Council for Innovative Research

(1)

ISSN 2347-1921

5583 | P a g e D e c e m b e r 2 3 , 2 0 1 5

Linear System in general form for Hyperbolic type

Naser zomot

Department of Mathematics, Zarqa University ,Zarqa, Jordan

[email protected]

ABSTRACT

In this paper, we consider the linear system of partial differential equations hyperbolic type Which is solved by T.V.

Chekmarev [1] and solve the general form for this Hyperbolic type.

Indexing terms/Keywords

Hyperbolic; partial differential equations; linear system.

Council for Innovative Research

Peer Review Research Publishing System Journal: JOURNAL OF ADVANCES IN MATHEMATICS

Vol .11, No. 8

www.cirjam.com, [email protected]

(2)

ISSN 2347-1921

5584 | P a g e D e c e m b e r 2 3 , 2 0 1 5 INTRODUCTION

we consider the system of linear system of partial differential equations hyperbolic type

     

  x y u d   x y v g   x y c

y x f v y x b u y x a

v u

y x

, ,

,

, ,

,









(1)

with initial condition

   

  x   x

v

y y u

y x







0 0

, ,

(2)

T.V.CHEKMARIOV [1] solve this system when a=d=0

   

  x y u g   x y c

y x f v y x b

v u

y x

, ,









we can reduce this system to one differential equation of second order and we can solve it by Riemann method, but the solution is exist in case existence for b(x,y), c(x,y), f(x,y)and g(x,y)and continuity for derivatives.

The Main Result

If we take the system (1) with initial condition (2) , we want to solve it by using the result theory of integral Voltera.

Rewrite the system (1)

     

  x y v c   x y u g   x y d

y x f v y x b u y x a

v u

y x

, ,

,

, ,

,













Or

 

   

 

^ ^

 

   

 

^ ^

0

,

0 , ,

,

0 , ,

x

y

a y d

x

d x d

y

x

a y d

x b x y v f x y e

y

d x d

y c x y u g x y e

e u

e

 

 

 

 



  

 

  

 

 

 

 

 

 

 

Integrate by x and then by y , we get

 

           

 

           

1 1

0 0

1 1

0 0

0

, ,

0

, ,

0

, , [ , , , ]

x x

o

y y

a y d x a y d

d x d y d x d

e u x y u y b y v y f y e d

x

e v x y v x c x u x g x e d

y

x y

 

 

 

   

     

 

 

  

 

  



(3)

ISSN 2347-1921

5585 | P a g e D e c e m b e r 2 3 , 2 0 1 5

Rewrite (3) in form

   

⁰ ^ ^

     

¹ ¹

   

¹ ¹

0 0

, , ,

, , , ,

x x x

x

a y d x a y d x a y d

u x y y e b y v y e d f y e d

x x

 

   

        

    

   

⁰ ^ ^

     

¹ ¹

   

¹ ¹

0 0

, , ,

(4)

, , , ,

y y y

y

d x d y d x d y d x d

v x y x e c x u x e d g x e d

y y

 

   

        

    

Let

       

^ ^

       

^ ^

1 1

0

1 1

0

, ,

, , 1

, ,

x x

x

y y

y

a y d

x

d x d

y d x d

x y f y e d y e

x

x y g x e d x e

y





 

 

 

   

    



 

 

 

 



(5)

In (4),If we put u(x,y) in v(x,y) we get

     

¹ ¹

       

¹ ¹

0 0

, ,

,

1

, [ , , , ]

y x

y d x d x a d

v x y c x e x b v e d d

y x

 

  





        

  

    

Or

       

0 0

1 1

, , , , , ,

x y

v x y x y x y v d d

x y

k      

    

(6)

where

         

1 1

0

1 1 1 1

,

1 1

, ,

1

, , , ,

, , ,

y

y x

y d x d

a d d x d

x y x y c x x e d

y

x y c x b e

k



 



     

   

  





  

 



 

In (4), If we put v(x,y) in u(x,y) we get

       

¹ ¹

       

¹ ¹

0 0

, ,

, , , [

1

, , , ]

y x

a y d y d d

x

u x y x y b y e y c u e d d

x y

 

 



 

           

    

^Or

       

0 0

, , , , , ,

x y

u x y x y k x y u d d

x y

     

    

(7) where

(4)

ISSN 2347-1921

5586 | P a g e D e c e m b e r 2 3 , 2 0 1 5

         

1 1

0

1 1 1 1

, 1

, ,

, , , ,

, , , , ,

x

y x

a y d

x

a y d d d

x y x y b y y e d

x

k x y b y c e



 



 

   

   

    





   

 





The problem (1),(2) reduces to double integral (6),(7).

Let R and R1 Rezolvent kernel k and k1 then the solution of (6),(7) is

       

0 0

1 1 1

, , , , , ,

x y

u x y x y R x y d d

x y

v x y x y x y d d

x y

R

     

   

 

 

   

(8)

Then, (8) give us the solution for the problem (1) ,(2).

ACKNOWLEDGMENTS

This research is funded by the Deanship of research in Zarqa university -Jordan

REFERENCES

[1] T.V. Chekmarev ,solution in quadratures Cauchy and Gursat problems for a linear

System of differential equations in partial derivetives . scientists notes GSU, 80 release, the second part , Gorkuu 1967, page 63-69.

[2] V.S. Surikov , I.P. Mischenko , the construction of the Riemann formulation for the Equations

u

_xy

      x  y u  0

. Scientific works of the Krasnodar polytechnic Institute, 1970 issue 30 , page 19-25.

[3] A.V.Bitsadze, some classes of partial differential equations. Moscow, science 1981 page 448