In this paper, we derive two novel finite difference schemes for two types of time-space fractional diffusion equations by adopting weighted and shifted Grünwald operator, which is used to approximate the Riemann-Liouville fractional derivative to the second order accuracy. The stability and convergence of the schemes are analyzed via mathematical induction. Moreover, the illustrative numerical examples are carried out to verify the accuracy and effectiveness of the schemes.
We develop a formula for matching a Taylor series about the origin and an asymptotic exponential expansion for large values of the coordinate. We test it on the expansion of the generating functions for the moments and connected moments of the Hamiltonian operator. In the former case the formula produces the energies and overlaps for the Rayleigh-Ritz method in the Krylov space. We choose the harmonic oscillator and a strongly anharmonic oscillator as illustrative examples for numerical test. Our results reveal some features of the connected-moments expansion that were overlooked in earlier studies and applications of the approach.
In this manuscript, a reliable scheme based on a general functional transformation is applied to construct the exact travelling wave solution for nonlinear differential equations. Our methodology is investigated by means of the modified homotopy analysis method which contains two convergence-control parameters. The obtained results reveal that the proposed approach is a very effective. Several illustrative examples are investigated in detail.
In this paper, the problem of laminar, isothermal, incompressible and viscous flow in a rectangular domain bounded by two moving porous walls, which enable the fluid to enter or exit during successive expansions or contractions is solved analytically by using the homotopy analysis method (HAM). Graphical results are presented to investigate the influence of the nondimensional wall dilation rate α and permeation Reynolds number Re on the velocity, normal pressure distribution and wall shear stress. The obtained solutions, in comparison with the numerical solutions, demonstrate remarkable accuracy. The present problem for slowly expanding or contracting walls with weak permeability is a simple model for the transport of biological fluids through contracting or expanding vessels.
In this paper we study a class of new Generalized Fractional Advection-Diffusion Equations (GFADEs) with a new Generalized Fractional Derivative (GFD) proposed last year. The new GFD is defined in the Caputo sense using a weight function and a scale function. The GFADE is discussed in a bounded domain, and numerical solutions for two examples consisting of a linear and a nonlinear GFADE are obtained using an implicit finite difference approach. The stability of the numerical scheme is investigated, and the order of convergence is estimated numerically. Numerical results illustrate that the finite difference scheme is simple and effective for solving the GFADEs. We investigate the influence of weight and scale functions on the diffusion of GFADEs. Linear and nonlinear stretching and contracting functions are considered. It is found that an increasing weight function increases the rate of diffusion, and a scale function can stretch or contract the diffusion on the time domain.
INTRODUCTION: The aim of this study was to determine whether convergence varies with age and pseudophakia. MATERIAL AND METHODS: 86 patients, aged 21–85 (average age: 62.7) years were included in the study group: 39 patients with binocular pseudophakia and 41 with their own lens, as well as 68 people with their own lens, at the age of 19–25 (average 22.1) years. Examinations were carried out at distances of 50 and 10 cm, and for distant vision. Interpupillary distance measurements were used to calculate the difference of convergence capabilities. The exami-nation results of phakic and pseudophakic eyes were compared. RESULTS: The mean convergence of the pseudophakic patients was 3.52 mm (± 0.42 mm), in the patients with their own lens: 3.46 (± 0.74 mm). No statistically significant differences between the groups were found. No statistically significant difference was found in the mean convergence between the two groups of patients aged 19–25 (group 1) and over 65 years (group 2), either. The average value of convergence in group 1 was 3.47 mm (3.84–2.93 mm). Ho-wever, in the second group the values were 3.46 mm (3.96–2.48 mm). There was no evidence of statistical differences in the average value of convergence in both groups for distances of 50 cm and 10 cm. CONCLUSIONS: Convergence does not depend on age. Presbyopia does not lead to a loss of convergence in phakic or pseudophakic eyes.
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WSTĘP: Celem badania było stwierdzenie, czy konwergencja zależy od wieku i pseudofakii. MATERIAŁ I METODY: Do grupy badanej włączono 86 pacjentów w wieku 21–85 lat (średnia: 62,7): 39 pacjentów z obuoczną pseudofakią, 41 z własną soczewką oraz 68 osób z własną soczewką w wieku 19–25 lat (średnia 22,1). Badanie zostało wykonane w odległości 50 i 10 cm oraz dla spojrzenia w dal. Wykorzystując uzyskane pomiary rozstawu źrenic, obliczono różnicę zdolności konwergencji oraz porównano obie grupy badanych. WYNIKI: Średnia wartość konwergencji u pacjentów z pseudofakią wyniosła 3,52 mm (± 0,42 mm), a w grupie pacjentów z własną soczewką 3,46 mm (± 0,74 mm). Na podstawie przeprowadzanych analiz nie stwierdzono istotnych statystycznie różnic pod względem średniej konwergencji pomiędzy grupami pacjentów w wieku od 19 do 25 lat (grupa 1.) i powyżej 65 lat (grupa 2). Średnia wartość konwergencji w grupie 1. wynosiła 3,47 mm (maksymalna 3,84 mm, minimalna 2,93 mm). Natomiast w grupie 2 – 3,46 mm (maksymalna 3,96 mm, minimalna 2,48 mm). Średnia wartość konwergencji przy patrzeniu na przedmiot znajdujący się w odległości 50 cm i 10 cm różniła się istotnie statystycznie pomiędzy badanymi grupami. WNIOSKI: Dane z uzyskanych analiz statycznych pozwalają stwierdzić, że konwergencja nie zależy od wieku oraz starczowzroczność, nie prowadzi do utraty zdolności konwergencji w oczach fakijnych i pseudofakijnych.