Direct solution of initial and boundary value problems of third order ODEs using maximal-order fourth-derivative block method

Adeyeye, Oluwaseun; Omar, Zurni

doi:10.1063/1.5121039

Cited by 5 publications

(8 citation statements)

References 12 publications

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“…For the first point, y n+1 , let s = t−t n+1 h and dt = hds be substituted into (2), ( 5) and (8). By evaluating the integral from −3 to −2 using MAPLE gives the following…”

Section: Methodsmentioning

confidence: 99%

“…A wide variety of real life situations are represented by mathematical models as third order ordinary differential equations (ODEs), such as chemical engineering, biology, electromagnetic waves, quantum mechanics, the motion of rocket, and thin film flow [1][2][3][4]. Nevertheless, the theoretical solutions for most of these equations are undefined; therefore, third-order ODEs have gained significant attention and the need to develop numerical methods with more accurate approximations is eminent [5][6][7][8]. In the classical way, solving higher order ODEs is done by reducing the equation into an equivalent system of first-order ODEs, but this process is too rigorous compared to the direct methods [9][10][11].…”

Section: Introductionmentioning

confidence: 99%

“…Where the points need to be collocated and interpolated after which a system of linear equations must be resolved. Moreover, scholars have looked lately at methods that involved derivatives in solving ODEs, which lead to a more accurate numerical results and to increase the order of the methods [8,[19][20][21]. In fact, the accuracy of a method increases with the increase in the order of the method.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Implicit Three-Point Block Numerical Algorithm for Solving Third Order Initial Value Problem Directly with Applications

Allogmany

Ismail

2020

Mathematics

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Recently, direct methods that involve higher derivatives to numerically approximate higher order initial value problems (IVPs) have been explored, which aim to construct numerical methods with higher order and very high precision of the solutions. This article aims to construct a fourth and fifth derivative, three-point implicit block method to tackle general third-order ordinary differential equations directly. As a consequence of the increase in order acquired via the implicit block method of higher derivatives, a significant improvement in efficiency has been observed. The new method is derived in a block mode to simultaneously evaluate the approximations at three points. The derivation of the new method can be easily implemented. We established the proposed method’s characteristics, including order, zero-stability, and convergence. Numerical experiments are used to confirm the superiority of the method. Applications to problems in physics and engineering are given to assess the significance of the method.

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“…For the first point, y n+1 , let s = t−t n+1 h and dt = hds be substituted into (2), ( 5) and (8). By evaluating the integral from −3 to −2 using MAPLE gives the following…”

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Implicit Three-Point Block Numerical Algorithm for Solving Third Order Initial Value Problem Directly with Applications

Allogmany

Ismail

2020

Mathematics

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show abstract

“…Much and considerable attention have been dedicated to solving higher order ordinary differential equations of the form (1.1) directly without being reduced to system of first order ordinary differential equation. For instance, [17][18][19][20][21] etc. proposed block methods for direct solution of third order ordinary differential equation, the outcome is better when reduced to first order ordinary differential equation.…”

Section: Original Research Articlementioning

confidence: 99%

One–Step Hybrid Block Scheme for the Numerical Approximation for Solution of Third Order Initial Value Problems

Sabo

Kyagya

Solomon

2021

JSRR

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In this research, we have proposed the simulation of linear block algorithm for modeling third order highly stiff problem without reduction to a system of first order ordinary differential equation, to address the weaknesses in reduction method. The method is derived using the linear block method through interpolation and collocation. The basic properties of the block method were recovered and was found to be consistent, convergent and zero-stability. The new block method is been applied to model third order initial value problems of ordinary differential equations without reducing the equations to their equivalent systems of first order ordinary differential equations. The result obtained on the process on some sampled modeled third order linear problems give better approximation than the existing methods which we compared our result with.

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“…Therefore, in order to overcome these challenges, it will be appropriate and more efficient if direct method of solving (1) is employed as suggested by Dahlquist [7], Hall & Suleiman [8], Omar [9] and Kayode [10]. Some authors who adopt solving (1) directly are Adeyeye & Omar [11,12], Kuboye, Elusakin & Quadri [13], Raymond, Skwame & Adiku [14], Sabo, Althemai & Hamadina [15], Abdelrahim [16], Tumba, Skwame & Raymond [17].…”

Section: Original Research Articlementioning

confidence: 99%

The Direct Simulation of Third Order Linear Problems on Single Step Block Method

Sabo

Skwame

Kyagya

et al. 2021

AJRCoS

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In this article, the direct simulation of third order linear problems on single step block method has been proposed. In order to overcoming the setbacks in reduction method, direct method has been proposed using power series to reduce computational burden that occur in the reduction method. Numerical properties for the block method are established and the method developed is consistent, convergent and zero-stable. To validate the accuracy of the block method, certain numerical test problems were considered, the results shown that the accuracy of our method are more accurate over the existing method in literature.

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Direct solution of initial and boundary value problems of third order ODEs using maximal-order fourth-derivative block method

Cited by 5 publications

References 12 publications

Implicit Three-Point Block Numerical Algorithm for Solving Third Order Initial Value Problem Directly with Applications

Implicit Three-Point Block Numerical Algorithm for Solving Third Order Initial Value Problem Directly with Applications

One–Step Hybrid Block Scheme for the Numerical Approximation for Solution of Third Order Initial Value Problems

The Direct Simulation of Third Order Linear Problems on Single Step Block Method

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