Numerical solution of Bratu’s and related problems using a third derivative hybrid block method

Rufai, Mufutau Ajani; Ramos, Higinio

doi:10.1007/s40314-020-01372-8

Cited by 16 publications

(7 citation statements)

References 19 publications

(18 reference statements)

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“…Remark 3. By the definitions of sequences given in Equations (20)- (22), and as a consequence of Lemma 1, it follows that t (k) ≤ s (k) ≤ u (k) ≤ t (k+1) ≤ r0 for each k = 0, 1, 2, . .…”

Section: Semilocal Convergence Analysismentioning

confidence: 88%

“…An important observation is that the technique, which is utilized in the analysis, can also be applied in general to the other methods in order to extend their applicability in a similar way. Furthermore, as a future work, the applicability of given iterative methods can be investigated for the solution of problems considered in [21][22][23].…”

Section: Discussionmentioning

confidence: 99%

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A Unified Local-Semilocal Convergence Analysis of Efficient Higher Order Iterative Methods in Banach Spaces

2022

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To deal with the estimation of the locally unique solutions of nonlinear systems in Banach spaces, the local as well as semilocal convergence analysis is established for two higher order iterative methods. The given methods do not involve the computation of derivatives of an order higher than one. However, the convergence analysis was carried out in earlier studies by using the assumptions on the higher order derivatives as well. Such types of assumptions limit the applicability of techniques. In this regard, the convergence analysis is developed in the present study by imposing the conditions on first order derivatives only. The central idea for the local analysis is to estimate the bounds on convergence domain as well as the error approximations of the iterates along with the formulation of sufficient conditions for the uniqueness of the solution. Based on the choice of initial estimate in the given domain, the semilocal analysis is established, which ensures the convergence of iterates to a unique solution in that domain. Further, some applied problems are tested to certify the theoretical deductions.

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Section: Semilocal Convergence Analysismentioning

confidence: 88%

Section: Discussionmentioning

confidence: 99%

A Unified Local-Semilocal Convergence Analysis of Efficient Higher Order Iterative Methods in Banach Spaces

2022

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“…In this technique, the integral form of Bratu’s problem is converted into a system of non-linear equations. Rufai and Ramos (2020) used the third derivative hybrid block method to solve the problem (1). This approach deals with the three intermediate points that are carefully selected over the optimization of the local truncation errors.…”

Section: Introductionmentioning

confidence: 99%

Numerical simulation of Bratu’s problem using a new form of the Adomian decomposition technique

Umesh

2023

HFF

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Purpose This paper aims to discuss a new form of the Adomian decomposition technique for the numerical treatment of Bratu’s type one-dimensional boundary value problems (BVPs). Moreover, the author also addresses convergence and error analysis for the completeness of the proposed technique. Design/methodology/approach First, the author discusses the standard Adomian decomposition method and an algorithm based on Duan’s corollary and Rach’s rule for the fast calculation of the Adomian polynomials. Then, a new form of the Adomian decomposition technique is present for the numerical simulation of Bratu’s BVPs. Findings The reliability and validity of the proposed technique are examined by calculating the absolute errors of Bratu’s problem for some different values of Bratu parameter λ. Numerical simulation demonstrates that the proposed technique yields higher accuracy than the Bessel collocation and other known methods. Originality/value Unlike the other methods, the proposed technique does not need linearization, discretization or perturbation to handle the non-linear problems. So, the results obtained by the present technique are more physically realistic.

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“…Some of these approximation techniques comprise embedded Runge-Kutta type methods, general linear methods, Numerov-type methods, spline methods, finite difference, Nyström methods, space-time numerical methods, block methods, and various collocation methods. These methods are extensively discussed in various research papers and books, such as [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

A New Hybrid Block Method for Solving First-Order Differential System Models in Applied Sciences and Engineering

Rufai,

Carpentieri,

Ramos

2023

Fractal Fract

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This paper presents a new hybrid block method formulated in variable stepsize mode to solve some first-order initial value problems of ODEs and time-dependent partial differential equations in applied sciences and engineering. The proposed method is implemented considering an adaptive stepsize strategy to maintain the estimated error in each step within a specified tolerance. In order to evaluate the performance and usefulness of the proposed technique in real-world applications, several differential problems from applied sciences and engineering, such as the SIR model, Jacobi elliptic function problem, and chemical reactions problems, are solved numerically. The results of numerical simulations in this work demonstrate that the proposed method is more efficient than other existing numerical methods used for comparisons.

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Numerical solution of Bratu’s and related problems using a third derivative hybrid block method

Cited by 16 publications

References 19 publications

A Unified Local-Semilocal Convergence Analysis of Efficient Higher Order Iterative Methods in Banach Spaces

A Unified Local-Semilocal Convergence Analysis of Efficient Higher Order Iterative Methods in Banach Spaces

Numerical simulation of Bratu’s problem using a new form of the Adomian decomposition technique

A New Hybrid Block Method for Solving First-Order Differential System Models in Applied Sciences and Engineering

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