Estimating the derivative of the Legendre polynomial

Антонов, В. А.; Холшевников, К. В.; Шайдулин, В. Ш.

doi:10.3103/s1063454110040011

Cited by 10 publications

(13 citation statements)

References 3 publications

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“…Moreover, the first (positive) maximum m l of Q l attained at x l is larger than the first positive maximum m 2 l of u 2 l attained at x 2 l and smaller than the first positive maximum m 1 l of u 1 l attained at x 1 l . Moreover, we have

x_{2 l} < x_{l} < x_{1 l} .

As it is shown in , the function u 1 l is in fact given by the Bessel equation. Thus, defining

η = ν_{1} x, u_{1 l} = 2 \sqrt{η} u_{1 B},

we conclude that u 1 B ( η ) satisfies the Bessel equation

η^{2} {\overset{. .}{u}}_{1 B} + η {\dot{u}}_{1 B} + (η^{2} - 1) u_{1 B} = 0

with

{\dot{u}}_{1 B}

meaning differentiation with respect to η .…”

Section: Proof Of Theorems 1 Andmentioning

confidence: 83%

“…Remark The normalization implies that

∥ P_{l} ∥_{C^{0} ([0, π])} = \sqrt{l + 1 / 2}

. In the standard normalization, used, for example, in , the right‐hand side of is multiplied by

\frac{1}{l + 1 / 2}

, so the supremum of the absolute value of P l is equal to 1.…”

Section: Mild Solutions To the Initial‐boundary Value Problemmentioning

confidence: 99%

“…Proof The estimations for λ =0 and λ =3/4 are given in for different normalizations (see Remark after ). Next, let us note that for λ ∈[0,3/4), we can write

\begin{matrix} 1.61em | {(1 - y^{2})}^{λ} \frac{d}{d y} P_{l} (y) 1.61em | = {((), {(1 - y^{2})}^{3 / 4} 1.61em | \frac{d}{d y} P_{l} (y) 1.61em |)}^{4 λ / 3} 1.61em | \frac{d}{d y} P_{l} (y) {1.61em |}^{1 - 4 λ / 3} . \end{matrix}

Using the estimates for

λ = \frac{3}{4}

and λ =0, we have

\begin{matrix} 1.61em | {(1 - y^{2})}^{λ} \frac{d}{d y} P_{l} (y) 1.61em | {(l + 1 / 2)}^{- 1 / 2} < 1.61em (A (\frac{3}{4}) {(l + a (\frac{3}{4}))}^{1 / 2} {1.61em)}^{4 λ / 3} 1.61em (A (0) {(l + a (0))}^{2} {1.61em)}^{1 - 4 λ / 3} \\ = A \end{matrix}

…”

Section: Mild Solutions To the Initial‐boundary Value Problemmentioning

confidence: 99%

“…To analyze the properties of the function Q l , in particular to find the position and the value of its first (closest to zero) maximum, we will use the results of . The function Q l satisfies the equation

Q_{l}^{''} + q (x) Q_{l} = 0

where

q (x) = ν^{2} - \frac{3}{4 \sin^{2} x}, ν = ν (l) = (l + 1 / 2),

and ′ denotes the differentiation with respect to x .…”

Section: Proof Of Theorems 1 Andmentioning

confidence: 99%

“…Remark The asymptotic expansion of the function Q l close to x =0 was proposed in in the form:

Q_{l} (x) = \frac{l (l + 1)}{2} x^{3} \sum_{k = 0}^{\infty} a_{k} x_{k}^{2} .

Such a form is implied by the fact that

\frac{d {\overset{P}{true}}_{l}}{d y} |_{y = 1} = \frac{l (l + 1)}{2}

and that

\sin^{3} x \approx x^{3}

close to x =0.

□

…”

Section: Proof Of Theorems 1 Andmentioning

confidence: 99%

See 4 more Smart Citations

Regularity of solutions to a reaction–diffusion equation on the sphere: the Legendre series approach

Bialecki

Kaźmierczak

Nowicka

et al. 2017

Math Methods in App Sciences

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In the paper, we study some ‘a priori’ properties of mild solutions to a single reaction–diffusion equation with discontinuous nonlinear reaction term on the two‐dimensional sphere close to its poles. This equation is the counterpart of the well‐studied bistable reaction–diffusion equation on the Euclidean plane. The investigation of this equation on the sphere is mainly motivated by the phenomenon of the fertilization of oocytes or recent studies of wave propagation in a model of immune cells activation, in which the cell is modeled by a ball. Because of the discontinuous nature of reaction kinetics, the standard theory cannot guarantee the solution existence and its smoothness properties. Moreover, the singular nature of the diffusion operator near the north/south poles makes the analysis more involved. Unlike the case in the Euclidean plane, the (axially symmetric) Green's function for the heat operator on the sphere can only be represented by an infinite series of the Legendre polynomials. Our approach is to consider a formal series in Legendre polynomials obtained by assuming that the mild solution exists. We show that the solution to the equation subject to the Neumann boundary condition is C1 smooth in the spatial variable up to the north/south poles and Hölder continuous with respect to the time variable. Our results provide also a sort of ‘a priori’ estimates, which can be used in the existence proofs of mild solutions, for example, by means of the iterative methods. Copyright © 2017 John Wiley & Sons, Ltd.

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x_{2 l} < x_{l} < x_{1 l} .

As it is shown in , the function u 1 l is in fact given by the Bessel equation. Thus, defining

η = ν_{1} x, u_{1 l} = 2 \sqrt{η} u_{1 B},

we conclude that u 1 B ( η ) satisfies the Bessel equation

η^{2} {\overset{. .}{u}}_{1 B} + η {\dot{u}}_{1 B} + (η^{2} - 1) u_{1 B} = 0

with

{\dot{u}}_{1 B}

meaning differentiation with respect to η .…”

Section: Proof Of Theorems 1 Andmentioning

confidence: 83%

“…Remark The normalization implies that

∥ P_{l} ∥_{C^{0} ([0, π])} = \sqrt{l + 1 / 2}

. In the standard normalization, used, for example, in , the right‐hand side of is multiplied by

\frac{1}{l + 1 / 2}

, so the supremum of the absolute value of P l is equal to 1.…”

Section: Mild Solutions To the Initial‐boundary Value Problemmentioning

confidence: 99%

“…Proof The estimations for λ =0 and λ =3/4 are given in for different normalizations (see Remark after ). Next, let us note that for λ ∈[0,3/4), we can write