Uniform $L^1$ behavior in classes of integro-differential equations with convex kernels

Noren, Richard

doi:10.1216/jie-1988-1-3-385

Cited by 13 publications

(18 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To estimate v for t > 1/e, we employ the technical condition (1.17) to obtain the estimate (proved in [11]), \D\ -fo J2^'a'(s^ds' 0<z<£, I E,^(t) + E,0Kt) a continuous function on e < t < p. The estimates for (1 /t2) J + K di are done exactly as in [2], Thus, for t > l/e, we have u'{t,X\,...,Xn) < G(t), where G{t) is some function in L'(l/e,oo).…”

Section: / 0mentioning

confidence: 99%

“…Included in [11] where A(t) = /0' a\(s) H ha"(s) ds, a condition stated in terms of the Fourier transform of the functions a,, i = and a mild technical condition. In [7], (1.8) is…”

mentioning

confidence: 99%

“…shown for functions a, that are completely monotonic on (0, oo), ((-1 )ja\j\t) > 0, j = 0,1,2,3,...) and satisfy conditions that are similar to (1.10) and the transform condition used in [11]. By contrast in Theorems 1.2 and 1.3 and in the corollary, no condition like (1.10) is used to prove (1.4).…”

mentioning

confidence: 99%

See 2 more Smart Citations

Uniform 𝐿¹ behavior in a class of linear Volterra equations

Noren¹

1989

Quart. Appl. Math.

Self Cite

View full text Add to dashboard Cite

show abstract

Section: / 0mentioning

confidence: 99%

“…Included in [11] where A(t) = /0' a\(s) H ha"(s) ds, a condition stated in terms of the Fourier transform of the functions a,, i = and a mild technical condition. In [7], (1.8) is…”

mentioning

confidence: 99%

See 1 more Smart Citation

Uniform 𝐿¹ behavior in a class of linear Volterra equations

Noren¹

1989

Quart. Appl. Math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The asymptotic analysis of the problem have been a very active field over the past several decades, and there are numerous very ingenious results; see, for example . When

n = 1

, Carr and Hannsgen establish the estimates

\int_{0}^{\infty} | | u (t) | | d t \leq C | | u_{0} | |,

where

C

is a positive constant independent of

u (t)

, and

| | . | |

denotes the norm in

boldH

, when the kernel

a_{1} (t)

satisfies

a_{1} (t) \in C (0, \infty) \cap L^{1} (0, 1) is nonconstant, nonnegative, nonincreasing, convex,

and - a_{1}^{'} (t) is convex on (0, \infty) .

But, when

n \geq 2

it is a conjecture that holds if all

a_{j} (t)

satisfy (1.5), see [8, p. 390] or , p. 550].…”

Section: Introductionmentioning

confidence: 99%

“…We note that the completely monotonic kernels (1.3) satisfies (1.5). Noren , Theorem 1] gives sufficient conditions such that holds for

n > 1

.…”

Section: Introductionmentioning

confidence: 99%

The time discretization in classes of integro-differential equations with completely monotonic kernels: Weighted asymptotic convergence

2015

Numer. Methods Partial Differential Eq.

View full text Add to dashboard Cite

We consider the time discretization for the solution of the equationHere, the operators L j are densely defined positive self-adjoint linear operator on a Hilbert space H and have spectral decompositions with respect to a common resolution of the identity {E λ } in H. The kernel functions a j (t), 1 ≤ j ≤ n, are assumed to be completely monotonic on (0,∞) and locally integrable, but not constant. The considered time discretization method comes from [Da Xu, Science China Mathematics 56 (2013), 395-424], where the backward Euler method is combined with order one convolution quadrature for approximating the integral term. In this article, the convergence properties of the time discretization are given in the weighted l 1 (ρ; 0, ∞; H) and l ∞ (ρ; 0, ∞; H) norm, where ρ is a given weighted function. Numerical experiments show the theoretical results.

show abstract

An efficient Sinc‐collocation method by the single exponential transformation for the nonlinear fourth‐order partial integro‐differential equation with multiterm kernels

Jiang

2021

Math Methods in App Sciences

View full text Add to dashboard Cite

In this article, a Sinc-collocation method is proposed and analyzed for solving the nonlinear fourth-order partial integro-differential equation with the multiterm kernels. The time derivative is discretized by the backward Euler method, and the Riemann-Liouville (R-L) fractional integral terms are treated by means of the first-order convolution quadrature. Then, the nonlinear term is linearized.A fully discrete scheme is established with the space discretization by the Sinc approximation based on the single exponential (SE) transformation. The spatial exponential convergence of the proposed method is derived. Several numerical examples are provided to illustrate the accuracy and effectiveness of our method.

show abstract

Uniform $L^1$ behavior in classes of integro-differential equations with convex kernels

Cited by 13 publications

References 0 publications

Uniform 𝐿¹ behavior in a class of linear Volterra equations

Uniform 𝐿¹ behavior in a class of linear Volterra equations

The time discretization in classes of integro-differential equations with completely monotonic kernels: Weighted asymptotic convergence

An efficient Sinc‐collocation method by the single exponential transformation for the nonlinear fourth‐order partial integro‐differential equation with multiterm kernels

Contact Info

Product

Resources

About