A Method of Moment Approach in Solving Boundary Value Problems

Misilmani, Hilal M. El; Kabalan, Karim Y.; Abou-Shahine, Mohamad Y.; Al‐Husseini, Mohammed

doi:10.4236/jemaa.2015.73007

Cited by 5 publications

(4 citation statements)

References 5 publications

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“…Partial Differential Equations (PDEs) are a useful tool for the mathematical expression of many natural phenomena and are useful in the solution of physical and other issues requiring functions of several variables. The transmission of heat/sound, fluid movement, turbulent flow, heat transfer analysis, elasticity, electrostatics, and electrodynamics are a few examples of these issues; see Ahsan et al [ 1 ], Wang and Guo [ 2 ], Arif et al [ 3 , 4 ], Adoghe et al [ 5 ], Nawaz et al [ 6 ], Animasaun et al [ 7 ], Devnath et al [ 8 ], Ahsan et al [ 9 ], Wang et al [ 10 ], Rufai et al [ 11 ], Nawaz and Arif [ 12 ], Ramakrishna et al [ 13 ], El Misilmani et al [ 14 ]). According to Quarteroni and Valli [ 15 ], numerical approximation techniques for partial differential equations (PDEs) constitute a cornerstone in diverse scientific and engineering disciplines.…”

Section: Background Informationmentioning

confidence: 99%

Unveiling the Power of Implicit Six-Point Block Scheme: Advancing numerical approximation of two-dimensional PDEs in physical systems

Olaoluwa Omole,

Olusheye Adeyefa,

Iyabo Apanpa

et al. 2024

PLoS ONE

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In the era of computational advancements, harnessing computer algorithms for approximating solutions to differential equations has become indispensable for its unparalleled productivity. The numerical approximation of partial differential equation (PDE) models holds crucial significance in modelling physical systems, driving the necessity for robust methodologies. In this article, we introduce the Implicit Six-Point Block Scheme (ISBS), employing a collocation approach for second-order numerical approximations of ordinary differential equations (ODEs) derived from one or two-dimensional physical systems. The methodology involves transforming the governing PDEs into a fully-fledged system of algebraic ordinary differential equations by employing ISBS to replace spatial derivatives while utilizing a central difference scheme for temporal or y-derivatives. In this report, the convergence properties of ISBS, aligning with the principles of multi-step methods, are rigorously analyzed. The numerical results obtained through ISBS demonstrate excellent agreement with theoretical solutions. Additionally, we compute absolute errors across various problem instances, showcasing the robustness and efficacy of ISBS in practical applications. Furthermore, we present a comprehensive comparative analysis with existing methodologies from recent literature, highlighting the superior performance of ISBS. Our findings are substantiated through illustrative tables and figures, underscoring the transformative potential of ISBS in advancing the numerical approximation of two-dimensional PDEs in physical systems.

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Section: Background Informationmentioning

confidence: 99%

Unveiling the Power of Implicit Six-Point Block Scheme: Advancing numerical approximation of two-dimensional PDEs in physical systems

Olaoluwa Omole,

Olusheye Adeyefa,

Iyabo Apanpa

et al. 2024

PLoS ONE

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“…Substituting in leads to the following equation:

\sum_{n = 1}^{N} ((), Z^{- 1} {γ_{n}}^{2} - {Z_{p}}^{- 1} γ_{n} - Y) a_{n} e^{- γ_{n} x} + ((), Z^{- 1} {γ_{n}}^{2} + {Z_{p}}^{- 1} γ_{n} - Y) b_{n} e^{γ_{n} x} = 0,

where

Z_{p}^{- 1} = \frac{italic∂ Z^{- 1} ((), x)}{∂x}

. To determine a n and b n coefficients, the Galerkin testing method is adopted. For this purpose, Equation is tested with F m ( x ) and B m ( x ) m = 1, …, N − 1, BFs leading to the calculation of the following inner products:

(〈〉,, Z^{- 1} e^{\pm γ_{n} x}, e^{\pm γ_{m} x}); (〈〉,, Z_{p}^{- 1} e^{\pm γ_{n} x}, e^{\pm γ_{m} x}); (〈〉,, {italicYe}^{\pm γ_{n} x}, e^{\pm γ_{m} x}),

where

(〈〉,, f ((), x), g ((), x)) = \int_{0}^{l} f ((), x) . g ((), x) italicdx

.…”

Section: Theorymentioning

confidence: 99%

Fast and low complexity frequency domain analysis of nonuniform substrate integrated waveguide‐based structures

Rabaani

Boulejfen

Kouki

et al. 2020

Int J RF Microw Comput Aided Eng

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A novel, low complexity approach for the analysis of nonuniform lossy substrate‐integrated waveguide transmission lines based on the method of moments is proposed. The approach uses frequency‐dependent basis functions derived from the structure's propagation characteristics. Two tapered structures are analyzed, fabricated, and measured to validate the proposed approach. The analytical results of the proposed approach for both structures are compared to those obtained by measurement and by three‐dimensional field simulation. Excellent agreement is observed between the three sets of results with simulation time savings on more than 98% and memory requirement reduction of more than 97%.

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“…Only Zhong-Xin Li, et al [10] presented a mathematical model for computing currents along a grounding grid buried in a horizontal multilayered earth model. This mathematical model was a hybrid of Galerkin's method of moment [11] and the conventional nodal analysis. An electromagnetic model to calculate the distribution of grounding grid potential subjected to lightning surges was presented in [12,13].…”

Section: Introductionmentioning

confidence: 99%

A proposed method for calculating the voltage and current distributions of grounded and isolated networks subjected to line‐to‐ground faults

Gouda,

Dein,

Mbasso

et al. 2024

IET Science Measure & Tech

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One crucial objective of this article is to present simplified and accurate approaches to study the current distribution and ground surface potential around the area surrounding an earthed and isolated system in case of a line‐to‐earth fault. The present study is done in the case of uniform and two‐layer soils. Suggested models for calculating the distribution of the earth's surface current density, step, and touch voltages in grounded and isolated systems are presented, discussed, and adapted. The contact and arc resistances of line to ground faulty conductors are considered. Rods and/or grounding grids usually do grounded systems. The impact of both step and touch potentials and current density are investigated in this article. The methods of the calculations are based on the electrical concepts, the charge simulation method, and the imaging procedure for the grounding system. The suggested method can be considered a novel and simple technique for calculating the current distributions of the ground surface during line to ground fault. The results agree with those obtained by others, with the advantage of the proposed algorithm for its ease of application and simplicity. 3‐D dimensions’ contours of the current density and the electric potential on the earth surface around the faulty point are presented in case of homogeneous and two layers’ soil. Finally, this article's novelty is to devise a method for calculating the step and touch potentials and current density around the point of contact of the conductor carrying the electric voltages in the event of faults occurring in the electrical network, whether the network is grounded or isolated. Such faults of electrical networks may cause human hazards.

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A Method of Moment Approach in Solving Boundary Value Problems

Cited by 5 publications

References 5 publications

Unveiling the Power of Implicit Six-Point Block Scheme: Advancing numerical approximation of two-dimensional PDEs in physical systems

Unveiling the Power of Implicit Six-Point Block Scheme: Advancing numerical approximation of two-dimensional PDEs in physical systems

Fast and low complexity frequency domain analysis of nonuniform substrate integrated waveguide‐based structures

A proposed method for calculating the voltage and current distributions of grounded and isolated networks subjected to line‐to‐ground faults

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