A convergence analysis of the numerical solution of boundary-value problems by using two-dimensional Haar wavelets

Wichailukkana, Narongpol; Novaprateep, Boriboon; Boonyasiriwat, Chaiwoot

doi:10.2306/scienceasia1513-1874.2016.42.346

Cited by 12 publications

(8 citation statements)

References 11 publications

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“…The key factors of any numerical method are the accuracy and convergence. These are studied for HWM in [43,45,65]. The convergence theorem is proven in [45], which also shows that the order of convergence of the HWM approach based on [15] is equal to two.…”

Section: Introductionmentioning

confidence: 93%

Application of Higher Order Haar Wavelet Method for Solving Nonlinear Evolution Equations

Ratas

Salupere

2020

Mathematical Modelling and Analysis

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Section: Introductionmentioning

confidence: 93%

Application of Higher Order Haar Wavelet Method for Solving Nonlinear Evolution Equations

Ratas

Salupere

2020

Mathematical Modelling and Analysis

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“…In the present section, we prove the convergence of the proposed Haar wavelet method. Initially, we introduce some lemmas needed to prove the convergence.Lemma The Haar waveforms and their integral functions are bounded above and their upper bounds are as follows :

h_{i} false(y false) \leq 1, \forall i and p_{i} false(y false) \leq \frac{1}{2^{j + 1}}, q_{i} false(y false) < ℭ {((), \frac{1}{2^{j + 1}})}^{2}, for i > 1,

with

ℭ = \frac{8}{3}

. Proof Refer [66]. Lemma Let

f false(y t_{l + 1} false) = \frac{\partial^{2} v false(y t_{l + 1} false)}{\partial y^{2}} \in L^{2} false(normalℝ false)

be a function defined on [ α , β ] at ( l + 1)‐ th time level and

f false(y t_{l + 1} false) \approx \sum_{i = 1}^{2 M_{1}} a_{i, l} h_{i} false(y false)

.…”

Section: Convergence Resultsmentioning

confidence: 99%

Haar‐wavelet based approximation for pricing American options under linear complementarity formulations

Kumar

Deswal

2020

Numerical Methods Partial

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In this manuscript, we present a novel and highly accurate wavelet‐based approximation technique to explore the sensitivities and value of American options diagnosed by linear complementarity problems. For a detailed analysis of such financially relevant problems, we transform the actual final value problem into a dimensionless initial value problem. To avoid the unacceptable large truncation error, the unbounded domain is trimmed into a bounded domain. A remarkable observation is that to investigate the various physical and numerical aspects of the options' sensitivities; the proposed scheme is efficient as it explicitly provides the numerical approximation of all the derivatives of the solution function. The multiresolution technique of the wavelets and the convergence of the proposed wavelet scheme are comprehensively analyzed. The wavelet analysis is accompanied by illustrative examples to demonstrate the proficiency and robustness of the present method coupled with graphical representations. It has been shown that the present method is efficient to solve free boundary problems. It is worthy to note that the highly accurate and promising computational results are enough to confirm the performance of the proposed method. The simulated results of options' Greeks analyzed and discussed have vast applications in different financial institutes and trading markets.

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“…From Wichailukkana et al, 46 it can be observed that the upper bounds of Q i,m (x) satisfy the following:…”

Section: Haar Waveletsmentioning

confidence: 99%

Numerical solution of a fourth‐order singularly perturbed boundary value problem with discontinuities via Haar wavelets

Chakravarthy

Sundrani

Ramos

2022

Math Methods in App Sciences

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In this paper, an efficient wavelet-based numerical scheme is presented for the solution of fourth-order singularly perturbed boundary value problems with discontinuous data. The fourth-order derivative of the solution function is expanded in Haar series and then integrated to get the approximations for the lower order derivatives of the solution function. The convergence of the proposed method is analyzed. The analysis shows that the error bound is inversely proportional to the wavelet resolution. The method is applied on three test problems to check its performance. The proposed method is computationally fast, cheap, and reliable, even considering low-order wavelet resolutions. Another important advantage of the proposed method is that it can be easily extended to similar classes of problems with a variety of boundary conditions.

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A convergence analysis of the numerical solution of boundary-value problems by using two-dimensional Haar wavelets

Abstract: ABSTRACT:This study provides an analysis of the convergence of the Haar wavelet-based method for solving twodimensional boundary value problems. The convergence analysis shows that the approximation method is of order 2. The analytical results are validated via two numerical examples.

Cited by 12 publications

References 11 publications

Application of Higher Order Haar Wavelet Method for Solving Nonlinear Evolution Equations

Application of Higher Order Haar Wavelet Method for Solving Nonlinear Evolution Equations

Haar‐wavelet based approximation for pricing American options under linear complementarity formulations

Numerical solution of a fourth‐order singularly perturbed boundary value problem with discontinuities via Haar wavelets

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