scholarly journals Efficient approximate analytical methods for nonlinear fuzzy boundary value problem

2022 ◽  
Vol 12 (2) ◽  
pp. 1916
Author(s):  
Ali Fareed Jameel ◽  
Hafed H Saleh ◽  
Amirah Azmi ◽  
Abedel-Karrem Alomari ◽  
Nidal Ratib Anakira ◽  
...  
Keyword(s):  
Boundary Value Problem ◽  
Analytical Methods ◽  
Boundary Value ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Numerical Comparison ◽  
Fuzzy Boundary Value Problems ◽  
Fuzzy Boundary ◽  
Approximate Analytical Methods

This paper aims to solve the nonlinear two-point fuzzy boundary value problem (TPFBVP) using approximate analytical methods. Most fuzzy boundary value problems cannot be solved exactly or analytically. Even if the analytical solutions exist, they may be challenging to evaluate. Therefore, approximate analytical methods may be necessary to consider the solution. Hence, there is a need to formulate new, efficient, more accurate techniques. This is the focus of this study: two approximate analytical methods-homotopy perturbation method (HPM) and the variational iteration method (VIM) is proposed. Fuzzy set theory properties are presented to formulate these methods from crisp domain to fuzzy domain to find approximate solutions of nonlinear TPFBVP. The presented algorithms can express the solution as a convergent series form. A numerical comparison of the mean errors is made between the HPM and VIM. The results show that these methods are reliable and robust. However, the comparison reveals that VIM convergence is quicker and offers a swifter approach over HPM. Hence, VIM is considered a more efficient approach for nonlinear TPFBVPs.

scholarly journals Analytical solutions to contact problem with fractional derivatives in the sense of Caputo

2020 ◽  
Vol 24 (Suppl. 1) ◽  
pp. 313-323
Author(s):  
Muhammad Noor ◽  
Muhammad Rafiq ◽  
Salah-Ud-Din Khan ◽  
Muhammad Qureshi ◽  
Muhammad Kamran ◽  
...  
Keyword(s):  
Boundary Value Problem ◽  
Iteration Method ◽  
Boundary Value ◽  
Variational Iteration Method ◽  
Contact Problems ◽  
Function Technique ◽  
Fractional Boundary Value Problem ◽  
Variational Iteration ◽  
Residual Errors ◽  
Fractional Boundary Value Problems

The current study extends the applications of the variational iteration method for the analytical solution of fractional contact problems. The problem involves Caputo sense while calculating the derivative of fractional order, we apply the Penalty function technique to transform it into a system of fractional boundary value problems coupled with a known obstacle. The variational iteration method is employed to find the series solution of fractional boundary value problem. For different values of fractional parameters, residual errors of solutions are plotted to make sure the convergence and accuracy of the solution. The reasonably accurate results show that one of the highly effective and stable methods for the solution of fractional boundary value problem is the method of variational iteration.

Surface waves on shear currents: solution of the boundary-value problem

1993 ◽  
Vol 252 ◽  
pp. 565-584 ◽  
Author(s):  
Victor I. Shrira
Keyword(s):  
Boundary Value Problem ◽  
Growth Rates ◽  
Wave Motion ◽  
Boundary Value ◽  
A Priori ◽  
Approximate Solutions ◽  
Short Wave ◽  
Capillary Waves ◽  
Potential Motion ◽  
Approximate Expressions

We consider a classic boundary-value problem for deep-water gravity-capillary waves in a shear flow, composed of the Rayleigh equation and the standard linearized kinematic and dynamic inviscid boundary conditions at the free surface. We derived the exact solution for this problem in terms of an infinite series in powers of a certain parameter e, which characterizes the smallness of the deviation of the wave motion from the potential motion. For the existence and absolute convergence of the solution it is sufficient that e be less than unity.The truncated sums of the series provide approximate solutions with a priori prescribed accuracy. In particular, for the short-wave instability, which can be interpreted as the Miles critical-layer-type instability, the explicit approximate expressions for the growth rates are derived. The growth rates in a certain (very narrow) range of scales can exceed the Miles increments caused by the wind.The effect of thin boundary layers on the dispersion relation was also investigated using an asymptotic procedure based on the smallness of the product of the layer thickness and wavenumber. The criterion specifying when and with what accuracy the boundary-layer influence can be neglected has been derived.

scholarly journals Rapid Convergence of Solution for Hybrid System with Causal Operators

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Peiguang Wang ◽  
Zhifang Li ◽  
Yonghong Wu
Keyword(s):  
Boundary Value Problem ◽  
Hybrid System ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Periodic Boundary Value Problem ◽  
Rapid Convergence ◽  
Causal Operators ◽  
Convergence Of Solution

We investigated the convergence of iterative sequences of approximate solutions to a class of periodic boundary value problem of hybrid system with causal operators and established two sequences of approximate solutions that converge to the solution of the problem with rate of orderk≥2.

Quasilinearization and boundary value problems at resonance

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Kareem Alanazi ◽  
Meshal Alshammari ◽  
Paul Eloe
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Approximate Solutions ◽  
Monotonicity Condition ◽  
Uniqueness Of Solutions ◽  
Shift Method ◽  
At Resonance ◽  
Problem At Resonance

Abstract A quasilinearization algorithm is developed for boundary value problems at resonance. To do so, a standard monotonicity condition is assumed to obtain the uniqueness of solutions for the boundary value problem at resonance. Then the method of upper and lower solutions and the shift method are applied to obtain the existence of solutions. A quasilinearization algorithm is developed and sequences of approximate solutions are constructed, which converge monotonically and quadratically to the unique solution of the boundary value problem at resonance. Two examples are provided in which explicit upper and lower solutions are exhibited.

scholarly journals Numerical Solution of Higher Order Boundary Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Shahid S. Siddiqi ◽  
Muzammal Iftikhar
Keyword(s):  
Boundary Value Problems ◽  
Homotopy Analysis Method ◽  
Present Method ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Higher Order ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Linear And Nonlinear

The aim of this paper is to use the homotopy analysis method (HAM), an approximating technique for solving linear and nonlinear higher order boundary value problems. Using HAM, approximate solutions of seventh-, eighth-, and tenth-order boundary value problems are developed. This approach provides the solution in terms of a convergent series. Approximate results are given for several examples to illustrate the implementation and accuracy of the method. The results obtained from this method are compared with the exact solutions and other methods (Akram and Rehman (2013), Farajeyan and Maleki (2012), Geng and Li (2009), Golbabai and Javidi (2007), He (2007), Inc and Evans (2004), Lamnii et al. (2008), Siddiqi and Akram (2007), Siddiqi et al. (2012), Siddiqi et al. (2009), Siddiqi and Iftikhar (2013), Siddiqi and Twizell (1996), Siddiqi and Twizell (1998), Torvattanabun and Koonprasert (2010), and Kasi Viswanadham and Raju (2012)) revealing that the present method is more accurate.

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