A second order numerical method for singularly perturbed Volterra integro-differential equations with delay

Erdoğan, Fevzi

doi:10.2478/ijmce-2024-0007

Cited by 7 publications

(1 citation statement)

References 16 publications

(25 reference statements)

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“…Fractional differential equations have seen a lot of use in physics and engineering over the last few decades. Thousands of efforts have been put into developing reliable and consistent numerical and analytical methodologies to solve these fractional equations over the past decade or more [15][16][17][18][19][20][21][22][23][24][25]. To find precise and approximative analytical solutions, certain potent techniques have been developed, few of them are Yang-Laplace decomposition approach [26], Sumudu decomposition in local fractional [27], the method of variational iteration [28,29], the method of homotopy analysis [30,31], and the method of fractional difference [32].…”

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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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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An approximate solution for stochastic Fitzhugh–Nagumo partial differential equations arising in neurobiology models

Uma,

Jafari,

Raja Balachandar

et al. 2024

Math Methods in App Sciences

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In this paper, approximate solutions for stochastic Fitzhugh–Nagumo partial differential equations are obtained using two‐dimensional shifted Legendre polynomial (2DSLP) approximation. The problem's suitability and solvability are confirmed. The convergence analysis for the proposed methodology and the error analysis in the norm are carried out. Using Maple software, an algorithm is created and implemented to arrive at the numerical solution. The solution thus obtained is compared with the exact solution and the solution obtained using the explicit order RK1.5 method.

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Numerical solution of fractional PDEs through wavelet approach

Yan,

Kumbinarasaiah,

Manohara

et al. 2024

Z. Angew. Math. Phys.

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To solve fractional partial differential equations (FPDEs) under various physical conditions, this study developed a novel method known as the Hermite wavelet method employing the functional integration matrix. The method that has been suggested is based on the Hermite wavelet collocation process. To determine the solution of the fractional differential equations, the Caputo fractional derivative operator of order $$\alpha \in (0,1]$$ α ∈ ( 0 , 1 ] is used. With the use of appropriate grid points, this method converts FPDEs into a system of nonlinear algebraic equations. We achieve a solution by solving these nonlinear algebraic equations by the Newton–Raphson method. Tables and graphs show that the suggested method produces superior results. We provide various illustrative examples to establish the effectiveness of the suggested concept, and the outcomes support the applicability of the suggested strategy. Obtained results are numerically expressed in terms of absolute errors. Finally, convergence analyses are discussed as some theorem with proof.

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A second order numerical method for singularly perturbed Volterra integro-differential equations with delay

Cited by 7 publications

References 16 publications

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

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

An approximate solution for stochastic Fitzhugh–Nagumo partial differential equations arising in neurobiology models

Numerical solution of fractional PDEs through wavelet approach

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