

In Eq.(3.16)we change variables s=c-ce-tand x=c-ce-yand denote ρ(y)=τ(cce-y)e(α-1/2)y.As a result we obtain

The function K(x)is continuous and has the exponential order of decrease at infinity. Eq.(3.18)isanintegralWiener-Hopf,theFouriertransformationturnsit intotheRiemann boundary-value problem which is solvable in quadratures.The Fredholm theorems for integral equations of convolution type are valid only in a particular case,when their index equals zero.The index of Eq.(3.18)is that of the expression

with opposite sign,where

With the help of the residue theory,calculating the Fourier integral(3.20)([25],p.198), we obtain

Now let us calculate the index of expression(3.19).Since

we get Re(1-λμk1-αcosαπK∧(x))＞0 and,consequently,

i.e.,with the change of the argument of 1-λμk1-αcosαπK∧(x)on the real axis the index of Eq.(3.18)equals zero([27],p.55).Hence,taking into account the uniqueness of the solution to Problem TF,we conclude that(3.17)is uniquely solvable,and hence so is Problem TF.

Acknowledgments

The author would like to thank anonymous referees for their useful suggestions.

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10.4208/jpde.v27.n2.8 June 2014

?Corresponding author.Email address:mruziev@mail.ru,ruzievmkh@gmail.com(M.Kh.Ruziev)

AMS Subject Classifications:35M10

Chinese Library Classifications:O175.28