Implicit Methods for Numerical Solution of Singular Initial Value Problems

Rahaman, H. M. Saidur; Hasan, MJ; Ali, Ayub; Alam, Md. Shamsul

doi:10.2478/amns.2020.2.00001

Cited by 46 publications

(35 citation statements)

References 17 publications

(33 reference statements)

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“…In the very recent time, by choosing 1 (16 6) w = + , 3 4 w = and using fourth order Lagrange's interpolation formula, Rahaman et al [17] has been derive an integral formula as…”

Section: Recent Existing Formulaementioning

confidence: 99%

“…where 0, 1, 2,.. i = . Also utilizing the fifth order Lagrange's interpolation formula, Rahaman et al [17] derived the fourth order integral formula as . Based on formula (9), Rahaman et al [17] derived the fourth order implicit formula as…”

Section: W F X Ph Y P Y Y W F X Qh Y Q Y Y W F X H Ymentioning

confidence: 99%

“…In [17] (formulae (Eq. (7), ( 9) and ( 11))), we observe that the existing higher order formula is giving better result than the lower order because the starting point of the higher order formula is nearer to the singular point than lower order formula.…”

Section: Derivation Of the Proposed Formulamentioning

confidence: 99%

“…(17) reduces to Simpson's 1/3 rule with half spacing. Based on formula (17), a new third order implicit method has been proposed for solving initial value problem having initial singularity at 0…”

Section: Derivation Of the Proposed Formulamentioning

confidence: 99%

“…Recently, Hasan et al [12][13][14][15][16] derived second and third order implicit formulae based on the integral formulae for solving initial value problems having an initial singularity. Most Recently, Rahaman et al [17] derived the third, fourth and fifth order implicit methods for solving initial value problems having initial singular point based on integral formulae. These implicit formulae give more accurate results than those obtained by the implicit Euler and implicit Runge-Kutta (RK2, RK3 and RK4) methods for first order singular initial value problems.…”

Section: Introductionmentioning

confidence: 99%

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Approximate Numerical Solution of Singular Integrals and Singular Initial Value Problems

Rahaman¹,

Hasan²,

Ali³

et al. 2020

AJAM

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Numerical integration is one of the important branch of mathematics. Singular integrals arises in different applications in applied and engineering mathematics. The evaluation of singular integrals is one of the most challenging jobs. Earlier different techniques were developed for evaluating such integrals, but these were not straightforward. Recently various order straightforward formulae have been developed for evaluating such integrals but; all these integral formulae depend on Romberg technique for more accurate results. Based on these integral formulae, different order (up to fifth) implicit methods have been developed for solving singular initial value problems. These implicit methods give better results than those obtained by implicit Runge-Kutta methods but; the derivation of such higher order formulae are not so easy. In this article, a new third order straightforward integral formula has been proposed for evaluating singular integrals. This new formula is able to evaluate more efficiently than others existing formulae, moreover it has the independent ability to calculate very near accurate result to the exact value of the numerical integrals. Based on this new integral formula a new third order implicit method has been proposed for solving singular initial value problems. The new method provides significantly better results than other existing methods.

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“…In the very recent time, by choosing 1 (16 6) w = + , 3 4 w = and using fourth order Lagrange's interpolation formula, Rahaman et al [17] has been derive an integral formula as…”

Section: Recent Existing Formulaementioning

confidence: 99%

Section: W F X Ph Y P Y Y W F X Qh Y Q Y Y W F X H Ymentioning

confidence: 99%

Section: Derivation Of the Proposed Formulamentioning

confidence: 99%

Section: Derivation Of the Proposed Formulamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Approximate Numerical Solution of Singular Integrals and Singular Initial Value Problems

Rahaman¹,

Hasan²,

Ali³

et al. 2020

AJAM

Self Cite

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Extraction of the gravitational potential and high‐frequency wave perturbation properties of nonlinear (3 + 1)‐dimensional Vakhnenko–Parkes equation via novel approach

Baskonus

Cattani

et al. 2022

Math Methods in App Sciences

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The (3 + 1)‐dimensional Vakhnenko–Parkes mathematical model has a wide range of applications in science and engineering. In this paper, the model studied is investigated and analyzed by using two effective schemes such as sine‐Gordon expansion method and its newly developed version, rational SGEM. Moreover, many novel properties of model studied are extracted in detail. Furthermore, their wave distributions properties and graphs are also plotted under the strain conditions. Interactions of the gravitational potential and high‐frequency wave perturbation properties are also reported in a detailed manner.

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A novel numerical approach for the third order Emden–Fowler type equations

Sakar,

Saldır,

Aydın

et al. 2024

Z Angew Math Mech

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This article aims to achieve robust numerical results by applying the Chebyshev reproducing kernel method without homogenizing the initial‐boundary conditions of the Emden–Fowler (E‐F) equation, thereby introducing a new perspective to the literature. A novel numerical approach is presented for solving the initial‐boundary value problem of third‐order E‐F equations using Chebyshev reproducing kernel theory. Unlike previous applications, which were confined to homogeneous initial‐boundary value problems or required homogenization, the proposed method is effective for both homogeneous and nonhomogeneous cases. To handle the initial‐boundary conditions of the E‐F equations, additional basis functions are introduced rather than imposing conditions on the reproducing kernel Hilbert space. The method's effectiveness is demonstrated through five examples, which validate the theoretical analysis. Overall, the results emphasize the method's efficiency.

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Implicit Methods for Numerical Solution of Singular Initial Value Problems

Cited by 46 publications

References 17 publications

Approximate Numerical Solution of Singular Integrals and Singular Initial Value Problems

Approximate Numerical Solution of Singular Integrals and Singular Initial Value Problems

Extraction of the gravitational potential and high‐frequency wave perturbation properties of nonlinear (3 + 1)‐dimensional Vakhnenko–Parkes equation via novel approach

A novel numerical approach for the third order Emden–Fowler type equations

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