scholarly journals Solving linear PDEs with the aid of two-dimensional Legendre wavelets

2012 ◽  
Author(s):  
Fukang Yin ◽  
Junqiang Song
Keyword(s):  
Two Dimensional ◽  
Legendre Wavelets ◽  
Linear Pdes

Two-Dimensional Legendre Wavelets for Solving Variable-Order Fractional Nonlinear Advection-Diffusion Equation with Variable Coefficients

2018 ◽  
Vol 19 (7-8) ◽  
pp. 793-802 ◽  
Author(s):  
M. Hosseininia ◽  
M. H. Heydari ◽  
Z. Avazzadeh ◽  
F. M. Maalek Ghaini
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Main Idea ◽  
Variable Coefficients ◽  
Variable Order ◽  
Two Dimensional ◽  
Legendre Wavelets ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Nonlinear Advection

AbstractThis article studies a numerical scheme for solving two-dimensional variable-order time fractional nonlinear advection-diffusion equation with variable coefficients, where the variable-order fractional derivative is in the Caputo type. The main idea is expanding the solution in terms of the 2D Legendre wavelets (2D LWs) where the variable-order time fractional derivative is discretized. We describe the method using the matrix operators and then implement it for solving various types of fractional advection-diffusion equations. The experimental results show the computational efficiency of the new approach.

Numerical solution of two-dimensional first kind Fredholm integral equations by using linear Legendre wavelet

2016 ◽  
Vol 14 (01) ◽  
pp. 1650004 ◽  
Author(s):  
M. Tahami ◽  
A. Askari Hemmat ◽  
S. A. Yousefi
Keyword(s):  
Integral Equation ◽  
Tensor Product ◽  
Numerical Solution ◽  
Fredholm Integral Equation ◽  
Fredholm Integral Equations ◽  
Wavelet Basis ◽  
Two Dimensional ◽  
Legendre Wavelets ◽  
Numerical Examples ◽  
One Dimensional

In one-dimensional problems, the Legendre wavelets are good candidates for approximation. In this paper, we present a numerical method for solving two-dimensional first kind Fredholm integral equation. The method is based upon two-dimensional linear Legendre wavelet basis approximation. By applying tensor product of one-dimensional linear Legendre wavelet we construct a two-dimensional wavelet. Finally, we give some numerical examples.

Two-Dimensional Legendre Wavelets for Solving Time-Fractional Telegraph Equation

2014 ◽  
Vol 6 (2) ◽  
pp. 247-260 ◽  
Author(s):  
M. H. Heydari ◽  
M. R. Hooshmandasl ◽  
F. Mohammadi
Keyword(s):  
Numerical Solution ◽  
Fractional Integration ◽  
Telegraph Equation ◽  
Operational Matrices ◽  
Two Dimensional ◽  
Legendre Wavelets ◽  
Fractional Telegraph Equation ◽  
Manageable Method

AbstractIn this paper, we develop an accurate and efficient Legendre wavelets method for numerical solution of the well known time-fractional telegraph equation. In the proposed method we have employed both of the operational matrices of fractional integration and differentiation to get numerical solution of the time-telegraph equation. The power of this manageable method is confirmed. Moreover the use of Legendre wavelet is found to be accurate, simple and fast.

Solving two-dimensional linear partial differential equations based on Chebyshev neural network with extreme learning machine algorithm

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Min Liu ◽  
Muzhou Hou ◽  
Juan Wang ◽  
Yangjin Cheng
Keyword(s):  
Neural Network ◽  
Partial Differential Equations ◽  
Extreme Learning Machine ◽  
Two Dimensional ◽  
Content Type ◽  
Linear Partial Differential Equations ◽  
Chebyshev Neural Network ◽  
Learning Machine ◽  
Hidden Layer ◽  
Linear Pdes

Purpose This paper aims to develop a novel algorithm and apply it to solve two-dimensional linear partial differential equations (PDEs). The proposed method is based on Chebyshev neural network and extreme learning machine (ELM) called Chebyshev extreme learning machine (Ch-ELM) method. Design/methodology/approach The network used in the proposed method is a single hidden layer feedforward neural network. The Kronecker product of two Chebyshev polynomials is used as basis function. The weights from the input layer to the hidden layer are fixed value 1. The weights from the hidden layer to the output layer can be obtained by using ELM algorithm to solve the linear equations established by PDEs and its definite conditions. Findings To verify the effectiveness of the proposed method, two-dimensional linear PDEs are selected and its numerical solutions are obtained by using the proposed method. The effectiveness of the proposed method is illustrated by comparing with the analytical solutions, and its superiority is illustrated by comparing with other existing algorithms. Originality/value Ch-ELM algorithm for solving two-dimensional linear PDEs is proposed. The algorithm has fast execution speed and high numerical accuracy.

scholarly journals The Pricing of Asian Options in Uncertain Volatility Model

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Yulian Fan ◽  
Huadong Zhang
Keyword(s):  
Numerical Algorithms ◽  
Reduction Technique ◽  
Nonlinear Pdes ◽  
Asian Options ◽  
Two Dimensional ◽  
One Dimensional ◽  
Arithmetic Average ◽  
Volatility Model ◽  
Expectation Theory ◽  
Linear Pdes

This paper studies the pricing of Asian options when the volatility of the underlying asset is uncertain. We use the nonlinear Feynman-Kac formula in the G-expectation theory to get the two-dimensional nonlinear PDEs. For the arithmetic average fixed strike Asian options, the nonlinear PDEs can be transferred to linear PDEs. For the arithmetic average floating strike Asian options, we use a dimension reduction technique to transfer the two-dimensional nonlinear PDEs to one-dimensional nonlinear PDEs. Then we introduce the applicable numerical computation methods for these two classes of PDEs and analyze the performance of the numerical algorithms.

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