Interval Analytic Method in Existence Result for Hyperbolic Partial Differential Equation ()
1. Introduction
In this paper, we utilize interval analytic methods in the investigation of the existence of solution of the hyperbolic partial differential equation
(1.1)
with characteristic initial values
(1.2)
prescribed in a two-dimensional rectangle
where
and
![](https://www.scirp.org/html/htmlimages\6-5300562x\126575b6-a37a-4fbc-b83e-c2a435a4ec77.png)
, ![](https://www.scirp.org/html/htmlimages\6-5300562x\a61f153f-55ba-4c33-bd0a-e25e02f7fc04.png)
and
where
means that z is continuous on
and possesses continuous partial derivatives
on ![](https://www.scirp.org/html/htmlimages\6-5300562x\6b61da14-1ee5-4b2a-abe2-a34eb4062879.png)
Without the assumption of monotonicity on the function
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. With the use of a variation of parameters formula used in [1] and theorem 5.7 of [2] on interval iterative technique we generate a nested sequence of interval functions which converges to an interval solution. This interval solution is thus a majorant of the solution of the equation and it coincides with the real valued solution if it is degenerate. 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. The result in this paper generalizes those of [1] [8] as the monotonicity condition imposed on the function
is not in any way necessary.
The basic results in interval analysis used in this work are found in [2] [6] [7] [9] -[13] for readers who may not be familiar with them.
2. Differential Inequalities and Majorisation of Solution
Definition 2.1: A function
is said to be an upper solution of the hyperbolic initial value problem (1.1) and (1.2) on
if
![](https://www.scirp.org/html/htmlimages\6-5300562x\8a1a33c3-c9dc-49b9-a366-0df9af38a0e8.png)
![](https://www.scirp.org/html/htmlimages\6-5300562x\e5ad12b9-6d96-46d6-bbcc-f9b02e0957b3.png)
![](https://www.scirp.org/html/htmlimages\6-5300562x\fdf4399a-4d05-4c22-aa31-2031c42bb8ad.png)
.
Definition 2.2: A function
is said to be a lower solution of the hyperbolic initial value problem (1.1) and (1.2) on
if the reversed inequalities hold true with
in place of
in the specified intervals.
Next, we shall consider some results concerning the upper and lower solutions of Equation (1.1) and conditions (1.2).
Theorem 2.1: Suppose that
and
(2.1)
(2.2)
(2.3)
Then we have
(2.4)
where the inequality is componentwise.
Proof: We shall establish this theorem by contradiction. From assumption (2.3) we see clearly that the theorem is true for the point (0,0) on ![](https://www.scirp.org/html/htmlimages\6-5300562x\d9ff6a58-93aa-40fc-92b9-d1cf0e89f606.png)
Suppose that inequality (2.4) is not true at a point
and assume that
(2.5)
then by assumption (2.3)
and
cannot both be zero.
Let
be such that
then
and so
![](https://www.scirp.org/html/htmlimages\6-5300562x\335da937-07df-4434-a677-dfbd8f857c25.png)
Thus, we have, for
(or
),
![](https://www.scirp.org/html/htmlimages\6-5300562x\47fc040f-a08a-41a2-a4e8-dbefd1897b87.png)
and this contradicts assumption (2.5).
If
then
(or vice-versa) and for
we have
![](https://www.scirp.org/html/htmlimages\6-5300562x\75400a9f-470f-429a-831b-db94b020e36b.png)
If
and
a similar argument can be advanced to obtain
Hence,
![](https://www.scirp.org/html/htmlimages\6-5300562x\f6ed348f-65dc-48ae-bf94-ebc217ea902f.png)
and this is still a contradiction to our earlier assumption (2.5).
Suppose instead that
(2.6)
Then
otherwise condition (2.3) would immediately give the required contradiction.
Now for
let
such that
, we have
so
![](https://www.scirp.org/html/htmlimages\6-5300562x\7a0ca5dd-571c-4ee0-860d-2a03caa01ec5.png)
![](https://www.scirp.org/html/htmlimages\6-5300562x\e69affcd-2ddf-4a5e-bd44-65385936972e.png)
This contradicts assumptions (2.6).
Similarly, if we assume that
, we would also arrive at a contradiction. At
and
left hand derivatives are used to obtain the result.
Hence, we conclude that, the assertion (2.4) holds true on
and this proves the theorem.
Theorem 2.2: Let
and
be functions defined on
which satisfy assumptions (2.1), (2.2) and (2.3) of Theorem 2.1. Suppose in addition that they satisfy the following conditions,
(2.7)
Then the solution
of problem (1.1) and (1.2) together with its derivatives
satisfy
![](https://www.scirp.org/html/htmlimages\6-5300562x\5bfdb7a4-5c88-4bb5-9393-3f13feb2f8e4.png)
On the rectangle,
, where the inclusion is componentwise.
Proof: Notice that the lower endpoints of the intervals in Equation (2.7) satisfy assumption (2.3) of Theorem 2.1 when
is replaced by
. Therefore
and
satisfy the hypothesis of Theorem 2.1 and hence
(2.8)
Similarly, replacing
by
in assumption (2.3) we obtain the upper endpoints of the intervals in conditions (2.7) and so by Theorem 2.1 we also have
(2.9)
Combining inequalities (2.8) and (2.9) we have the desired result.
3. Construction and Existence of Solution
Our purpose in this section is to establish the existence of solution to the problem (1.1) satisfying initial values (1.2) by means of interval analytic method. To this end an integral operator is constructed, the solution of the resulting operator equation is equivalent to the solution of the initial value problem under consideration. An interval extension of this operator is then used to generate a sequence of interval functions which converges to the required solution.
Let
be such that
on
and a function
, defined by
(3.1)
where
is the function in Equation (1.1) and
is a constant suitably chosen such that
Clearly it can be seen that
is continuous on ![](https://www.scirp.org/html/htmlimages\6-5300562x\c1438dc2-3a21-4de0-9f83-684a7163ee0c.png)
With this new function
, Equation (1.1) becomes
(3.2)
By using the variation of constant formula of Lemma 4.1 in [1] , we obtain the solution of Equation (3.2), satisfying initial values (1.2) as:
![](https://www.scirp.org/html/htmlimages\6-5300562x\513db05f-ab05-4e8b-827f-c58ab198473d.png)
Differentiating with respect to
, we obtain
![](https://www.scirp.org/html/htmlimages\6-5300562x\bfcad9a0-4c32-4e90-a795-8300ece4b0af.png)
and similarly by differentiating with respect to
we obtain
![](https://www.scirp.org/html/htmlimages\6-5300562x\685408c7-eaa0-49f9-89af-c91b44f0f8af.png)
Eliminating the derivatives
and
by introducing the function
and
into the integro-differential equations we obtain the system of integral equations
(3.3)
(3.4)
(3.5)
which is equivalent to the problem (3.2) and initial values (1.2).
Denoting the right hand side of these integral equations by
and
respectively, we have the following:
(3.6)
With these we prove the following result.
Lemma 3.1: Let
and
satisfy conditions (2.7) of Theorem 2.2. Suppose that for functions
with
on
, we have
(3.7)
where
is the constant appearing in Equation (3.1). Then the following hold true.
(3.8)
for all ![](https://www.scirp.org/html/htmlimages\6-5300562x\d9dc2b94-066e-4195-b814-c85b351e8ecd.png)
Proof: We first consider the lower endpoints of the inclusions and differentiating we have, from Equation (3.3)
![](https://www.scirp.org/html/htmlimages\6-5300562x\9a81359d-5254-40a8-8534-988a52dc4106.png)
differentiating again with respect to
we obtain
![](https://www.scirp.org/html/htmlimages\6-5300562x\24564f6d-098c-417f-bbee-9d497883c395.png)
This, by Equation (3.1) and assumption (3.7), gives
![](https://www.scirp.org/html/htmlimages\6-5300562x\3f03302b-cba1-4564-895a-07e8d1e405d8.png)
Similarly by differentiating Equation (3.3) with respect to
, we obtain
![](https://www.scirp.org/html/htmlimages\6-5300562x\c760549d-016c-4ead-9e5a-57eb9e220736.png)
By conditions (2.1) and (3.7) we have
for
and
for
From these we see that
satisfies the assumptions of Lemma 4.2 of [1] since ![](https://www.scirp.org/html/htmlimages\6-5300562x\02ed5a48-e099-44b8-b029-688d68420066.png)
Thus
![](https://www.scirp.org/html/htmlimages\6-5300562x\490ab164-37a0-4a66-8a86-159a8e5a21a1.png)
![](https://www.scirp.org/html/htmlimages\6-5300562x\c1d530c7-27c0-4c69-810f-371d89a3c44e.png)
![](https://www.scirp.org/html/htmlimages\6-5300562x\f3e87dcd-9cb4-43dd-9b24-c4ed787c279b.png)
![](https://www.scirp.org/html/htmlimages\6-5300562x\bdf4f02b-c4a9-4c33-bed4-890833d50706.png)
![](https://www.scirp.org/html/htmlimages\6-5300562x\95380829-31fa-4569-85bd-b0105b1c73ab.png)
![](https://www.scirp.org/html/htmlimages\6-5300562x\c9248e9f-b144-4959-aebc-94c51a13e346.png)
It could similarly be proved that
![](https://www.scirp.org/html/htmlimages\6-5300562x\ee79f2eb-487a-47ff-87e3-4196a2cba253.png)
![](https://www.scirp.org/html/htmlimages\6-5300562x\d29cb026-b94f-4c72-8680-117921957a8e.png)
and
![](https://www.scirp.org/html/htmlimages\6-5300562x\5acb85be-763f-4e14-8cf2-5911f2136e28.png)
Hence the lemma is established.
Theorem 3.1: Let the functions
satisfy conditions (2.7). Suppose that the function
is such that
![](https://www.scirp.org/html/htmlimages\6-5300562x\7fe70357-8111-4089-a7dd-db1311763963.png)
and
for function
satisfying
![](https://www.scirp.org/html/htmlimages\6-5300562x\919c4a42-25ba-479f-b857-16e22216c575.png)
and
, constant, suitably chosen in Equation (3.1).
Then there exists a convergent nested sequence of interval functions
such that the unique solution
of Equations (1.1) and (1.2) satisfies
![](https://www.scirp.org/html/htmlimages\6-5300562x\6e92fa92-768d-40d2-9eb9-dea1388a1b16.png)
with
,
degenerate where the initial interval
is given by
![](https://www.scirp.org/html/htmlimages\6-5300562x\ea02d615-7263-4827-877e-1c02052a35d3.png)
Proof: From the construction earlier considered, we see that any solution of Equation (1.1) which satisfies condition (1.2) solves the integral Equation (3.3). Conversely if
solves the integral Equation (3.3) we have that
![](https://www.scirp.org/html/htmlimages\6-5300562x\3d750ed8-be59-4ca7-819a-8c17a080bf6c.png)
which by Equations (3.1) and (3.6) gives
![](https://www.scirp.org/html/htmlimages\6-5300562x\b3ce551a-ed54-4c71-81d0-dd033e107bc0.png)
with
.
and these imply that
again solves the Equation (1.1) and satisfies condition (1.2). Therefore, we shall seek the solution of the integral equation given by (3.3) which is transformed to the operator equation
![](https://www.scirp.org/html/htmlimages\6-5300562x\1d320a90-61d3-406f-a75d-ec88a63dd0d0.png)
Let
be an interval function defined on
such that
for
and the interval function
an interval extension of the function
defined in Equation (3.1). Then the interval integral operator
defined by
![](https://www.scirp.org/html/htmlimages\6-5300562x\3c41fb82-f93c-4308-bc6b-d31c4151bab6.png)
is an interval majorant of
.
Then the problem reduces to solving the interval operator equation
![](https://www.scirp.org/html/htmlimages\6-5300562x\1013be9d-85d1-4b70-b8e7-fc444d6dfb66.png)
However to determine
we need to also determine
and
which are respectively interval extensions to the function
and
. This is done by solving the interval operator equations
![](https://www.scirp.org/html/htmlimages\6-5300562x\c029c621-3951-4266-9773-0a49b96c109c.png)
With
and
defined respectively by
![](https://www.scirp.org/html/htmlimages\6-5300562x\38917151-74ce-42b7-ae68-4f9c30afa43d.png)
and
![](https://www.scirp.org/html/htmlimages\6-5300562x\dd2652bb-b14d-4389-9e2c-c4b9219be150.png)
which majorise the real operators
and
respectively.
Define the sequences
by
with
![](https://www.scirp.org/html/htmlimages\6-5300562x\57e19628-00fe-483f-a410-40fcab899d18.png)
with
and
with
![](https://www.scirp.org/html/htmlimages\6-5300562x\436a0b66-46b3-4afb-b894-6bd6b9e1f452.png)
We have the sequence
as required.
We shall show that
convergences to a limit. But this can only be so if the sequence
and
also converge.
By Theorem 5.7 of [2] , these sequences converge if
![](https://www.scirp.org/html/htmlimages\6-5300562x\cdb0357d-e2d5-4461-b66b-e3606dc62cc1.png)
and
![](https://www.scirp.org/html/htmlimages\6-5300562x\c181f2ce-9b4c-47b6-aceb-48470b81c910.png)
Now for
![](https://www.scirp.org/html/htmlimages\6-5300562x\acbc7348-605e-4c63-8e20-c02eec1d6e47.png)
![](https://www.scirp.org/html/htmlimages\6-5300562x\94175bd0-3b87-4ec4-99df-180d4c738f2d.png)
by the first inclusion of Equation (3.8). Hence
![](https://www.scirp.org/html/htmlimages\6-5300562x\8ec3e7e8-eaac-4b21-bef6-17d2d6c63e4a.png)
Similarly we have by the result given in Equation (3.8) of Lemma 3.1
![](https://www.scirp.org/html/htmlimages\6-5300562x\bdec95ab-3c02-4397-a788-36eea01eaac5.png)
and
![](https://www.scirp.org/html/htmlimages\6-5300562x\a18a6096-5e57-4d10-bfa9-75fd6f8484c7.png)
Since these initial intervals satisfy the hypothesis of Theorem 5.7 of [2] , the result of the theorem implies that
and
converge as sequences and are equally nested. Furthermore, the solution
of Equation (1.1) satisfying condition (1.2) belongs to the limit function
of the sequence
that is,
![](https://www.scirp.org/html/htmlimages\6-5300562x\c853a3fa-c68b-4326-89be-5c52d738ea62.png)
and this proves the theorem.
Lemma 3.2: Assume that the functions
satisfy conditions (2.7) and in addition they also satisfy conditions (2.1) and (2.2). Suppose further that the function f appearing on the right hand side of Equation (1.1) satisfies:
(3.9)
whenever the functions
and
are such that
![](https://www.scirp.org/html/htmlimages\6-5300562x\199bed2d-98ef-4af6-bb86-d1404197b70b.png)
for constant
suitably chosen. Then we have
(3.10)
for any function
satisfying
![](https://www.scirp.org/html/htmlimages\6-5300562x\57fa9e6d-1dce-48be-b4ee-02ce9cf9c11e.png)
Proof: From inequality (2.1) we have
![](https://www.scirp.org/html/htmlimages\6-5300562x\8a418516-3aaa-4c89-8585-11b55ac79da2.png)
Since
![](https://www.scirp.org/html/htmlimages\6-5300562x\2fb71963-b764-44f8-88d3-bac551754d3a.png)
From inequality (3.9) we have
![](https://www.scirp.org/html/htmlimages\6-5300562x\13bc37c3-2d75-4a81-a270-6b7fd8be82c7.png)
and so
![](https://www.scirp.org/html/htmlimages\6-5300562x\a0eb77b8-a96f-4ff7-9498-be932b7e1c27.png)
which is the first inequality in (3.7).
Also from inequality (2.2) we have
![](https://www.scirp.org/html/htmlimages\6-5300562x\cbf2c82c-a533-4688-8eb6-240d2d83f602.png)
and using inequality (3.9) we have
![](https://www.scirp.org/html/htmlimages\6-5300562x\697d3dd0-be7d-461e-b7ad-317e4b3c85c1.png)
Therefore
![](https://www.scirp.org/html/htmlimages\6-5300562x\a2b6e8ac-67ad-45f8-b81d-c01360951134.png)
which also is the second inequality in (3.7). Since all the other conditions of Lemma 3.1 are also satisfied, the proof of this lemma follows as for Lemma 3.1 to obtain the desired result.
Remark 3.1: If
in inequality (3.9) then we have
![](https://www.scirp.org/html/htmlimages\6-5300562x\43b00e5b-fb44-47ba-a028-34762cda0bd7.png)
for
and this implies that
is monotone increasing in its domain of definition. Therefore the result of lemma 3.2 also holds for a monotone function ![](https://www.scirp.org/html/htmlimages\6-5300562x\9b0b0252-47fc-42cd-a4c1-1d055186a2e4.png)
Theorem 3.2: Suppose that the function
satisfies conditions (2.1), (2.2) and (2.7). If in addition the function
appearing in Equation (1.1) satisfies
![](https://www.scirp.org/html/htmlimages\6-5300562x\a4be4a35-df12-4c35-b661-419bf183da36.png)
whenever
![](https://www.scirp.org/html/htmlimages\6-5300562x\4bb69795-b49e-4319-8389-b8582836fbc9.png)
for some constant
, suitably chosen.
Then there exists a nested sequence of interval function
with each term majorising the unique solution
of Equation (1.1) satisfying condition (1.2) such that the limit
of this sequence also contains
, that is,
![](https://www.scirp.org/html/htmlimages\6-5300562x\886621b2-df98-430b-a160-90f7df05136f.png)
Proof: As it has been shown in the proof of Lemma 3.2, the conditions prescribed in this theorem can equally be linked with those of Theorem 3.1. Therefore the proof can be established in a manner similar to that of Theorem 3.1.