Optimization of One‐Step Block Method for Solving Second‐Order Fuzzy Initial Value Problems

Al-Refai, Safa; Syam, Muhammed I.; Al-Refai, Mohammed

doi:10.1155/2021/6650413

Cited by 3 publications

(3 citation statements)

References 28 publications

(43 reference statements)

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“…x−axis shows the value of the approximation solution, y−axis show the value of α-level values, Y, Y are the lower and upper bounds of the exact solution respectively, y, y are the lower and upper bounds of the approximate solution respectively, E = Y − y computes the absolute error of the lower bound approximation, E = Y − y computes the absolute error of the upper bound approximation, h is the step size, TSBM: Two-step Block Method with Third and Fourth Derivatives, EBHDEF: Extended Block Hybrid Backward Differentiation Formula [16], BDF: Block Differentiation Formula [15], BBDF: Block Backward Differentiation Formula [15], OOMB: Optimization of One-Step Block Method [17], RK5: Runge Kutta Method Order Five [14], OHAM: Optimal Homotopy Asymptotic Method [7], FDM: Finite Difference Method [30].…”

Section: Resultsmentioning

confidence: 99%

“…The biggest drawback of these approaches is the reduction of the second-order FODEs to the system of first-order FODEs, which leads to computational burden and also impacts solution accuracy. To bypass the rigor of reduction, block methods were introduced for the direct solution of second-order FODEs in [15][16][17]. However, due to the order of the block methods developed by these studies, it is observed that there is still room to improve the accuracy of their obtained results in terms of absolute error.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Numerical Solution of Second Order Fuzzy Ordinary Differential Equations using Two-Step Block Method with Third and Fourth Derivatives

Hussain

Adeyeye

Ahmad

2023

J. Nig. Soc. Phys. Sci.

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Fuzzy differential equation models are suitable where uncertainty exists for real-world phenomena. Numerical techniques are used to provide an approximate solution to these models in the absence of an exact solution. However, existing studies that have developed numerical techniques for solving second-order fuzzy ordinary differential equations (FODEs) possess an absolute error accuracy that could be improved. Therefore, this article developed a more accurate higher derivative self-starting block scheme for the numerical solution of second-order FODEs with fuzzy initial and boundary conditions imposed. Linear block approach using Taylor series expansion is adopted for the derivation of the proposed method and the basic properties are established using the definitions of stability and consistency for block methods. According to the numerical results, when compared to the exact solution in terms of absolute error, the new method proposed in this article outperformed existing numerical methods. It is thus concluded that the proposed method is effective for solving second-order FODEs directly.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Second Order Fuzzy Ordinary Differential Equations using Two-Step Block Method with Third and Fourth Derivatives

Hussain

Adeyeye

Ahmad

2023

J. Nig. Soc. Phys. Sci.

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show abstract

“…gr (x, β, α z ) ∂x4 = z gr (x, β, α z ), (5.20)z gr (0) = z gr (0) = z gr (0) = z gr (0) = z (4) gr (0) = −1 + β + 2(1 − β)α 0 ,(5.21)whereβ, α z = α 0 ∈ [0, 1]. The solution of IVP (5.20)-(5.21) is z gr (x, β, α 0 ) = e x [−1 + β + 2(1 − β)α 0 ] .…”

mentioning

confidence: 99%

Existence and uniqueness of solutions for fuzzy initial value problems under granular differentiability

Soma¹,

Grande²,

Agarwal³

et al. 2023

J. Math. Computer Sci.

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In this paper, we introduce the notion of second and higher-order granular differentiability for fuzzy number-valued functions. A weighted granular metric is defined for continuously granular differentiable mappings and proves that it is a complete metric space. Fuzzy initial value problems are investigated for second and higher-order fuzzy differential equations under granular differentiability. Sufficient conditions are established for the existence and uniqueness of solutions for the fuzzy initial value problems. An algorithm is developed to determine the solution to the fuzzy initial value problem under granular differentiability. Moreover, examples are presented to verify our theoretical results and algorithm.

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An Efficient Method for Solving Second-Order Fuzzy Order Fuzzy Initial Value Problems

Dallashi

Syam

2022

Symmetry

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In this paper, we present an accurate numerical approach based on the reproducing kernel method (RKM) for solving second-order fuzzy initial value problems (FIVP) with symmetry coefficients such as symmetric triangles and symmetric trapezoids. Finding the exact solution of FIVP is not an easy task since the definition will produce a complicated optimization problem. To overcome this difficulty, a numerical method is developed to solve this type of problems. We start by introducing the necessary definitions and theorems about the fuzzy logic. Then, we derived the kernels for two Hilbert spaces. The RKM is derived for the second-order IVP in the Boolean sense, and then we generalize it for the fuzzy sense. Numerical and theoretical results will be given to obtain the accuracy of the developed technique. We solved four linear and non-linear fuzzy IVPs numerically using the proposed method, and we compute the error in each case to show the efficiency of the method. The absolute error was very small in the four examples.

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Optimization of One‐Step Block Method for Solving Second‐Order Fuzzy Initial Value Problems

Cited by 3 publications

References 28 publications

Numerical Solution of Second Order Fuzzy Ordinary Differential Equations using Two-Step Block Method with Third and Fourth Derivatives

Numerical Solution of Second Order Fuzzy Ordinary Differential Equations using Two-Step Block Method with Third and Fourth Derivatives

Existence and uniqueness of solutions for fuzzy initial value problems under granular differentiability

An Efficient Method for Solving Second-Order Fuzzy Order Fuzzy Initial Value Problems

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