Abstract
The wellposedness of difference schemes of the initial value problem for delay differential equations with unbounded operators acting on delay terms in an arbitrary Banach space is studied. Theorems on the wellposedness of these difference schemes in fractional spaces are proved. In practice, the coercive stability estimates in Hölder norms for the solutions of difference schemes of the mixed problems for delay parabolic equations are obtained.
Keywords:
wellposedness; delay parabolic equations; fractional spaces; coercive stability1 Introduction
Approximate solutions of the delay differential equations have been studied extensively in a series of works (see, for example, [16] and the references therein) and developed over the last three decades. In the literature mostly the sufficient condition
was considered for the stability of the following test delay differential equation:
with the initial condition
It is known that delay differential equations can be solved by applying standard numerical
methods for ordinary differential equations without the presence of delay. However,
it is difficult to generalize any numerical method to obtain a high order of accuracy
algorithms, because highorder methods may not lead to efficient results. It is well
known that even if
Delay partial differential equations arise from various applications, like in climate models, biology, medicine, control theory, and many others (see, for example, [7] and the references therein).
The theory of approximate solutions of delay partial differential equations has received less attention than delay ordinary differential equations. A situation which occurs in delay partial differential equations when the delay term is an operator of lower order with respect to the other operator term is widely investigated (see, for example, [79] and the references therein). In the case where the delay term is an operator of the same order with respect to other operator term, this is studied mainly in a Hilbert space (see, for example, [10] and the references therein). In fact there are very few papers where the delay term is an operator of the same order with respect to the other operator term, this being investigated in a general Banach space (see [1114]) and in these works, the authors look only for partial differential equations under regular data. Additionally, approximate solutions of the delay parabolic equations in the case where the delay term is a simple operator of the same order with respect to the other operator term were studied recently in papers [1519].
It is known that various initialboundary value problems for linear evolutionary delay partial differential equations can be reduced to an initial value problem of the form
in an arbitrary Banach space E with the unbounded linear operators A and
for some
The strongly positive operator A defines the fractional spaces
As noted in [19], it is important to study the stability of solutions of the initial value problem (4) for delay differential equations and of difference schemes for approximate solutions of problem (4) under the assumption that
holds for every
for every
in an arbitrary Banach space E with the small positive parameter ε in the high derivative and with the unbounded linear operators A and
Additionally, using the first and second order of the accuracy implicit difference schemes for differential equations without the presence of delay, the first and second order of the accuracy implicit difference schemes,
are presented for approximate solutions of the initial value problem (4). Here, we
will put
The main aim of present paper is to study the wellposedness of the difference schemes
(9) and (10). We establish the coercive stability estimates in fractional spaces
The paper is organized as follows. In Section 2, theorems on coercive stability of difference schemes (9) and (10) are established. In Section 3, the coercive stability estimates in Hölder norms for the solutions of difference schemes for the approximate solutions of delay parabolic equations are obtained. Finally, Section 4 is our conclusion.
2 The wellposedness of difference schemes (9) and (10)
First, we consider the difference scheme (9) when
Theorem 1Assume that the condition (7) holds for every
holds for any
Proof Let us consider
where
Let us estimate
condition (11) and estimates (5) and (7), we obtain
where
Making the substitution
Therefore
for every
for every
Applying the inequality
we get
for every
for every
Applying mathematical induction, one can easily show that it is true for every k. Actually, suppose that the estimate (16) is true for
Using the estimate (16), we obtain
for every
Now, we consider the difference scheme (9) when
for some
Recall that (see, for example, [[23], Chapter 2, p.116]) A is a strongly positive operator in a Banach space E iff its spectrum
for some
Lemma 1For anyzon the edges of the sector,
and outside it the estimate
holds for any
Lemma 2Let for all
Here
Suppose that
holds for every
The use of Lemmas 1 and 2 enables us to establish the following statement.
Theorem 2Assume that the condition
holds for every
Proof Let us consider
Using the estimates (5), (17), and condition (19), we obtain
for every
for every
for every
for every
In a similar manner as Theorem 1, applying mathematical induction, one can easily show that it is true for every k. Theorem 2 is proved. □
Now we consider the difference scheme (10). We have not been able to obtain the same
result for the solution of the difference scheme (10) in spaces
Theorem 3Suppose that the following estimates hold:
and
Then for the solution of the difference scheme (10), the coercive estimate (12) holds.
Proof Let us consider
where
Let us estimate
for every
for every
for every
for every
In a similar manner as Theorem 1, applying mathematical induction, one can easily show that it is true for every k. Theorem 3 is established. □
Now, we consider the difference scheme (10) when
for some
holds for every
Theorem 4Assume that all conditions of Theorem 2 and Theorem 3 are satisfied. Then for the solution of the difference scheme (10), the estimate (12) holds.
Proof Let us consider
Using the estimates (5), (17), and the condition (19), we obtain
for every
for every
for every
for every
In a similar manner as Theorem 1, applying mathematical induction, one can easily show that it is true for every k. Theorem 4 is established. □
Note that these abstract results are applicable to the study of the coercive stability
of various delay parabolic equations with local and nonlocal boundary conditions with
respect to the space variables. However, it is important to study structure of
3 Applications
First, the initialboundary value problem for onedimensional delay parabolic equations is considered:
where
is defined. To formulate the result, one needs to introduce the Banach space
Here and in the future,
To the differential operator
acting in the space of grid functions
for the system of ordinary differential equations. In the second step, problem (28) is replaced by the firstorder accuracy in the difference scheme in t,
Theorem 5Assume that
Then for the solution of the difference scheme (29) the following coercive stability estimates hold:
for all
The proof of Theorem 5 is based on the estimate
and on the abstract Theorem 1, the positivity of the operator
Theorem 6For any
Second, the initial nonlocal boundary value problem for onedimensional delay parabolic equations is considered:
where
To the differential operator
acting in the space of grid functions
for the system of ordinary differential equations. In the second step, problem (34) is replaced by the firstorder accuracy of the difference scheme in t
Theorem 7Assume that all the conditions of Theorem 5 are satisfied. Then for the solution of the difference scheme (35) the coercive stability estimate (31) holds.
The proof of Theorem 7 is based on the estimate
and on the abstract Theorem 1, the positivity of the operator
Theorem 8For any
Third, the initial value problem on the range
for 2mthorder multidimensional delay differential equations of parabolic type is considered:
where
of the differential operator of the form
acting on the functions defined on the space
for
The difference operator
acts on functions defined on the entire space
where
An infinitely differentiable function
holds for any smooth function
The function
It will be assumed that for
Suppose that the coefficient
With the help of
for an infinite system of ordinary differential equations. Now, problem (41) is replaced by the firstorder accuracy of the difference scheme in t
To formulate the result, one needs to introduce the spaces
Theorem 9Assume that all the conditions of Theorem 7 are satisfied. Then for the solution of the difference scheme (42) the following coercive stability estimates hold:
for all
The proof of Theorem 9 is based on the estimate
and on the abstract Theorem 1, the positivity of the operator
4 Conclusion
In the present paper, the wellposedness of the difference schemes for the approximate
solutions of the initial value problem for delay parabolic equations with unbounded
operators acting on delay terms in an arbitrary Banach space is established. Theorems
on the coercive stability of these difference schemes in fractional spaces are established.
In practice, the coercive stability estimates in Hölder norms for the solutions of
the difference schemes for the approximate solutions of the mixed problems for delay
parabolic equations are obtained. Note that in the present paper
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Each of the authors read and approved the final version of the manuscript.
Acknowledgements
This work is supported by Trakya University Scientific Research Projects Unit (Project No: 201091). We would like to thank to the reviewers, whose careful reading, helpful suggestions, and valuable comments helped us to improve the manuscript.
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