The generalized fractional order of the Chebyshev functions on nonlinear boundary value problems in the semi-infinite domain

2017 ◽  
Vol 6 (3) ◽  
Author(s):  
Kourosh Parand ◽  
Mehdi Delkhosh
Keyword(s):  
Boundary Value Problems ◽  
Collocation Method ◽  
Fractional Order ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Numerical Results ◽  
Algebraic Equations ◽  
Nonlinear Boundary Value Problems ◽  
Infinite Domain ◽  
Nonlinear Boundary Value

AbstractA new collocation method, namely the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) collocation method, is given for solving some nonlinear boundary value problems in the semi-infinite domain, such as equations of the unsteady isothermal flow of a gas, the third grade fluid, the Blasius, and the field equation determining the vortex profile. The method reduces the solution of the problem to the solution of a nonlinear system of algebraic equations. To illustrate the reliability of the method, the numerical results of the present method are compared with several numerical results.

Newton’s Method Applied to Problems in Nonlinear Mechanics

1965 ◽  
Vol 32 (2) ◽  
pp. 383-388 ◽  
Author(s):  
G. A. Thurston
Keyword(s):  
Boundary Value Problems ◽  
Newton’S Method ◽  
Newton's Method ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Practical Method ◽  
Algebraic Equations ◽  
Nonlinear Mechanics ◽  
Nonlinear Boundary Value Problems ◽  
Nonlinear Boundary Value

Many problems in mechanics are formulated as nonlinear boundary-value problems. A practical method of solving such problems is to extend Newton’s method for calculating roots of algebraic equations. Three problems are treated in this paper to illustrate the use of this method and compare it with other methods.

scholarly journals An efficient numerical approach for solving two-point fractional order nonlinear boundary value problems with Robin boundary conditions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hyunju Kim ◽  
Junseo Lee ◽  
Bongsoo Jang
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Fractional Order ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Robin Boundary Conditions ◽  
Nonlinear Boundary Value Problems ◽  
Nonlinear Boundary Value ◽  
And Performance ◽  
Robin Boundary

AbstractThis article proposes new strategies for solving two-point Fractional order Nonlinear Boundary Value Problems (FNBVPs) with Robin Boundary Conditions (RBCs). In the new numerical schemes, a two-point FNBVP is transformed into a system of Fractional order Initial Value Problems (FIVPs) with unknown Initial Conditions (ICs). To approximate ICs in the system of FIVPs, we develop nonlinear shooting methods based on Newton’s method and Halley’s method using the RBC at the right end point. To deal with FIVPs in a system, we mainly employ High-order Predictor–Corrector Methods (HPCMs) with linear interpolation and quadratic interpolation (Nguyen and Jang in Fract. Calc. Appl. Anal. 20(2):447–476, 2017) into Volterra integral equations which are equivalent to FIVPs. The advantage of the proposed schemes with HPCMs is that even though they are designed for solving two-point FNBVPs, they can handle both linear and nonlinear two-point Fractional order Boundary Value Problems (FBVPs) with RBCs and have uniform convergence rates of HPCMs, $\mathcal{O}(h^{2})$ O ( h 2 ) and $\mathcal{O}(h^{3})$ O ( h 3 ) for shooting techniques with Newton’s method and Halley’s method, respectively. A variety of numerical examples are demonstrated to confirm the effectiveness and performance of the proposed schemes. Also we compare the accuracy and performance of our schemes with another method.

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