New Modification on Heun’s Method Based on Contraharmonic Mean for Solving Initial Value Problems with High Efficiency

Workie, Abushet Hayalu

doi:10.1155/2020/6650855

Cited by 3 publications

(4 citation statements)

References 5 publications

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“…An initial value problem (IVP) is a differential equation with an initial condition that specifies the value of an unknown function at a specific point in the domain. Differential equations are commonly used in physics, chemistry, biology, and economics in science and engineering to solve a variety of physical problems [1,3].…”

Section: Introductionmentioning

confidence: 99%

“…Numerical methods have been improved on occasion to increase performance in accordance with our requirements. By assuming the tangent slope as an average of the arithmetic mean and contra-harmonic mean, [1] proposed to improve the Improved Euler's (or Heun's) method. [10] proposed a hybrid numerical method that combines the Modified Euler method, the Improved Euler's method, and the 2 nd -order contra harmonic mean method to solve initial value problems.…”

Section: Introductionmentioning

confidence: 99%

“…Solve the initial value problem (IVP) y ′ = x 3 e −2x − 2y ; yð0Þ = 1 which is reported in[1] using the MEM, IMEM, and the proposed method with step sizes h = 0:1 and h = 0:05 and compare the errors. Solve the initial value problem (IVP) y′ = x + y ; yð0Þ = 1 which is reported in[2].…”

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confidence: 99%

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Small Modification on Modified Euler Method for Solving Initial Value Problems

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Small Modification on Modified Euler Method for Solving Initial Value Problems

Workie

2021

Abstract and Applied Analysis

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“…Heun's method, sometimes called the explicit trapezoidal rule, is probably the most common second-order Runge-Kutta (RK) scheme for ODEs and ODE systems [39], so it is straightforward to use it as a component of the method of lines. It starts with a predictor step, which is an explicit Euler stage.…”

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confidence: 99%

Comparison of Old and New Stable Explicit Methods for Heat Conduction, Convection, and Radiation in an Insulated Wall with Thermal Bridging

2022

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Using efficient methods to calculate heat transfer in building components is an important issue. In the current work, 14 numerical methods are examined to solve the heat transfer problem inside building walls. Not only heat conduction but convection and radiation are considered as well, in addition to heat generation. Five of the used methods are recently invented explicit algorithms, which are unconditionally stable for conduction problems. First, the algorithms are verified in a 1D case by comparing the results of the methods to an analytical solution. Then they are tested on real-life cases in the case of surface area (made of brick) and cross-sectional area (two-layer brick and insulator) walls with and without thermal bridging. Equidistant and non-equidistant grids are used as well. The goal was to determine how the errors depend on the properties of the materials, the mesh type, and the time step size. The results show that the best algorithms are typically the leapfrog-hopscotch and the modified Dufort–Frankel and odd–even hopscotch algorithms since they are quite accurate for larger time step sizes, even for 100 s as well.

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A Comparative Analysis of Numerical Methods for Solving the Leaky Integrate and Fire Neuron Model

et al. 2023

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The human nervous system is one of the most complex systems of the human body. Understanding its behavior is crucial in drug discovery and developing medical devices. One approach to understanding such a system is to model its most basic unit, neurons. The leaky integrate and fire (LIF) method models the neurons’ response to a stimulus. Given the fact that the model’s equation is a linear ordinary differential equation, the purpose of this research is to compare which numerical analysis method gives the best results for the simplified version of this model. Adams predictor and corrector (AB4-AM4) and Heun’s methods were then used to solve the equation. In addition, this study further researches the effects of different current input models on the LIF’s voltage output. In terms of the computational time, Heun’s method was 0.01191 s on average which is much less than that of the AB-AM4 method (0.057138) for a constant DC input. As for the root mean square error, the AB-AM4 method had a much lower value (0.0061) compared to that of Heun’s method (0.3272) for the same constant input. Therefore, our results show that Heun’s method is best suited for the simplified LIF model since it had the lowest computation time of 36 ms, was stable over a larger range, and had an accuracy of 72% for the varying sinusoidal current input model.

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New Modification on Heun’s Method Based on Contraharmonic Mean for Solving Initial Value Problems with High Efficiency

Cited by 3 publications

References 5 publications

Small Modification on Modified Euler Method for Solving Initial Value Problems

Small Modification on Modified Euler Method for Solving Initial Value Problems

Comparison of Old and New Stable Explicit Methods for Heat Conduction, Convection, and Radiation in an Insulated Wall with Thermal Bridging

A Comparative Analysis of Numerical Methods for Solving the Leaky Integrate and Fire Neuron Model

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