New development of homotopy analysis method for non-linear integro-differential equations with initial value problems

Eshkuvatov, Zainidin K.

doi:10.23939/mmc2022.04.842

Cited by 3 publications

(1 citation statement)

References 22 publications

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“…Then, this has been extended by Wazwaz [3] [4] to Volterra integral equation and to boundary value problems for higher-order integro-differential equations. There are many other methods developed by different researchers for linear and nonlinear IDEs with initial, boundary or mixed conditions, for instance homotopy analysis method (HAM) developed by Liao [5] [6] [7], modified HAM [8], q-HAM [9], new development of HAM [10], homotopy perturbation method (HPM) developed by Ji-Huan He [11] [12], HPM for nonlinear differential-difference equations [13], HPM for nth-Order Integro-Differential Equations [14], collocation method [15], new boundary element method [16], Linear Programming Method [17], Laplace Decomposition Algorithm [18], polynomial approximations [19], Wavelet Galerkin method [20], and so on.…”

Section: Introductionmentioning

confidence: 99%

Application of HAM for Nonlinear Integro-Differential Equations of Order Two

Eshkuvatov¹,

Khayrullaev²,

Nurillaev³

et al. 2023

JAMP

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In this work, we consider the second order nonlinear integro-differential Equation (IDEs) of the Volterra-Fredholm type. One of the popular methods for solving Volterra or Fredholm type IDEs is the method of quadrature while the problem of consideration is a linear problem. If IDEs are nonlinear or integral kernel is complicated, then quadrature rule is not most suitable; therefore, other types of methods are needed to develop. One of the suitable and effective method is homotopy analysis method (HAM) developed by Liao in 1992. To apply HAM, we firstly reduced the IDEs into nonlinear integral Equation (IEs) of Volterra-Fredholm type; then the standard HAM was applied. Gauss-Legendre quadrature formula was used for kernel integrations. Obtained system of algebraic equations was solved numerically. Moreover, numerical examples demonstrate the high accuracy of the proposed method. Comparisons with other methods are also provided. The results show that the proposed method is simple, effective and dominated other methods.

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Section: Introductionmentioning

confidence: 99%

Application of HAM for Nonlinear Integro-Differential Equations of Order Two

Eshkuvatov¹,

Khayrullaev²,

Nurillaev³

et al. 2023

JAMP

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Collisional breakage population balance equation: An analytical approach

Hussain,

Bariwal,

Kumar

2025

Journal of Mathematical Analysis and Applications

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An analytic approach for nonlinear collisional fragmentation model arising in bubble column

Hussain,

Arora,

Kumar

2024

Physics of Fluids

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The phenomenon of coagulation and breakage of particles plays a pivotal role in diverse fields. It aids in tracking the development of aerosols and granules in the pharmaceutical sector, coagulation or breakage of droplets in chemical engineering, understanding blood clotting mechanisms in biology, and facilitating cheese production through the action of enzymes within the dairy industry. A significant portion of research in this direction concentrates on coagulation or linear breakage processes. In the case of linear case, bubble particles break down due to inherent stresses or specific conditions of the breakage event. However, in many practical situations, particle division is primarily due to forces exerted during collisions between particles, necessitating an approach that accounts for nonlinear collisional breakage. Despite its critical role in a wide array of engineering and physical operations, the study of this nonlinear fragmentation phenomenon has not been extensively pursued. This article introduces an innovative semi-analytical method that leverages the beyond linear use of equation superposition function to address the nonlinear integro-partial differential model of collisional breakage population balance. This approach is versatile, allowing for the resolution of both linear/nonlinear equations while sidestepping the complexities associated with discretization of domain. To assess the precision of this method, we conduct a thorough convergence analysis. This process utilizes the principle of contractive mapping in the Banach space, a globally recognized strategy for verifying convergence. We explore a variety of kernel parameters associated with collisional kernels, alongside breakage and initial distribution functions, to derive novel iterative solutions. Comparing our findings with those obtained through the finite volume method regarding number density functions and their integral moments, we demonstrate the reliability and accuracy of our approach. The consistency and correctness of our method are further validated by depicting the errors between the exact and approximated solutions in graphical and tabular formats.

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New development of homotopy analysis method for non-linear integro-differential equations with initial value problems

Cited by 3 publications

References 22 publications

Application of HAM for Nonlinear Integro-Differential Equations of Order Two

Application of HAM for Nonlinear Integro-Differential Equations of Order Two

Collisional breakage population balance equation: An analytical approach

An analytic approach for nonlinear collisional fragmentation model arising in bubble column

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