Pure and Applied Mathematics Journal
Volume 4, Issue 6, December 2015, Pages: 259-263

On the Finding the Other Eigenvalues and Eigen Functions and Ortogonal Basis with a Nonlocal Parity Condition of the Third Kind

Naser Abbasi1, *, Hamid Mottaghi Golshan2, Mahmood Shakori1

1Department of Mathematics, Lorestan University, Khoramabad, Iran

2Department of Mathematics, Islamic Azad University, Ashtian Branch, Ashtian, Iran

Email address:

(N. Abbasi)
(H. M. Golshan)
(M. Shakori)

To cite this article:

Naser Abbasi, Hamid Mottaghi Golshan, Mahmood Shakori. On the Finding the Other Eigenvalues and Eigen Functions and Ortogonal Basis with a Nonlocal Parity Condition of the Third Kind.Pure and Applied Mathematics Journal.Vol.4, No. 6, 2015, pp. 259-263. doi: 10.11648/j.pamj.20150406.16


Abstract: In the present paper, we find out the eigenvalues and the corresponding eigenfunctions of the modified Frankl problem with a nonlocal parity condition, the completeness and the basis property in the elliptic part of the third kind of a domain in . We also consider a new boundaries condition and analyze the orthogonal basis of the eigenfunctions depending on parameters of the problem.

Keywords: Frankl Problem, Lebesgue Integral, Holder Inequality, Bessel Equation


1. Introduction

The classical Frankl problem was considered in [3]. The problem was further developed in [2, pp.339-345], [12, pp.235-252]. Several authors have also have investigated this issue (see [1-13]). On the solution of the Frankl prolem in a special domain in [12]. About spectrum of the gasedynamic problem of Frankl for the model equation of mixed type in [10]. About construction of the gasedynamic problem of Frankl in [11]. Basis property of eigen -functions of the generalized problem of Frankl with a nonlocal parity condition and with the discontinuity of the gradient of solution in [9].

In the present paper, we consider boundaries conditions of the third kind on the intervals (-1,0) and (0,1) of the axis OY for which the derivatives of functions with respect to on these intervals are related by linear dependence. We show that if the dependence coefficient exceeds -1 (the coefficient cannot be zero, since, otherwise, the problem will degenerate), then the systems of eigenfunctions of the problem forms a Riesz basis in the elliptic part of the domain.

2. Statement of the Modified Frankl Problem

Definition 1. Find a solution

of the modified generalized Frankl problem

(1)

in and the boundary conditions

(2)

in the polar coordinate system

(3)

(4)

(5)

here is the domain in the top half-plane bounded by a circle

and the segment [0,1] of the axis OY, and  is the domain in the bottom half space, where is bounded by the characteristic and  and the segment [0,1] of the axis OX and  is bounded by the characteristic and  and the segment [-1,0] of the axis OY.

Definition 2. .System is called complete in if

Definition 3. .System is called minimal in if

Remark 4. If the system is minimal inthen it is also minimal infor ,and if it is complete infor

Theorem 5 ([1,5]). The eigenvalues and eigenfunctions of problem (1-5) can be written out in two series.

In the first series, the eigenvaluesare found from the equation

where , are roots of the Bessel equation (6),, is the Bessel function [6,7, Russian translation], and the eigenfunctions are given by the formulas

(6)

where

 in ,

 in , and,  in .

In the second series, the eigenvaluesare found from the equation

where, and the corresponding eigenfunctions have the form

(7)

Theorem 6 ([5]). The function system

(8)

is complete and a Riesz basis in, provided that.

3. Main Results

Consider Frankl problem (1)-(5) with the new boundary condition

(9)

(10)

(11)

Theorem 7. The eigenvalues and eigenfunctions of problem (1-5) can be written out in two series.

In the first series, the eigenvaluesare found from the equation

(12)

where, are roots of the Bessel equation (6),, is the Bessel function [4], and the eigenfunctions are given by the formula

(13)

where

,for in , for,  in , and,  for, in .

In the second series, the eigenvaluesare found from the equation

(14)

where and theare the roots of the Bessel equation (8).

where , and

and .

Theorem 8. The function system

(15)

is complete and a Riesz basis in, provided that.

Proof. In order to prove this theorem we use the method in [1, 6] by considering convergence function

inand Riesz basis the systemfor .

Remark 9. For  the system (10) is not complete but is minimal, foris complete but is not minimal, and if , is complete and minimal.

Theorem 10. The system of eigenfunctions

is complete and basis in the space, therefore

and then in .

Proof. Using Fubini theorem for anywe have

again sinceso;

and since the system in is orthogonal and complete, it is enough to prove:

Using the Holder inequality

with the integration interval (0,1).

This inequality is equivalent to

Also system is orthogonal and complete inof relation

imply that

According to theorem 6, we conclude that in Similarly, if we consider the above calculations for sequencewe have

Because completenessin

The proof of the theorem is complete.

Remark 11. If  then the system becomes the system  which is basis in the space and an orthogonal basis in the space

The proof of remark 11 results from theorem 10.

Remark 12. In case and  then the system is complete but is not minimal.

In case  and  then the system is not complete but is minimal.

In case then the system is complete but is not minimal in the space and is not complete in the space

The proof of remark 12 results from theorem 8.

4. Conclusion

Consider Frankl problem (1)-(5) with the new boundary condition (9)-(10( and (11), so we find out the eigenvalues of the problem with a nonlocal parity condition ,the completeness and the basis property in the elliptic part of the third kind of a domin in

Acknowledgements

This research was partially supported by Lorestan University, Khoramabad and the authors are grateful to Academician E.I. Moiseev for his interest in this work.


References

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  2. A. V.Bitsadze, Nekotoryeklassyuravneni v chastnykhproizvodnykh. (Russian) [Some classes of partial differential equations] "Nauka", Moscow, 1981. 448pp.
  3. F. Frankl,On the problems of Chaplygin for mixed sub- and supersonic flows. (Russian) Bull. Acad. Sci. URSS. Sr.Math. [IzvestiaAkad. Nauk SSSR] 9, (1945).121-143.
  4. H.vanHaeringen,L.P. Kok, Higher transcendental functions, Vol. II [McGraw-Hill, New York, 1953; MR 15, 419] by A.Erdlyi, W. Magnus, F. Oberhettinger and F. G. Tricomi. Math.Comp. 41 (1983), no. 164, 778.
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  7. E. I.Moiseev,N. Abbasi, The basis property of an eigenfunction of the Frankl problem with a nonlocal parity conditionin the space Sobolev (W1p (0,π)). Integral Transforms Spec. Funct. (England) 22 (2011), no. 6, 415–421.
  8. E. I. Moiseev, On the solution of the Frankl problem in a special domain, Differ. Uravn. 8(4)(1992),pp.721-723.
  9. E. I. Moiseev and N. Abbasi, Basis property of eigen functions of the generalized problem of Frankl with a nonlocal parity condition and with the discontinuity of the gradient of solution , Differ.Uravn.44(10)(2009),pp.155 -160.
  10. K. B. Sabitov, About spectrum of the gasedynamic problem of Frankl for the model equation of mixed type, Differ. Uravn. 39(11) (2003),pp.1568-1570.
  11. K. B. Sabitov and V. V. Tikhomirov, About construction of the gasedynamic problem of Frankl,Differ.Uravn.23(10)(1990),pp.100-109.
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