Page 106 - IJOCTA-15-1

´
R. Temoltzi-Avila, J. Temoltzi-Avila / IJOCTA, Vol.15, No.1, pp.92-102 (2025)
y j and ˙y j satisfy the following inequality for each source, then we can verify that the function de-
t ∈ [0, T]: fined in (14) is a solution to the initial-boundary
value problem (1)–(3).

δ j κ 2
|y j (t)| ≤ , | ˙y j (t)| ≤ 2δ j 1 + . The proof of Theorem 1 is complete.
κ 1 β 2 κ 1
j
Acknowledgments
To prove Theorem 1, we verify that the series on
the right-hand side of (14) and the series defin- None.
ing the functions in (7) converge uniformly on Ω.
In the proof of Lemma 1 it has been shown that Funding
the series on the right-hand sides of (14), (15)
and (16) converge uniformly on Ω to the func- None.
tions y, ∂y and ∂y , respectively, so it remains to
∂x ∂t Conflict of interest
verify that this also holds for the series defining
2
2
the functions ∂ y 2 and cf D α ∂ y 2 . The authors declare that they have no conflict of
∂x 0 t ∂x
interest regarding the publication of this article.
We observe from (14) that
∞
2
∂ y X 2 Author Contributions
(x, t) = − β sin(β j x)y j (t), (18)
´
∂x 2 j Conceptualization: Ra´ul Temoltzi-Avila
j=1
Formal analysis: All authors
and therefore,
Methodology: All authors
2 ∞ ∞
∂ y X 1 X Writing – original draft: All authors
2
(x, t) ≤ β |y j (t)| ≤
j δ j .
∂x κ 1
2 Writing – review & editing: All authors
j=1 j=1
P ∞
We observe from this inequality that if j=1 j is
δ
convergent, then the series on the right-hand side Availability of data
2
∂ y
of (18) converges uniformly to the function
∂x 2 Not applicable.
on Ω, which follows from the Weierstrass M-test.
References
On the other hand, from the representation (18)
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2
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α
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α
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e
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0
t
(1 − α)κ 1 β 2 [3] Conejero, J. A., Franceschi, J. & Pic´o-Marco,
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(18), (19) and the series that defines the heat 4
100