Numerical Solution of Second Order Initial Value Problems of Bratu-type Equations using Sixth Order Runge-Kutta Seven Stages Method

Fenta, Hibist Bazezew; Derese, Getachew Adamu

doi:10.12962/j24775401.v5i1.3806

Cited by 5 publications

(11 citation statements)

References 1 publication

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“…To obtain the values of 𝑎 𝑗 , (𝑗 = 0,1, … ,8), Equation ( 4) is solved using Gaussian elimination and then substituted back into Equation (2) to get the approximate solution. The derivation of the new method can be described by the following procedures.…”

Section: Development Of the Methodsmentioning

confidence: 99%

“…The new three-step hybrid block method ( 6) is said to be consistent if its order is greater than or equal to one (1). Therefore, this new method is consistent since its order is greater than 1.…”

Section: Order Of the Methodsmentioning

confidence: 99%

“…Consider the following second order initial value problem (IVP) of ordinary differential equations (ODEs) 𝑦 ′′ = 𝑓(𝑥, 𝑦, 𝑦 ′ ), 𝑦(𝑎) = 𝜂 0 , 𝑦 ′ (𝑎) = 𝜂 1 (1) with 𝑥 ∈ [𝑎, 𝑏]. Equation (1) could either be linear or nonlinear depending on the properties of the dependent variable, and various numerical methods have been developed to solve either linear, nonlinear, or both.…”

Section: Introductionmentioning

confidence: 99%

“…Equation (1) could either be linear or nonlinear depending on the properties of the dependent variable, and various numerical methods have been developed to solve either linear, nonlinear, or both. Some of these numerical methods are seen in studies such as [1] where second order Bratu-type IVPs were solved using a sixth order Runge-Kutta method while [2] adopted predictor-corrector method of the same type of IVPs. [3] developed a new class of variable coefficients and two-step semi-hybrid methods to solve problems in the form of Equation ( 1), [4] enhanced the conventional Numerov method to solve second order IVPs with better accuracy, and [5] solved Equation (1) problems using a novel Lie group based neural network method.…”

Section: Introductionmentioning

confidence: 99%

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Numerical Solution of Linear and Nonlinear Second Order Initial Value Problems Using Three-Step Generalized Off-Step Hybrid Block Method

Mansor¹,

Adeyeye²,

Omar³

2023

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The numerical of second order initial value problems (IVPs) has garnered a lot of attention in literature, with recent studies ensuring to develop new methods with better accuracy than previously existing approaches. This led to the introduction of hybrid block methods which is a class of block methods capable of directly solving second order IVPs without reduction to a system of first order IVPs. Its hybrid characteristic features the addition of off-step points in the derivation of this block method, which has shown remarkable improvement in the accuracy of the block method. This article proposes a new three-step hybrid block method with three generalized off-step points to find the direct solution of second order IVPs. To derive the method, a power series is adopted as an approximate solution and is interpolated at the initial point and one off-step point while its second derivative is collocated at all points in the interval to obtain the main continuous scheme. The analysis of the method shows that the developed method is of order 7, zero-stable, consistent, and hence convergent. The numerical results affirm that the new method performs better than the existing methods it is compared with, in terms of error accuracy when solving the same IVPs of second order ordinary differential equations.

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Section: Development Of the Methodsmentioning

confidence: 99%

Section: Order Of the Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Numerical Solution of Linear and Nonlinear Second Order Initial Value Problems Using Three-Step Generalized Off-Step Hybrid Block Method

Mansor¹,

Adeyeye²,

Omar³

2023

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“…As the Runge-Kutta theory is well developed, a large family of algorithms, such as that of (16), exist [54]. Fenta and Derese [55] presented, for example, an algorithm of 6 th order, which is utilized together with ( 16) and ( 14) in Section IV. Owing to its size, it is not reproduced here, and it roughly duplicates the computing effort compared with (16).…”

Section: Runge-kutta Numerical Integrationmentioning

confidence: 99%

Reevaluation of Algorithmic Basics for ZUPT-Based Pedestrian Navigation

2022

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During the last 10 to 15 years, pedestrian navigation based on zero velocity updates (ZUPT) has become very popular. One of the main reasons for this is the increasing availability of small, low-cost inertial measurement units. However, the processing of the data from these units for pedestrian navigation is almost exclusively based on algorithmic features that originate from classical inertial navigation with highgrade sensors. In addition, the historical background of the ZUPT approach presupposes also sensors with high accuracy. This leads to the problem that neither the ZUPT approach nor the algorithmic features mentioned are consistent with the accuracy level of the inertial measurement units used normally for pedestrian navigation. Therefore, the question arises of whether the usual algorithmic basics and numerical procedures employed by ZUPT-based pedestrian navigation are adequate. Supported by a literature review showing the state of the art, this study investigates the effect of basics such as the system states and the usage of Runge-Kutta method on the navigation accuracy from pedestrian inertial data aided by ZUPT. To this end, a comparative processing of a well-known, published dataset of a short walk and of own data from the authors using different data fusion algorithms was employed. The important results of this study are that the oftenused omission of inertial sensor biases cannot be recommended, that a continuous-discrete Kalman filter in combination with a Runge-Kutta algorithm performs better than traditional filter configurations, and that total navigation states are an interesting alternative to classical navigation error-states.

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Mathematical model of epidemic disease COVID-19

Aziz

Mahmood²

2023

International Conference of Computational Methods in Sciences and Engineering Iccmse 2021

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Numerical Solution of Second Order Initial Value Problems of Bratu-type Equations using Sixth Order Runge-Kutta Seven Stages Method

Cited by 5 publications

References 1 publication

Numerical Solution of Linear and Nonlinear Second Order Initial Value Problems Using Three-Step Generalized Off-Step Hybrid Block Method

Numerical Solution of Linear and Nonlinear Second Order Initial Value Problems Using Three-Step Generalized Off-Step Hybrid Block Method

Reevaluation of Algorithmic Basics for ZUPT-Based Pedestrian Navigation

Mathematical model of epidemic disease COVID-19

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