id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-7095	Kamran; Zahra, Bibi; Ali, Zeeshan; Aloqaily, Ahmad ; Mlaiki, Nabil	A Laplace-Chebyshev Spectral Method for Multi-Dimensional Anomalous Transport	2025	30	.pdf	application/pdf	8640	533	72	Math, 18 (4) (2025), 7095 8 of 30 Next, we split the difference of integrals into three parts as follows:∣∣∣∣ ∫ τ1 0 (τ1 − ϑ)α−1 ( λ1∇2u(x̄1, ϑ)− λ2u(x̄1, ϑ) + f(x̄1, ϑ) ) dϑ − ∫ τ2 0 (τ2 − ϑ)α−1 ( λ1∇2u(x̄2, ϑ)− λ2u(x̄2, ϑ) + f(x̄2, ϑ) ) dϑ ∣∣∣∣ = [ ∫ τ1 0 [ (τ1 − ϑ)α−1 − (τ2 − ϑ)α−1 ] ( λ1∇2u(x̄1, ϑ)− λ2u(x̄1, ϑ) + f(x̄1, ϑ) ) dϑ︸ ︷︷ ︸ (iii) ] for the integral (i), using the mean value theorem, we have |(τ1 − ϑ)α−1 − (τ2 − ϑ)α−1| ≤ (α− 1)(τ2 − τ1)(τ1 − ϑ)α−2 Therefore: (i) ≤ (λ1∥∇2u∥∞ + λ2∥u∥∞ + ∥f∥∞)(α− 1)δ2 ∫ τ1 0 (τ1 − ϑ)α−2dϑ = (λ1∥∇2u∥∞ + λ2∥u∥∞ + ∥f∥∞)δ2τ α−1 1 .	cache/ejpam-7095.pdf	txt/ejpam-7095.txt