Wavelet-based lifting scheme for the numerical solution of some class of nonlinear partial differential equations

2018 ◽  
Vol 16 (05) ◽  
pp. 1850046 ◽  
Author(s):  
S. C. Shiralashetti ◽  
L. M. Angadi ◽  
A. B. Deshi
Keyword(s):  
Numerical Solution ◽  
Nonlinear Partial Differential Equations ◽  
Lifting Scheme ◽  
Nonlinear Pdes ◽  
Computational Time ◽  
Test Problems ◽  
Wavelet Filter ◽  
Science And Engineering ◽  
Partial Differential ◽  
Physical Phenomena

Partial differential equation (PDE) occurs in the mathematical modeling of many physical phenomena arising in science and engineering. In this paper, we present wavelet-based lifting scheme for the numerical solution of nonlinear PDEs using different wavelet filter coefficients. The numerical results obtained by this scheme are compared with the exact solution to demonstrate the accuracy and also speeds up convergence in lesser computational time as compared with the existing schemes. Some test problems are presented for the validity and applicability of the scheme.

Numerical Solution of Some Class of Nonlinear Partial Differential Equations Using Wavelet-Based Full Approximation Scheme

2019 ◽  
Vol 17 (06) ◽  
pp. 1950015
Author(s):  
S. C. Shiralashetti ◽  
L. M. Angadi ◽  
A. B. Deshi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solution ◽  
Approximation Scheme ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Pdes ◽  
Computational Time ◽  
Test Problems ◽  
Partial Differential ◽  
Almost All

In the last decades, wavelets have become a dominant tool for having applications in almost all the areas of engineering and science such as numerical simulation of partial differential equations (PDEs). The performance of the conventional numerical methods has been found to involve some difficulty to observe fast convergence in low computational time. To overcome this difficulty, we presented wavelet-based full approximation scheme (WFAS) for the numerical solution of some class of nonlinear PDEs using Daubechies wavelet intergrid operators. The numerical results obtained by this scheme are compared with the exact solution to reveal the accuracy and also speed up convergence in lesser computational time as compared with the existing schemes. Some test problems are presented to show the applicability and attractiveness of WFAS.

Wavelet Lifting Scheme for the Numerical Solution of Dynamic Reynolds Equation for Micropolar Fluid Lubrication

2021 ◽  
pp. 2150033
Author(s):  
S. C. Shiralashetti ◽  
M. H. Kantli ◽  
A. B. Deshi
Keyword(s):  
Numerical Solution ◽  
Micropolar Fluid ◽  
Reynolds Equation ◽  
Lifting Scheme ◽  
Computational Time ◽  
Biorthogonal Wavelets ◽  
Wavelet Theory ◽  
Science And Engineering ◽  
Promising Tool ◽  
Wavelet Lifting

Recently, wavelet theory has become a well recognized promising tool in science and engineering field; especially, wavelets are successfully used in fast algorithms for easy execution. In this paper, we developed wavelet lifting scheme using orthogonal and biorthogonal wavelets for the numerical solution of dynamic Reynolds equation for micropolar fluid lubrication. The numerical results gained through proposed scheme are compared with the exact solution to expose the accuracy with speed of convergence in lesser computational time as compared with the existing methods. The examples are given to demonstrate the applicability and attractiveness of proposed method.

Biorthogonal wavelet-based multigrid and full approximation scheme for the numerical solution of parabolic partial differential equations

2019 ◽  
Vol 12 (04) ◽  
pp. 1950054
Author(s):  
S. C. Shiralashetti ◽  
L. M. Angadi ◽  
A. B. Deshi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solution ◽  
Approximation Scheme ◽  
Computational Time ◽  
Parabolic Partial Differential Equations ◽  
Test Problems ◽  
Biorthogonal Wavelet ◽  
Partial Differential ◽  
Compression Performance

This paper presents biorthogonal wavelet-based multigrid (BWMG) and full approximation scheme (FAS) for the numerical solution of parabolic partial differential equations (PPDEs), which are working horse behind many commercial applications like finger print image compression. Performance of the proposed schemes is better than the existing ones in terms of super convergence with less computational time. Some of the test problems are taken to demonstrate the applicability and validity of the method.

scholarly journals Numerical Analysis and Comparison of Gridless Partial Differential Equations

2021 ◽  
Vol 15 ◽  
pp. 1223-1231
Author(s):  
Zhao Zhang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solution ◽  
Numerical Solutions ◽  
Absolute Error ◽  
High Accuracy ◽  
Science And Engineering ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Physical Phenomena

In the field of science and engineering, partial differential equations play an important role in the process of transforming physical phenomena into mathematical models. Therefore, it is very important to get a numerical solution with high accuracy. In solving linear partial differential equations, meshless solution is a very important method. Based on this, we propose the numerical solution analysis and comparison of meshless partial differential equations (PDEs). It is found that the interaction between the numerical solutions of gridless PDEs is better, and the absolute error and relative error are lower, which proves the superiority of the numerical solutions of gridless PDEs

scholarly journals Analytical Solution of Two-Dimensional Sine-Gordon Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Alemayehu Tamirie Deresse ◽  
Yesuf Obsie Mussa ◽  
Ademe Kebede Gizaw
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Test Problems ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Science And Engineering ◽  
Transform Method ◽  
Differential Transform ◽  
Good Agreement

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear sine-Gordon equations subject to appropriate initial conditions. Some lemmas which help us to solve the governing problem using the proposed method are proved. This scheme has the advantage of generating an analytical approximate solution or exact solution in a convergent power series form with conveniently determinable components. The method considers the use of the appropriate initial conditions and finds the solution without any discretization, transformation, or restrictive assumptions. The accuracy and efficiency of the proposed method are demonstrated by four of our test problems, and solution behavior of the test problems is presented using tables and graphs. Further, the numerical results are found to be in a good agreement with the exact solutions and the numerical solutions that are available in literature. We have showed the convergence of the proposed method. Also, the obtained results reveal that the introduced method is promising for solving other types of nonlinear partial differential equations (NLPDEs) in the fields of science and engineering.

scholarly journals A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1336
Author(s):  
Constantin Bota ◽  
Bogdan Căruntu ◽  
Dumitru Ţucu ◽  
Marioara Lăpădat ◽  
Mădălina Sofia Paşca
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Least Squares ◽  
Least Squares Method ◽  
Differential Quadrature Method ◽  
Nonlinear Partial Differential Equations ◽  
Differential Quadrature ◽  
Test Problems ◽  
Quadrature Method ◽  
Partial Differential

In this paper a new method called the least squares differential quadrature method (LSDQM) is introduced as a straightforward and efficient method to compute analytical approximate polynomial solutions for nonlinear partial differential equations with fractional time derivatives. LSDQM is a combination of the differential quadrature method and the least squares method and in this paper it is employed to find approximate solutions for a very general class of nonlinear partial differential equations, wherein the fractional derivatives are described in the Caputo sense. The paper contains a clear, step-by-step presentation of the method and a convergence theorem. In order to emphasize the accuracy of LSDQM we included two test problems previously solved by means of other, well-known methods, and observed that our solutions present not only a smaller error but also a much simpler expression. We also included a problem with no known exact solution and the solutions computed by LSDQM are in good agreement with previous ones.

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