An optimal homotopy continuation method: Convergence and visual analysis

Gdawiec, Krzysztof; Argyros, Ioannis K.; Qureshi, Sania; Soomro, Amanullah

doi:10.1016/j.jocs.2023.102166

Cited by 8 publications

(8 citation statements)

References 30 publications

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“…The proposed method (7) has the sixth order of convergence and is denoted as Proposed6. The leading term ϕ of the error equation is -3f 5 , the order of convergence (p) is 6, the efficiency…”

Section: Proposed Methodsmentioning

confidence: 99%

“…Multiplying ( 9) and ( 10) and putting the result in the first step of the proposed method given in (7), we get…”

Section: Order Of Convergencementioning

confidence: 99%

“…This section has addressed the topic of numerical simulations of scalar non-linear equations with the sixth-order numerical method proposed in (7). When comparing, we will consider parameters such as the number of iterations (k), the computational cost (COC = kd), the absolute error (ε) at the last iteration, the absolute functional value (χ) at the last iteration, the approximate root, and the CPU time in seconds to run the entire code for producing the required simulation results.…”

Section: Numerical Simulationsmentioning

confidence: 99%

“…Being higher-order accurate and cost-efficient is crucial for a numerical approach, achieved by minimizing the number of evaluations and iterations required. The study of efficient techniques for solving non-linear equations is carried out in many studies; see for example some of the recently published research papers sania.qureshi@faculty.muet.edu.pk VFAST Transactions on Mathematics [7,15,[19][20][21]. Nonlinear equations frequently arise through the application of mathematical methods, and finding their solution is a crucial objective in numerical analysis.…”

Section: Introductionmentioning

confidence: 99%

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An Improved Blended Numerical Root-Solver for Nonlinear Equations

Asad Ali Chandio,

Asif Ali Shaikh,

Qureshi

et al. 2024

VFAST trans. math.

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This study presents a novel three-step iterative approach for solving nonlinear equations inthe domains of science and engineering. It represents a notable change from traditional methodslike Halley by eliminating the need for second derivatives. The suggested method exhibits asixth order of convergence and only requires five function evaluations, showcasing its efficiencywith an index of roughly 1.430969. The suggested method effectively solves nonlinear problemsinvolving equations with algebraic and transcendental terms. Comparative analysis againstexisting root-solving algorithms demonstrates their superior performance. The results not onlyconfirm the strength and effectiveness of the three-step iterative approach but also highlight itspotential for wide-ranging use in many scientific and technical situations.

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Section: Proposed Methodsmentioning

confidence: 99%

“…Multiplying ( 9) and ( 10) and putting the result in the first step of the proposed method given in (7), we get…”

Section: Order Of Convergencementioning

confidence: 99%

Section: Numerical Simulationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An Improved Blended Numerical Root-Solver for Nonlinear Equations

Asad Ali Chandio,

Asif Ali Shaikh,

Qureshi

et al. 2024

VFAST trans. math.

Self Cite

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“…A growing area of mathematics called fractional calculus (FC) has diverse applications in all connected sectors of science and problems of engineering. Some of the findings were published in books or related review articles [4][5][6][7][8][9][10][11]. It is interesting to note that these derivatives and integrals are not just mathematical oddities, these models are increasingly being employed in fields such as engineering fields, electrical circuits, digital control systems, fluid movement, and many more [12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Unveiling new insights: taming complex local fractional Burger equations with the local fractional Elzaki transform decomposition method

Alhamzi,

Prasad,

Alkahtani

et al. 2024

Front. Appl. Math. Stat.

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This study aims to address the difficulties in solving coupled generalized non-linear Burger equations using local fractional calculus as a framework. The methodology used in this work, particularly in the area of local fractional calculus, combines the Elzaki transform with the Adomian decomposition method. This combination has proven to be a highly effective strategy for addressing non-linear partial differential equations within the local fractional context, which finds numerous practical applications. The proposed method offers a systematic and easily understandable procedure for tackling both linear and non-linear partial differential equations (PDEs). It provides an easy-to-follow path to solve these problems. We offer a real-world example that exhibits the method's successful use in resolving issues to corroborate its efficacy. The obtained solution is visually represented to illustrate the practical utility of this approach.2010 Mathematics Subject Classification34A34, 65M06, 26A33.

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