Research Journal of Engineering and Technology

Gurjinder Singh ¹, V. Kanwar ¹, Saurabh Bhatia ¹

Affiliations
1 University Institute of Engineering and Technology, Panjab University, Chandigarh 160 014, IN

Abstract

In this paper, we propose a class of the backward Euler's method for the numerical solution of initial value problems of ordinary differential equations. The proposed class is constructed by considering a suitable interpolating function. The accuracy and stability of the proposed class are considered. A comparison of numerical results obtained by some of the proposed methods and the existing classical backward Euler's method is given which demonstrate that for the problems tested here the proposed methods outperform the existing classical method.

Keywords

Ordinary Differential Equations, Initial Value Problems, Stability, Interpolation.

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Mona Narang ¹, Saurabh Bhatia ², V. Kanwar ²

Affiliations
1 D.A.V. College, Chandigarh, 160010, IN
2 University Institute of Engineering and Technology, Panjab University, Chandigarh 160014, IN

Abstract

In this paper, we present a one-parameter family of fifth-order methods by extending Nedzhibov's third-order methods for solving systems of nonlinear equations. For a particular value of parameter, the new fifth-order method is more efficient as compared to the existing methods as its computational cost is less. Further, it requires two function evaluations, two first order Fr´echet derivatives and one matrix inversion per iteration. Numerical examples confirm that the proposed method is highly efficient and useful in solving systems of nonlinear equations.

Keywords

System of Nonlinear Equations, Order of Convergence, Newton’s Method, Higher Order Methods, Computational Efficiency.

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Munish Kansal ¹, V. Kanwar ¹, Saurabh Bhatia ¹

Affiliations
1 University Institute of Engineering and Technology, Panjab University, Chandigarh 160014, IN

Abstract

This paper presents an improvement of the existing eighth-order derivative involved method [14] into derivative-free scheme, holding the order of convergence of the original method. Each member of the family requires only four function evaluations per iteration to achieve the eighth-order of convergence, while they are totally free from derivative evaluation. Hence, they agree with the optimality conjecture of Kung-Traub for providing multipoint iterations without memory. The proposed methods are compared with their closest competitors in a series of numerical experiments. Numerical experiments show that such derivativefree, high order schemes offer significant advantages over the derivative involved methods.

Keywords

Nonlinear Equations, Steffensen’s Method, King’s Method, Ostrowski’s Method, Efiiciency Index, Optimal Order of Convergence.

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