A simple algorithm for numerical solution of nonlinear parabolic partial differential equations

Saleem, Sidra; Aziz, Imran; Hussain, Malik Zawwar

doi:10.1007/s00366-019-00796-z

Cited by 9 publications

(6 citation statements)

References 26 publications

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“…For i = 1, the Haar wavelet is the scaling function which takes value 1 over I and elsewhere zero. We need the following integrals to solve a nth order PDE, [39][40][41]…”

Section: Haar Waveletsmentioning

confidence: 99%

“…The function of the i ‐Haar wavelet for x ∈ I is defined as 36‐38

h_{i} false(x false) = ({, \begin{matrix} array 1, & array for x \in [Ω_{1} (i), Ω_{2} (i)), \\ array - 1, & array for x \in [Ω_{2} (i), Ω_{3} (i)), \\ array 0, & array otherwise, \end{matrix})

where

Ω_{1} (i) = a + (b - a) \frac{l}{m}, Ω_{2} (i) = a + (b - a) \frac{l + 0.5}{m}, Ω_{3} (i) = a + (b - a) \frac{l + 1}{m} .

For

i = 1

, the Haar wavelet is the scaling function which takes value 1 over I and elsewhere zero. We need the following integrals to solve a n th order PDE, 39‐41

\begin{matrix} p_{i, 1} false(x false) = & \int_{a}^{x} h_{i} false(τ false) d τ, \\ p \end{matrix}

…”

Section: Haar Waveletsmentioning

confidence: 99%

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A computational approach for finding the numerical solution of modified unstable nonlinear Schrödinger equation via Haar wavelets

Abdullah

Rafiq

2021

Math Methods in App Sciences

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In this work, we combined the Haar wavelet collocation method with the backward Euler difference formula to determine the approximate solutions of the modified unstable nonlinear Schrödinger equation. The backward Euler difference formula estimates the time derivative term and the Haar wavelet collocation method estimate the space derivative terms of the modified unstable nonlinear Schrödinger equation. This approach reduces the modified unstable nonlinear Schrödinger equation into a finite system of linear equations. In addition, we substantiate the efficiency and accuracy of the method graphically and numerically with the help of four examples.

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“…For i = 1, the Haar wavelet is the scaling function which takes value 1 over I and elsewhere zero. We need the following integrals to solve a nth order PDE, [39][40][41]…”

Section: Haar Waveletsmentioning

confidence: 99%

“…The function of the i ‐Haar wavelet for x ∈ I is defined as 36‐38

h_{i} false(x false) = ({, \begin{matrix} array 1, & array for x \in [Ω_{1} (i), Ω_{2} (i)), \\ array - 1, & array for x \in [Ω_{2} (i), Ω_{3} (i)), \\ array 0, & array otherwise, \end{matrix})

where

Ω_{1} (i) = a + (b - a) \frac{l}{m}, Ω_{2} (i) = a + (b - a) \frac{l + 0.5}{m}, Ω_{3} (i) = a + (b - a) \frac{l + 1}{m} .

For

i = 1

, the Haar wavelet is the scaling function which takes value 1 over I and elsewhere zero. We need the following integrals to solve a n th order PDE, 39‐41

\begin{matrix} p_{i, 1} false(x false) = & \int_{a}^{x} h_{i} false(τ false) d τ, \\ p \end{matrix}

…”

Section: Haar Waveletsmentioning

confidence: 99%

A computational approach for finding the numerical solution of modified unstable nonlinear Schrödinger equation via Haar wavelets

Abdullah

Rafiq

2021

Math Methods in App Sciences

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“…As a result, the interpretation of any physical phenomena could be improved by using numerical approximations that are more precise. It is efficient and effective to obtain the numerical approximations of 1D or 2D linear or nonlinear parabolic partial differential equations using a wide variety of methodologies and schemes, such as the finite difference approach [1], the Haar wavelet collocation method [2], the Fourier spectral method [3], and many more. The referenced books [4][5][6][7] provide a review of linear and nonlinear parabolic partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

On the Numerical Treatment of 2D Nonlinear Parabolic PDEs by the Galerkin Method with Bivariate Bernstein Polynomial Bases

Pranta¹,

Ali²,

Shafiqul³

et al. 2022

JAMC

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In this study, numerical solutions are obtained for the time-dependent two-dimensional nonlinear parabolic partial differential equations (PDEs) with initial and Dirichlet boundary conditions. In assessing spatial derivatives, we employ the modified Galerkin method with the aid of Green's theorem, which minimizes the derivatives' order and incorporates boundary conditions. In the trial function, we use bivariate Bernstein polynomial bases. All the initial and boundary conditions are handled carefully by suitable transformation. Further, we exploit an iterative α-family approximation, especially the Crank Nicolson scheme, to take care the time derivative. Applying the proposed technique to a variety of nonlinear 2D parabolic PDEs, such as the 2D Burger's equation and the 2D Convection-Diffusion Reaction equation, the numerical results are presented in the form of tables and figures. The numerical results provide conclusive evidence that the technique being proposed is accurate and effective.

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“…Note that recently the authors have been trying to find alternative approaches for solving differential equations. In paper [18], the numerical solution of nonlinear twodimensional parabolic partial differential equations with initial and Dirichlet boundary conditions is considered. The time Continuous local splines of the fourth order of approximation and boundary value problem derivative is approximated using a finite difference scheme whereas space derivatives are approximated using the Haar wavelet collocation method.…”

Section: Introductionmentioning

confidence: 99%

Continuous Local Splines of the Fourth Order of Approximation and Boundary Value Problem

2020

International Journal of Circuits, Systems and Signal Processing

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This paper discusses the construction of polynomialand non-polynomial splines of the fourth order of approximation.The behavior of the Lebesgue constants for the left, the right, andthe middle continuous cubic polynomial splines are considered.The non-polynomial splines are used for the construction of thespecial central difference approximation. The approximation offunctions, and the solving of the boundary problem with thepolynomial and non-polynomial splines are discussed. Numericalexamples are done.

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A simple algorithm for numerical solution of nonlinear parabolic partial differential equations

Cited by 9 publications

References 26 publications

A computational approach for finding the numerical solution of modified unstable nonlinear Schrödinger equation via Haar wavelets

A computational approach for finding the numerical solution of modified unstable nonlinear Schrödinger equation via Haar wavelets

On the Numerical Treatment of 2D Nonlinear Parabolic PDEs by the Galerkin Method with Bivariate Bernstein Polynomial Bases

Continuous Local Splines of the Fourth Order of Approximation and Boundary Value Problem

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