Generalized mapped nodal Laguerre spectral collocation method for Volterra delay integro-differential equations with non
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Generalized mapped nodal Laguerre spectral collocation method for Volterra delay integro-differential equations with noncompact kernels Zhuyan Tang1 · Emran Tohidi2,3 · Fuli He4 Received: 25 May 2020 / Revised: 29 July 2020 / Accepted: 3 October 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract This paper is devoted to solve a class of weakly singular Volterra integro-differential equations with noncompact kernels. Since the solution of such these models may has singularities, so the classical spectral methods lose their high accuracy for solving them. Innovation of this article is that, we use the generalized mapped nodal Laguerre spectral collocation method to deal with the singularity. Therefore, we can make good use of the advantages of the mapped Laguerre functions. The best advantage of the proposed method is its robustness for solving problems that have singularity near the left (or right) boundary of the computational interval. We present the construction and analysis of the generalized log orthogonal Laguerre functions collocation method in this paper and some numerical examples with nonsmooth solutions are included to show the efficiency of the suggested numerical scheme with respect to the classical Jacobi spectral methods. Keywords Generalized mapped Laguerre functions · Spectral collocation method · Noncompact kernels · Weak singularity · Volterra integro-differential equations Mathematics Subject Classification 34A12 · 74S25 · 65L60 · 65L70 · 33C45 · 65B99
Communicated by Jose Alberto Cuminato.
B
Fuli He [email protected]; [email protected] Zhuyan Tang [email protected] Emran Tohidi [email protected]
1
School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei, People’s Republic of China
2
Informetrics Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam
3
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
4
School of Mathematics and Statistics, Central South University, Changsha 410083, People’s Republic of China 0123456789().: V,-vol
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1 Introduction In this paper, we consider the following Volterra delay integro-differential equation with noncompact kernels: φ (x) = p(x)φ(x) + q(x)φ(εx) +
0
+ g(x), x ∈ I : = (0, X ),
x
s μ−1 K 1 (x, s)φ(s) ds + xμ
0
εx
τ μ−1 K 2 (x, τ )φ(τ ) dτ (εx)μ
(1)
φ(0) = φ0 ,
where 0 < μ < 1, 0 < ε < 1 and φ(x) is the unknown function which our aim is to find it in an appropriate manner numerically. Taking into account that p(x), q(x) ∈ C(I ), K 1 , K 2 ∈ C(I × I ), meanwhile K 1 (x, x), K 2 (x, εx) = 0. Many researchers have given several effective methods to solve Volterra-type integrodifferential equations. In Jiang and Ma (2013), the Chebyshev collocation method have been put forward for solving (1) without the vanishing delay integral term. Legendre spectral collocation approach was considered to solve the second-order Volterra integro-differential equations with regular kernels (without co
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