Implicit Partial Hyperbolic Functional Differential Equations

In this chapter, we shall present existence results for some classes of initial value problems for partial hyperbolic implicit differential equations with fractional order.

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Implicit Partial Hyperbolic Functional Differential Equations

7.1 Introduction In this chapter, we shall present existence results for some classes of initial value problems for partial hyperbolic implicit differential equations with fractional order.

7.2 Darboux Problem for Implicit Differential Equations 7.2.1 Introduction This section concerns the existence results to fractional order IVP , for the system r

r

D 0 u.x; y/ D f .x; y; u.x; y/; D  u.x; y//I if.x; y/ 2 J WD Œ0; a  Œ0; b; 8 ˆ ˆ 0; D 0 is the mixed regularized derivative of order r D .r1 ; r2 / 2 .0; 1  .0; 1; f W J  Rn  Rn ! Rn is a given function, ' 2 AC.Œ0; a; Rn / and 2 AC.Œ0; b; Rn /: We present two results for the problems (7.1)–(7.2), the first one is based on Banach’s contraction principle and the second one on the nonlinear alternative of Leray–Schauder type.

S. Abbas et al., Topics in Fractional Differential Equations, Developments in Mathematics 27, DOI 10.1007/978-1-4614-4036-9 7, © Springer Science+Business Media New York 2012

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7 Implicit Partial Hyperbolic Functional Differential Equations

7.2.2 Riemann–Liouville and Caputo Partial Fractional Derivatives For a function h 2 L1 .Œ0; b; Rn /I b > 0 and ˛ 2 .0; 1: The connection between D0˛ and c D0˛ is given by c

D0˛ h.t/ D D0˛ h.t/ 

t ˛ h.0C /; for almost all t 2 Œ0; b:  .1  ˛/

(7.3)

For more detail see [166]. Corollary 7.1. For a function u 2 L1 .J; Rn / and r D .r1 ; r2 / 2 .0; 1  .0; 1: The r1 r1 connection between D0;x u.x; y/ and c D0;x u.x; y/ with respect to x is given by c

 r1  r1  D0;x u .x; y/ D D0;x u .x; y/ 

  x r1 u 0C ; y :  .1  r1 /

(7.4)

r2 r2 Analogously, the connection between D0;y u.x; y/ and c D0;y u.x; y/ with respect to y is given by



c

   r2 r2 D0;y u .x; y/ D D0;y u .x; y/ 

  y r2 u x; 0C :  .1  r2 /

(7.5)

Now, let us give the relation between D0r and c D0r ; where r D .r1 ; r2 / 2 .0; 1  .0; 1: Theorem 7.2. For u.x; y/ 2 AC.J; Rn / and r D .r1 ; r2 / 2 .0; 1  .0; 1 we have c

   r D0r u .x; y/ D D 0 u.x; y/ D D0r u .x; y/  

x r1  r2  D0;y u .0; y/  .1  r1 /

y r2  r1  x r1 y r2 D0;x u .x; 0/ C u.0; 0/:  .1  r2 /  .1  r1 / .1  r2 / r

Proof. According to ([246], Lemma 1) .D 0 u/.x; y/ D .c D0r u/.x; y/: Then  r    D 0 u .x; y/ D D0r q .x; y/ D Dxy I01r q.x; y/; q.x; y/ D u.x; y/  .x; y/; .x; y/ D u.x; 0/ C u.0; y/  u.0; 0/; I01r q.x; y/ D I01r u.x; y/  I01r .x; y/: As u.x; y/ D .x; y/ C I0 v.x; y/; then q.x; y/ D I0 v.x; y/; where v.x; y/ D Dxy u.x; y/:

7.2 Darboux Problem for Implicit Differential Equations

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Hence I1r q.x; y/ D I1r .I v.x; y// 1 D  .1  r1 / .1  r2 / 1

Zx Zy 0

0

0 s t Z Z @ .s  z/r1 .t  /r2 0

0

v.; z/dzd A dtds 2 AC.J /;

I01r .x; y/ D

y 1r2 I 1r1 u.x; 0/ .1  r2 / .1  r2 / 0;x C

x 1r1 I 1r2 u.0; y/ .1  r1 / .1  r1 / 0;y



x 1r1 y 1r2 u.0; 0/; .1  r1 /.1  r2 / .1  r1 / .1  r2 /

besides ([225], Lemma 2.1) I01r .x; y/ 2 AC.J; Rn /: Then I01r u.x; y/ D I01r q.x; y/ C I01r .x; y/