Successive iteration technique for singular nonlinear system with four-point boundary conditions

Barnwal, Amit K.; Pathak, Priti

doi:10.1007/s12190-019-01285-8

Cited by 3 publications

(8 citation statements)

References 25 publications

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“…Figures 9, 10, 11, 12 show the nature, as well as accuracy of the computed solution. We have compared our result with a recent work by Barnwal and Pathak (2019). In our case, L 1 error is negligible while in case of Barnwal and Pathak (2019) it is of the order 10 À6 or 10 À7 .…”

Section: Remarkmentioning

confidence: 54%

“…In this section, we discuss two test examples based on a system of nonlinear singular fourpoint boundary value problems (Barnwal and Pathak, 2019). 5.3.1 Example 5.…”

Section: Haar Wavelets Collocation Methodsmentioning

confidence: 99%

“…In this section, we develop the Haar wavelets collocation method for solving a system of nonlinear singular four-point BVPs and solve two test problems, considered by Barnwal and Pathak (2019). We also compare our results with the results of Barnwal and Pathak (2019).…”

Section: System Of Nonlinear Singular Four-point Boundary Value Problemsmentioning

confidence: 99%

“…Very recently, Barnwal and Pathak (2019) considered the following singular nonlinear four-point BVP:…”

Section: Introductionmentioning

confidence: 99%

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Haar wavelets collocation method for a system of nonlinear singular differential equations

Verma

Kumar

Tiwari

2020

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Purpose The purpose of this paper is to propose an efficient computational technique, which uses Haar wavelets collocation approach coupled with the Newton-Raphson method and solves the following class of system of Lane–Emden equations: −(tk1y′(t))′=t−ω1f1(t,y(t),z(t)), −(tk2z′(t))′=t−ω2f2(t,y(t),z(t)),where t > 0, subject to the following initial values, boundary values and four-point boundary values: y(0)=γ1, y′(0)=0, z(0)=γ2, z′(0)=0, y′(0)=0, y(1)=δ1, z′(0)=0, z(1)=δ2, y(0)=0, y(1)=n1z(v1), z(0)=0, z(1)=n2y(v2),where n1,n2,v1,v2∈(0,1) and k1≥0, k2≥0, ω1<1, ω2<1, γ1, γ2, δ1, δ2 are real constants. Design/methodology/approach To deal with singularity, Haar wavelets are used, and to deal with the nonlinear system of equations that arise during computation, the Newton-Raphson method is used. The convergence of these methods is also established and the results are compared with existing techniques. Findings The authors propose three methods based on uniform Haar wavelets approximation coupled with the Newton-Raphson method. The authors obtain quadratic convergence for the Haar wavelets collocation method. Test problems are solved to validate various computational aspects of the Haar wavelets approach. The authors observe that with only a few spatial divisions the authors can obtain highly accurate solutions for both initial value problems and boundary value problems. Originality/value The results presented in this paper do not exist in the literature. The system of nonlinear singular differential equations is not easy to handle as they are singular, as well as nonlinear. To the best of the knowledge, these are the first results for a system of nonlinear singular differential equations, by using the Haar wavelets collocation approach coupled with the Newton-Raphson method. The results developed in this paper can be used to solve problems arising in different branches of science and engineering.

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Section: Remarkmentioning

confidence: 54%

“…In this section, we discuss two test examples based on a system of nonlinear singular fourpoint boundary value problems (Barnwal and Pathak, 2019). 5.3.1 Example 5.…”

Section: Haar Wavelets Collocation Methodsmentioning

confidence: 99%

Section: System Of Nonlinear Singular Four-point Boundary Value Problemsmentioning

confidence: 99%

“…Very recently, Barnwal and Pathak (2019) considered the following singular nonlinear four-point BVP:…”

Section: Introductionmentioning

confidence: 99%

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Haar wavelets collocation method for a system of nonlinear singular differential equations

Verma

Kumar

Tiwari

2020

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“…1.0716E-38 3.0023E-39 Successive iteration (Barnwal and Pathak, 2019) 4.9806E-07 7.5635E-07 HWCM(J 5 3) (Verma et al, 2020) 1.1102E-16 5.5511E-17 HWCM(J 5 4) (Verma et al, 2020) 1. BFMCM for solving singular nonlinear system BFMCM is compared with the exact solution.…”

Section: Methodsmentioning

confidence: 99%

Biorthogonal flatlet multiwavelet collocation method for solving the singular nonlinear system with initial and boundary conditions

Mohseni,

Rostamy

2023

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PurposeThe numerical methods are of great importance for approximating the solutions of a system of nonlinear singular ordinary differential equations. In this paper, the authors present the biorthogonal flatlet multiwavelet collocation method (BFMCM) as a numerical scheme for a class of system of Lane–Emden equations with initial or boundary or four-point boundary conditions.Design/methodology/approachThe approach is involved in combining the biorthogonal flatlet multiwavelet (BFM) with the collocation method. The authors investigate the properties and procedure of the BFMCM for first time on this class of equations. By using the BFM and the collocation points, the method is constructed and it transforms the nonlinear differential equations problem into a system of nonlinear algebraic equations. The unknown coefficients of the assuming solution are determined by solving the obtained system. Additionally, convergence analysis and numerical stability of the suggested method are provided.FindingsAccording to the attained results, the proposed BFMCM has more accurate results in comparison with results of other methods. The maximum absolute errors are calculated by using the BFMCM for comparison purposes provided.Originality/valueThe key desirable properties of BFMCM are its efficiency, simple applicability and minimizes errors. Therefore, the proposed method can be used to solve nonlinear problems or problems with singular points.

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New Approach Based on Collocation and Shifted Chebyshev Polynomials for a Class of Three-Point Singular BVPS

Sriwastav¹,

Barnwal²,

Ramos³

et al. 2023

jaac

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Successive iteration technique for singular nonlinear system with four-point boundary conditions

Cited by 3 publications

References 25 publications

Haar wavelets collocation method for a system of nonlinear singular differential equations

Haar wavelets collocation method for a system of nonlinear singular differential equations

Biorthogonal flatlet multiwavelet collocation method for solving the singular nonlinear system with initial and boundary conditions

New Approach Based on Collocation and Shifted Chebyshev Polynomials for a Class of Three-Point Singular BVPS

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