Abstract
Without the usual assumption of monotonicity, we establish some results on the theory of hyperbolic differential inequalities which enable us to produce a majorising interval function for the solution of the hyperbolic initial value problem. Using this function, a variation of parameters formula and interval iterative technique, the existence of solution to the problem is established.
Highlights
In this paper, we utilize interval analytic methods in the investigation of the existence of solution of the hyperbolic partial differential equation ( ) = zxy f x, y, z, zx, zy,( x, y) ∈ Iab (1.1)with characteristic initial values= z ( x, 0) σ ( x); x ∈ Ia. = z (0, y) τ ( y); y ∈ Ib. (1.2) σ= (0) z (0=, 0) τ= (0) z0prescribed in a two-dimensional rectangle Iab= Ia × Ib, wh= ere Ia [= 0, a], Ib [0,b] a,b ∈ and = zx ∂∂= xz, zy
Without the assumption of monotonicity on the function f we establish some results on the theory of hyperbolic differential inequalities which enable us to produce a majorizing interval function for the solution of the equation
Similar interval methods had earlier been used by some authors in [3]-[7] for solution to differential equation but not for hyperbolic initial value problems
Summary
Similar interval methods had earlier been used by some authors in [3]-[7] for solution to differential equation but not for hyperbolic initial value problems. ( ) Definition 2.1: A function v ∈ C1,2 Ia,b , is said to be an upper solution of the hyperbolic initial value problem (1.1) and (1.2) on Iab if ( ) Definition 2.2: A function u ∈ C1,2 Ia,b , is said to be a lower solution of the hyperbolic initial value problem (1.1) and (1.2) on Iab if the reversed inequalities hold true with u in place of v in the specified intervals.
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