• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.23 no.4
• /
• pp.283-300
• /
• 2019

In this paper, we propose a new semi-analytic approach based on the generalized Taylor series for solving nonlinear differential equations of fractional order. Assuming the solution is expanded as the generalized Taylor series, the coefficients of the series can be computed by solving the corresponding recursive relation of the coefficients which is generated by the given problem. This method is called the generalized differential transform method(GDTM). In several literatures the standard GDTM was applied in each sub-domain to obtain an accurate approximation. As noticed in [19], however, a direct application of the GDTM in each sub-domain loses a term of memory which causes an inaccurate approximation. In this work, we derive a new recursive relation of the coefficients that reflects an effect of memory. Several illustrative examples are demonstrated to show the effectiveness of the proposed method. It is shown that the proposed method is robust and accurate for solving nonlinear differential equations of fractional order.

• East Asian mathematical journal
• /
• v.17 no.2
• /
• pp.197-215
• /
• 2001

A generalised Taylor series with integral remainder involving a convex combination of the end points of the interval under consideration is investigated. Perturbed generalised Taylor series are bounded in terms of Lebesgue p-norms on $[a,b]^2$ for $f_{\Delta}:[a,b]^2{\rightarrow}\mathbb{R}$ with $f_{\Delta}(t,s)=f(t)-f(s)$.

• Journal for History of Mathematics
• /
• v.27 no.2
• /
• pp.127-138
• /
• 2014

In this paper we investigate how Newton discovered the generalized binomial theorem. Newton's binomial theorem, or binomial series can be found in Calculus text books as a special case of Taylor series. It can also be understood as a formal power series which was first conceived by Euler if convergence does not matter much. Discovered before Taylor or Euler, Newton's binomial theorem must have a good explanation of its birth and validity. Newton learned the interpolation method from Wallis' famous book ${\ll}$Arithmetica Infinitorum${\gg}$ and employed it to get the theorem. The interpolation method, which Wallis devised to find the areas under a family of curves, was by nature arithmetrical but not geometrical. Newton himself used the method as a way of finding areas under curves. He noticed certain patterns hidden in the integer binomial sequence appeared in relation with curves and then applied them to rationals, finally obtained the generalized binomial sequence and the generalized binomial theorem.

• Communications of the Korean Mathematical Society
• /
• v.29 no.2
• /
• pp.269-283
• /
• 2014

The aim of the paper is to present several new relationships involving the hyperharmonic function introduced by Mez$\ddot{o}$ in (I. Mez$\ddot{o}$, Analytic extension of hyperharmonic numbers, Online J. Anal. Comb. 4, 2009) which is an analytic extension of the hyperharmonic numbers. These relations are obtained by using some fractional calculus theorems as Leibniz rules and Taylor like series expansions.

• Korean Journal of Mathematics
• /
• v.21 no.3
• /
• pp.271-283
• /
• 2013

In this paper, a generalized Adams-Moulton method that is a $m$-step method derived by using the Taylor's series is proposed to solve the satellite orbit determination problem. We show that our proposed method has produced much smaller error than the original Adams-Moulton method. Finally, the accuracy performance is demonstrated in the satellite orbit correction problem by giving a numerical example.

• Communications for Statistical Applications and Methods
• /
• v.18 no.5
• /
• pp.645-655
• /
• 2011

Chandrasekar et al. (2004) introduced a generalized Type-II hybrid censoring. In this paper, we propose generalized doubly Type-II hybrid censoring. In addition, this paper presents the statistical inference on the scale parameter for the half logistic distribution when samples are generalized doubly Type-II hybrid censoring. The approximate maximum likelihood(AMLE) method is developed to estimate the unknown parameter. The scale parameter is estimated by the AMLE method using two di erent Taylor series expansion types. We compar the AMLEs in the sense of the mean square error(MSE). The simulation procedure is repeated 10,000 times for the sample size n = 20; 30; 40 and various censored samples. The $AMLE_I$ is better than $AMLE_{II}$ in the sense of the MSE.

• Honam Mathematical Journal
• /
• v.32 no.3
• /
• pp.481-491
• /
• 2010

In this paper, we investigate a generalization of the Adams-Bashforth method by using the Taylor's series. In case of m-step method, the local truncation error can be expressed in terms of m - 1 coefficients. With an appropriate choice of coefficients, the proposed method has produced much smaller error than the original Adams-Bashforth method. As an application of the generalized Adams-Bashforth method, the accuracy performance is demonstrated in the satellite orbit prediction problem. This implies that the generalized Adams-Bashforth method is applied to the orbit prediction of a low-altitude satellite. This numerical example shows that the prediction of the satellite trajectories is improved one order of magnitude.

• East Asian mathematical journal
• /
• v.37 no.3
• /
• pp.319-331
• /
• 2021

In this paper, we aim to introduce two new families of analytic and bi-univalent functions associated with the Attiya's operator, which is defined by the Hadamard product of a generalized Mittag-Leffler function and analytic functions on the open unit disk. Then we estimate the second and third coefficients of the Taylor-Maclaurin series expansions of functions belonging to these families. Also, we investigate Fekete-Szegö problem for these families. Some relevant connections of certain special cases of the main results with those in several earlier works are also pointed out. Two naturally-arisen problems are given for further investigation.

• KSCE Journal of Civil and Environmental Engineering Research
• /
• v.33 no.2
• /
• pp.409-422
• /
• 2013

For stability design and P-${\Delta}$ analysis of steel frames with semi-rigid connections, the explicit form of the exact tangential stiffness matrix of a generalized semi-rigid frame element having rotational and translational connections is firstly derived using the stability functions. And its elastic and geometric stiffness matrix is consistently obtained by Taylor series expansion. Next depending on connection types of semi-rigidity, the corresponding tangential stiffness matrices are degenerated based on penalty method and static condensation technique. And then numerical procedures for determination of effective buckling lengths of generalized semi-rigid frames members and P-${\Delta}$ and shortly addressed. Finally three numerical examples are presented to demonstrate the validity and accuracy of the proposed method. Particularly the minimum braced frames and coupled buckling modes of the corresponding frames are investigated.

• Structural Engineering and Mechanics
• /
• v.32 no.5
• /
• pp.609-620
• /
• 2009

In this paper the results of a series of experimental tests upon three-point bending specimens made of polystyrene and containing re-entrant corners are firstly described. Tests involved different notch angles, different notch depths and finally different sizes of the samples. All the specimens broke at the defect, as expected because of the material brittleness and, hence, the generalized stress intensity factor was expected to be the governing failure parameter. Recorded failure loads are then compared with the predictions provided by a fracture criterion recently introduced in the framework of Finite Fracture Mechanics: fracture is assumed to propagate by finite steps, whose length is determined by the contemporaneous fulfilment of energy balance and stress requirements. This fracture criterion allows us to achieve the expression of the generalized fracture toughness as a function of the tensile strength, the fracture toughness and the notch opening angle. Comparison between theoretical predictions and experimental data turns out to be more than satisfactory.