scholarly journals A New Algorithm Based on Bernstein Polynomials Multiwavelets for the Solution of Differential Equations Governing AC Circuits

2021 ◽  
Vol 18 (21) ◽  
pp. 33
Author(s):  
Shweta Pandey ◽  
Sandeep Dixit ◽  
Sag R Verma
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Numerical Solution ◽  
Bernstein Polynomial ◽  
Error Function ◽  
Bernstein Polynomials ◽  
Operational Matrix ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Operational Matrix Of Integration

We extend the application of multiwavelet-based Bernstein polynomials for the numerical solution of differential equations governing AC circuits (LCR and LC). The operational matrix of integration is obtained from the orthonormal Bernstein polynomial wavelet bases, which diminishes differential equations into the system of linear algebraic equations for easy computation. It appeared that fewer wavelet bases gave better results. The convergence and exactness were examined by comparing the calculated numerical solution and the known analytical solution. The error function was calculated and illustrated graphically for the reliability and accuracy of the proposed method. The proposed method examined several physical issues that lead to differential equations. HIGHLIGHTS Differential equations governing AC circuits are converted into the system of linear algebraic equations using Bernstein polynomial multiwavelets operational matrix of integration for easy computation The convergence and exactness were examined by comparing the calculated numerical solution and the known analytical solution The error function is calculated and shown graphically GRAPHICAL ABSTRACT

scholarly journals Numerical Approach Based on Two-Dimensional Fractional-Order Legendre Functions for Solving Fractional Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Qingxue Huang ◽  
Fuqiang Zhao ◽  
Jiaquan Xie ◽  
Lifeng Ma ◽  
Jianmei Wang ◽  
...  
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Fractional Differential Equations ◽  
Operational Matrix ◽  
Numerical Approach ◽  
Principal Characteristic ◽  
Legendre Functions ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Fractional Differential

In this paper, a robust, effective, and accurate numerical approach is proposed to obtain the numerical solution of fractional differential equations. The principal characteristic of the approach is the new orthogonal functions based on shifted Legendre polynomials to the fractional calculus. Also the fractional differential operational matrix is driven. Then the matrix with the Tau method is utilized to transform this problem into a system of linear algebraic equations. By solving the linear algebraic equations, the numerical solution is obtained. The approach is tested via some examples. It is shown that the FLF yields better results. Finally, error analysis shows that the algorithm is convergent.

scholarly journals Approximate Solutions of Differential Equations by Using the Bernstein Polynomials

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Y. Ordokhani ◽  
S. Davaei far
Keyword(s):  
Differential Equations ◽  
Bernstein Polynomial ◽  
Numerical Solutions ◽  
Bernstein Polynomials ◽  
Approximate Solutions ◽  
High Accuracy ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Bernstein Polynomial Basis

A numerical method for solving differential equations by approximating the solution in the Bernstein polynomial basis is proposed. At first, we demonstrate the relation between the Bernstein and Legendre polynomials. By using this relation, we derive the operational matrices of integration and product of the Bernstein polynomials. Then, we employ them for solving differential equations. The method converts the differential equation to a system of linear algebraic equations. Finally some examples and their numerical solutions are given; comparing the results with the numerical results obtained from the other methods, we show the high accuracy and efficiency of the proposed method.

scholarly journals Bivariate Chebyshev polynomials of the fifth kind for variable-order time-fractional partial integro-differential equations with weakly singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Khadijeh Sadri ◽  
Kamyar Hosseini ◽  
Dumitru Baleanu ◽  
Ali Ahmadian ◽  
Soheil Salahshour
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Chebyshev Polynomials ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Variable Order ◽  
Algebraic Equations ◽  
Weakly Singular Kernel ◽  
Weakly Singular ◽  
Linear Algebraic Equations

AbstractThe shifted Chebyshev polynomials of the fifth kind (SCPFK) and the collocation method are employed to achieve approximate solutions of a category of the functional equations, namely variable-order time-fractional weakly singular partial integro-differential equations (VTFWSPIDEs). A pseudo-operational matrix (POM) approach is developed for the numerical solution of the problem under study. The suggested method changes solving the VTFWSPIDE into the solution of a system of linear algebraic equations. Error bounds of the approximate solutions are obtained, and the application of the proposed scheme is examined on five problems. The results confirm the applicability and high accuracy of the method for the numerical solution of fractional singular partial integro-differential equations.

scholarly journals Two matrix methods for solution of nonlinear and linear Lane-Emden type equations with mixed condition by operational matrix

2015 ◽  
Vol 4 (3) ◽  
pp. 420 ◽  
Author(s):  
Behrooz Basirat ◽  
Mohammad Amin Shahdadi
Keyword(s):  
Bernstein Polynomials ◽  
Operational Matrix ◽  
Numerical Procedure ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Mixed Condition ◽  
Matrix Methods ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
The Given

<p>The aim of this article is to present an efficient numerical procedure for solving Lane-Emden type equations. We present two practical matrix method for solving Lane-Emden type equations with mixed conditions by Bernstein polynomials operational matrices (BPOMs) on interval [<em>a; b</em>]. This methods transforms Lane-Emden type equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equations. We also give some numerical examples to demonstrate the efficiency and validity of the operational matrices for solving Lane-Emden type equations (LEEs).</p>

scholarly journals Solving Some Differential Equations Arising in Electric Engineering Using Hermite Polynomials

2020 ◽  
Vol 12 (4) ◽  
pp. 517-523
Author(s):  
G. Singh ◽  
I. Singh
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Collocation Method ◽  
Hermite Polynomials ◽  
Electric Circuit ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Electric Engineering ◽  
Collocation Points ◽  
Linear Algebraic Equations

In this paper, a collocation method based on Hermite polynomials is presented for the numerical solution of the electric circuit equations arising in many branches of sciences and engineering. By using collocation points and Hermite polynomials, electric circuit equations are transformed into a system of linear algebraic equations with unknown Hermite coefficients. These unknown Hermite coefficients have been computed by solving such algebraic equations. To illustrate the accuracy of the proposed method some numerical examples are presented.

scholarly journals A New Scheme for Solving Multiorder Fractional Differential Equations Based on Müntz–Legendre Wavelets

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Haifa Bin Jebreen ◽  
Fairouz Tchier
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Operational Matrix ◽  
Pseudospectral Method ◽  
Algebraic Equations ◽  
Legendre Wavelets ◽  
Derivative Operator ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
Fractional Differential

In this study, we apply the pseudospectral method based on Müntz–Legendre wavelets to solve the multiorder fractional differential equations with Caputo fractional derivative. Using the operational matrix for the Caputo derivative operator and applying the Chebyshev and Legendre zeros, the problem is reduced to a system of linear algebraic equations. We illustrate the reliability, efficiency, and accuracy of the method by some numerical examples. We also compare the proposed method with others and show that the proposed method gives better results.

Application of Bernstein Polynomial Multiwavelets for Solving Non Linear Variational Problems with Moving and Fixed Boundaries

2020 ◽  
Vol 13 ◽  
Author(s):  
Sandeep Dixit ◽  
Shweta Pandey ◽  
S.R. Verma
Keyword(s):  
Variational Problem ◽  
Direct Method ◽  
Bernstein Polynomials ◽  
Operational Matrix ◽  
Variational Problems ◽  
Equation System ◽  
Primary Objective ◽  
Necessary Condition ◽  
Algebraic Equations ◽  
Operational Matrix Of Integration

Background: In this article, an efficient direct method has been proposed in order to solve physically significant variational problems. The proposed technique finds its basis in Bernstein polynomials multiwavelets (BPMWs). The mechanism of the proposed method is to transform the variational problem into an algebraic equation system through the use of BPMWs. Objective: Since the necessary condition of extremization consists of a differential equation that cannot be easily integrated in complex cases, an approximated numerical solution becomes a necessity. Our primary objective is to establish a wavelet based method for solving variational problems of physical interest. Besides being computationally more effective, the proposed approach yields relatively more accurate results than other comparable methods. The approach employs fewer basis elements, which in turn increases the simplicity, decreases the calculation time, and furnishes better results. Methods: An operational matrix of integration, which is based on the BPMWs, is presented. We substitute the approximated values of , unknown function and their derivative functions with BPMWs operational matrix of integration and BPMWs. On substituting the respective values in the given variational problem, it gets converted into a system of algebraic equations. The obtained system is further solved using the Lagrange multiplier. Results: The results obtained yield a greater degree of convergence as compared to other existing numerical methods. Numerical illustrations based on physical variational problems and the comparisons of outcomes with exact solutions demonstrate that the proposed method yields better efficiency, applicability, and accuracy. Conclusion: The proposed method gives better results than other comparable methods, even with the use of a fewer number of basis elements. The large order of matrices, such as 32, 64, and 512, obtained by using other available methods is far too high to achieve accuracy in results in comparison to the ones we obtain by using matrices of relatively lower orders, such as 7, 8 and 13, in the proposed method. This method can also be used for extremization functional occurring in electrical circuits and mechanical physical problems.

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