]>
2018
11
11
ISSN 2008-1898
78
Improved convergence analysis of the Secant method using restricted convergence domains with real-world applications
Improved convergence analysis of the Secant method using restricted convergence domains with real-world applications
en
en
In this paper, we are concerned with the problem of approximating a solution of a nonlinear equations by means of using the Secant method. We present a new semilocal convergence analysis for Secant method using restricted convergence domains. According to this idea we find a more precise domain where the inverses of the operators involved exist than in earlier studies. This way we obtain smaller Lipschitz constants leading to more precise majorizing sequences. Our convergence criteria are weaker and the error bounds are more precise than in earlier studies. Under the same computational cost on the parameters involved our analysis includes the computation of the bounds on the limit points of the majorizing sequences involved. Different real-world applications are also presented to illustrate the theoretical results obtained in this study.
1215
1224
Ioannis K.
Argyros
Department of Mathematics Sciences
Cameron University
USA
iargyros@cameron.edu
Alberto
Magreñán
Escuela Superior de Ingeniería y Tecnología
Universidad Internacional de La Rioja
Spain
alberto.magrenan@unir.net
Íñigo
Sarría
Escuela Superior de Ingeniería y Tecnología
Universidad Internacional de La Rioja
Spain
inigo.sarria@unir.net
Juan Antonio
Sicilia
Escuela Superior de Ingeniería y Tecnología
Universidad Internacional de La Rioja, Av de la Paz, 137, 26002 Logroño, Spain
Spain
juanantonio.sicilia@unir.net
Secant method
Banach space
majorizing sequence
divided difference
local convergence
semilocal convergence
Article.1.pdf
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S. Amat, M. A. Hernández-Vern, M. J. Rubio, Improving the applicability of the Secant method to solve nonlinear systems of equations, Appl. Math. Comput., 247 (2014), 741-752
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I. K. Argyros, Computational theory of iterative methods, Elsevier B. V., Amsterdam (2007)
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I. K. Argyros, Y. J. Cho, S. Hilout , Numerical method for equations and its applications, CRC Press, Boca Raton (2012)
##[7]
I. K. Argyros, D. González, Á . A. Magreñán , A semilocal convergence for a uniparametric family of efficient Secant-like methods , J. Funct. Spaces, 2014 (2014), 1-10
##[8]
I. K. Argyros, S. Hilout, Weaker conditions for the convergence of Newton’s method , J. Complexity, 28 (2012), 364-387
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I. K. Argyros, Á . A. Magreñán , A unified convergence analysis for secant-type methods , J. Korean Math. Soc., 51 (2014), 1155-1175
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I. K. Argyros, Á . A. Magreñán, Expanding the applicability of the secant method under weaker conditions , Appl. Math. Comput., 266 (2015), 1000-1012
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I. K. Argyros, Á . A. Magreñán, Relaxed Secant-type methods, Nonlinear Stud., 21 (2014), 485-503
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R. Kaur, S. Arora, Nature Inspired Range BasedWireless Sensor Node Localization Algorithms, Int. J. Interact. Multimed. Artif. Intell., 4 (2017), 7-17
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P. P. Zabrejko, D. F. Nguen, The majorant method in the theory of Newton-Kantorovich approximations and the Pták error estimates, Numer. Funct. Anal. Optim., 9 (1987), 671-684
]
A viscosity iterative algorithm for split common fixed-point problems of demicontractive mappings
A viscosity iterative algorithm for split common fixed-point problems of demicontractive mappings
en
en
In this paper, we firstly introduce a new viscosity cyclic iterative algorithm for the split common fixed-point problem (SCFP) of demicontractive mappings. Next we prove the strong convergence of the sequence generated recursively by such a viscosity cyclic algorithm to a solution of the SCFP, which improves and extends some recent corresponding results.
1225
1234
Di
Gao
Department of Applied Mathematics, College of Natural Sciences
Pukyong National University
Republic of Korea
kekeke.444@163.com
Tae Hwa
Kim
Department of Applied Mathematics, College of Natural Sciences
Pukyong National University
Republic of Korea
taehwa@pknu.ac.kr
Yaqin
Wang
Department of Mathematics
Shaoxing University
China
wangyaqin0579@126.com
Multiple-set split equality common fixed-point problem
demicontractive mapping
viscosity cyclic iterative algorithm
strong convergence
Article.2.pdf
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Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239
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Y. Censor, A. Segal , The split common fixed point problem for directed operators, J. Convex Anal., 16 (2009), 587-600
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H. H. Cui, M. L. Su, F. H. Wang, Damped projection method for split common fixed point problems, J. Inequal. Appl., 2013 (2013), 1-10
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H. H. Cui, F. H. Wang, Iterative methods for the split common fixed point problem in Hilbert spaces, Fixed Point Theory Appl., 2014 (2014), 1-8
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H. M. He, S. Y. Liu, R. D. Chen, X. Y. Wang, Strong convergence results for the split common fixed point problem, J. Nonlinear Sci. Appl. , 9 (2016), 5332-5343
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A. Moudafi, The split common fixed-point problem for demicontractive mappings, Inverse Problems, 26 (2010), 1-6
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W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama (2000)
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Y. Q. Wang, T. H. Kim, Simultaneous iterative algorithm for the split equality fixed-point problem of demicontractive mappings, J. Nonlinear Sci. Appl., 10 (2017), 154-165
##[13]
Y. Q. Wang, T.-H. Kim, X. L. Fang, H. M. He, The split common fixed-point problem for demicontractive mappings and quasi-nonexpansive mappings, J. Nonlinear Sci. Appl., 10 (2017), 2976-2985
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]
Differentiability of pseudo-dual-quaternionic functions with a differential operator
Differentiability of pseudo-dual-quaternionic functions with a differential operator
en
en
This paper introduces the new concept of pseudo-dual-quaternions and some of their basic properties based on matrices. We extend the concept of differentiability to pseudo-dual-quaternionic functions. Also, we propose a corresponding Cauchy-Riemann formulas induced the properties of a holomorphic function of pseudo-dual-quaternionic variables.
1235
1242
Ji Eun
Kim
Department of Mathematics
Dongguk University
Republic of Korea
jeunkim@pusan.ac.kr
Dual-quaternion
differential operators
differentiability
Cauchy-Riemann formulas
Article.3.pdf
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H. H. Cheng, Programming with dual numbers and its applications in mechanisms design, Eng. Comput., 10 (1994), 212-229
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M. A. Clifford, Preliminary Sketch of Biquaternions , Proc. Lond. Math. Soc., 4 (1871/73), 381-395
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J. E. Kim, The corresponding inverse of functions of multidual complex variables in Clifford analysis, J. Nonlinear Sci. Appl., 9 (2016), 4520-4528
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J. E. Kim, A corresponding Cullen-regularity for split-quaternionic-valued functions, Adv. Difference Equ., 2017 (2017), 1-14
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]
On a \(q\)-analogue of the Hilbert's type inequality
On a \(q\)-analogue of the Hilbert's type inequality
en
en
In this paper, by introducing a parameter \(q\) and using the expression of the beta function establishing the inequality of
the weight coefficient, we give a \(q\)-analogue of the Hilbert's type inequality. As applications, a generalization
of Hardy-Hilbert's inequality are obtained.
1243
1249
Zhengping
Zhang
College of Mathematics and Physics
Chongqing University of Science and Technology
P. R. China
Gaowen
Xi
College of Mathematics and Physics
Chongqing University of Science and Technology
P. R. China
xigaowen@163.com
\(q\)-Analogue
Hilbert's type inequality
weight coefficient
Holder inequality
generalization
Article.4.pdf
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G. Gasper, M. Rahman , Basic Hypergeometric Series, Cambrideg University Press, Cambrideg (1990)
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G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, Cambrideg University Press, Cambridge (1934)
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]
Efficient approximations of finite and infinite real alternating \(p\)-series
Efficient approximations of finite and infinite real alternating \(p\)-series
en
en
For \(n\in\mathbb{N}\) and \(p\in\mathbb{R}\) the \(n\)th partial sum of the alternating
\(p\)-series, known also as alternating generalized harmonic number
of order \(p\),
\[
H^*(n,p):=\sum_{i=1}^n(-1)^{i+1}\frac{1}{i^p}
\]
is given in the form
\[
H^*(n,p)=S_q(k,n,p)+r^*_q(k,n,p),
\]
where \(k,q\in\mathbb{N}\) with \(k<\lfloor n/2\rfloor\) are parameters,
controlling the magnitude of the error term \(r^*_q(k,n,p)\). The
function \(S_q(k,n,p)\) consists of \(2(k+1)+q\) simple summands and
\(r^*_q(k,n,p)\) is estimated for \(q>-p+1\), as
\begin{equation*}
\big|r^*_q(k,n,p)\big| <
\frac{|p|(|p|+1)\cdots(|p|+q-1)\pi^{p+1}}{3(p+q-1)(2k\pi)^{p+q-1}}
.
\end{equation*}
Additionally, for \(p\in\mathbb{R}^+\) and \(k,q\in\mathbb{N}\), we have
\begin{equation*}
\left|r_q^*(k,\infty,p)\right|
\le\frac{p(p+1)\cdots(p+q-2)\pi^{p+1}}{3(2k\pi)^{p+q-1}}.
\end{equation*}
1250
1261
Vito
Lampret
University of Ljubljana
Slovenia, EU
vito.lampret@guest.arnes.si
Alternating
alternating generalized harmonic number
approximation
estimate
alternating \(p\)-series
Article.5.pdf
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[1]
E. K. Abalo, K. Y. Abalo, Convergence of p-series revisited with applications, Int. J. Math. Math. Sci., 2006 (2006), 1-8
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M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions: with formulas, graphs, and mathematical tables, Dover Publications, New York (1972)
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E. Chlebus, An approximate formula for a partial sum of the divergent p-series, Appl. Math. Lett., 22 (2009), 732-737
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J. Choi, H. M. Srivastava, Some summation formulas involving harmonic numbers and generalized harmonic numbers, Math. Comput. Modelling, 54 (2011), 2220-2234
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D. Cvijović, The Dattoli-Srivastava conjectures concerning generating functions involving the harmonic numbers, Appl. Math. Comput., 215 (2010), 4040-4043
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G. Dattoli, H. M. Srivastava , A note on harmonic numbers, umbral calculus and generating functions, Appl. Math. Lett., 21 (2008), 686-693
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Y. Hansheng, B. Lu, Another proof for the p-series test, College Math. J., 36 (2005), 235-237
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T. Kim, Y.-H. Kim, D.-H. Lee, D.-W. Park, Y. S. Ro, On the alternating sums of powers of consecutive integers, Proc. Jangjeon Math. Soc., 8 (2005), 175-178
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V. Lampret, Accurate double inequalities for generalized harmonic numbers, Appl. Math. Comput., 265 (2015), 557-567
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V. Lampret , Approximating real Pochhammer products: A comparison with powers, Cent. Eur. J. Math., 7 (2009), 493-505
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V. Lampret , Asymptotic inequalities for alternating harmonics, Bull. Math. Sci., 2017 (2017), 1-8
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V. Lampret, Even from Gregory-Leibniz series \(\pi\) could be computed: an example of how convergence of series can be accelerated, Lect. Mat., 27 (2006), 21-25
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V. Lampret, Wallis’ sequence estimated accurately using an alternating series, J. Number Theory, 172 (2017), 256-269
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D. A. MacDonald, A note on the summation of slowly convergent alternating series, BIT, 36 (1996), 766-774
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R. Meštrović, Proof of a congruence for harmonic numbers conjectured by Z.-W. Sun, Int. J. Number Theory, 8 (2012), 1081-1085
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T. M. Rassias, H. M. Srivastava, Some classes of infinite series associated with the Riemann zeta and polygamma functions and generalized harmonic numbers , Appl. Math. Comput., 131 (2002), 593-605
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A. Sîntămărian, Sharp estimates regarding the remainder of the alternating harmonic series, Math. Inequal. Appl., 18 (2015), 347-352
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A. Sofo, New families of alternating harmonic number sums, Tbilisi Math. J., 8 (2015), 195-209
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A. Sofo, Polylogarithmic connections with Euler sums , Sarajevo J. Math., 12 (2016), 17-32
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A. Sofo, Quadratic alternating harmonic number sums, J. Number Theory, 154 (2015), 144-159
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A. Sofo, H. M. Srivastava, A family of shifted harmonic sums, Ramanujan J., 37 (2015), 89-108
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Z.-W. Sun , Arithmetic theory of harmonic numbers , Proc. Amer. Math. Soc., 140 (2012), 415-428
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L. Tóth, J. Bukor, On the alternating series \(1 - \frac{1}{2} + \frac{1}{ 3} - \frac{1}{ 4} + ...\), J. Math. Anal. Appl., 282 (2003), 21-25
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Wolfram, Mathematica, Version 7.0, Wolfram Research, Inc., (1988–2009.), -
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T.-C. Wu, S.-T. Tu, H. M. Srivastava , Some combinatorial series identities associated with the digamma function and harmonic numbers, Appl. Math. Lett., 13 (2000), 101-106
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D.-Y. Zheng , Further summation formulae related to generalized harmonic numbers, J. Math. Anal. Appl., 335 (2007), 692-706
]
Symmetry Lie algebra and exact solutions of some fourth-order difference equations
Symmetry Lie algebra and exact solutions of some fourth-order difference equations
en
en
In this paper, all the Lie point symmetries of difference equations of the form
\[
u_{n+4}=\frac{u_n}{A_n +B_nu_nu_{n+2}},
\]
where, \((A_n)_{n \geq 0}\) and \((B_n)_{n \geq 0}\) are sequences of real numbers, are obtained. We perform reduction of order using the invariant of the group of transformations. Furthermore, we obtain their solutions. In particular, our work generalizes some results in the literature.
1262
1270
N.
Mnguni
School of Mathematics
University of the Witwatersrand
South Africa
734552@students.wits.ac.za
D.
Nyirenda
School of Mathematics
University of the Witwatersrand
South Africa
Darlison.Nyirenda@wits.ac.za
M.
Folly-Gbetoula
School of Mathematics
University of the Witwatersrand
South Africa
Mensah.Folly-Gbetoula@wits.ac.za
Difference equation
symmetry
group invariant solutions
Article.6.pdf
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[1]
M. Aloqeli , Dynamics of a rational difference equation, Appl. Math. Comput., 176 (2006), 768-774
##[2]
C. Cinar , On the Positive Solutions of the Difference Equation \(x_{n+1} = ax_{n-1}/(1 + bx_nx_{n-1})\), Appl. Math. Comput., 156 (2004), 587-590
##[3]
C. Cinar, On the Positive Solutions of the Difference Equation \(x_{n+1} = x_{n-1}/(1 + x_nx_{n-1})\), Appl. Math. Comput., 150 (2004), 21-24
##[4]
C. Cinar , On the Positive Solutions of the Difference Equation \(x_{n+1} = x_{n-1}/(-1 + x_nx_{n-1})\), Appl. Math. Comput., 158 (2004), 813-816
##[5]
C. Cinar , On the Positive Solutions of the Difference Equation \(x_{n+1} = x_{n-1}/(-1 + ax_nx_{n-1})\), Appl. Math. Comput., 158 (2004), 793-797
##[6]
C. Cinar, On the Positive Solutions of the Difference Equation \(x_{n+1}= x_{n-1}/(1 + ax_nx_{n-1})\) , Appl. Math. Comput., 158 (2004), 809-812
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E. M. Elsayed, On the Difference Equation \(x_{n-5}/(-1+x_{n-2}x_{n-5})\), Int. J. Contemp. Math. Sci., 33 (2008), 1657-1664
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E. M. Elsayed, On the solutions and periodic nature of some systems of difference equations, Int. J. Biomath, 7 (2014), 1-26
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E. M. Elsayed, On the Solution of Some Difference Equations, Eur. J. Pure Appl. Math., 3 (2011), 287-303
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E. M. Elsayed, Solutions of Rational Difference System of Order Two, Math. Comput. Modelling, 55 (2012), 378-384
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E. M. Elsayed, Solution and Attractivity for a Rational Recursive Sequence, Discrete Dyn. Nat. Soc., 2011 (2011), 1-17
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G. Feng, X. J. Yang, H. M. Srivasta, Exact traveling-wave solutions for linear and nonlinear heat-transfer equations, Thermal Science, 21 (2017), 2307-2311
##[13]
M. Folly-Gbetoula , Symmetry, reductions and exact solutions of the difference equation \(u_{n+2} = (au_n)/(1 + bu_nu_{n+1})\), J. Difference Equ. Appl., 23 (2017), 1017-1024
##[14]
M. Folly-Gbetoula, A. H. Kara , Symmetries, conservation laws, and Integrability of Difference Equations, Adv. Difference Equ., 2014 (2014), 1-14
##[15]
M. Folly-Gbetoula, A. H. Kara, The invariance, Conservation laws and Integration of some Higher-order Difference Equations, Advances and Applications in Discrete Mathematics, 18 (2017), 71-86
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P. E. Hydon , Difference Equations by Differential Equation Methods, Cambridge University Press, Cambridge (2014)
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Y. Ibrahim , On the Global Attractivity of Positive Solutions of a Rational Difference Equation , Selcuk J. Appl. Math., 9 (2008), 3-8
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T. F. Ibrahim, On the Third Order Rational Difference Equation \(x_{n+1} = x_nx_{n-2}/x_{n-1}(a+bx_nx_{n-2})\), Int. J. Contemp. Math. Sci., 27 (2009), 1321-1334
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D. Levi, L. Vinet, P. Winternitz, Lie group Formalism for Difference Equations , J. Phys. A, 30 (1997), 633-649
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G. R. W. Quispel, R. Sahadevan , Lie Symmetries and the Integration of Difference Equations, Phys. Lett. A, 184 (1993), 64-70
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X. J. Yang, F. Gao, H. M. Srivasta, Exact travelling wave solutions for local fractional two-dimensional Burgers-type equations, Comput. Math. Appl., 26 (2017), 203-210
##[22]
X. J. Yang, J. A. T. Machado, D. Baleanu, Exact traveling-wave solution for local fractional Boussinesq equation in fractal domain, Fractals, 25 (2017), 1-7
##[23]
X. J. Yang, J. A. T. Machado, D. Baleanu, C. Cattani, On exact traveling-wave solutions for local fractional Korteweg-de Vries equation, Chaos, 26 (2016), 1-5
]
The endpoint Fefferman-Stein inequality for the strong maximal function with respect to nondoubling measure
The endpoint Fefferman-Stein inequality for the strong maximal function with respect to nondoubling measure
en
en
Let \(d\mu(x_1, \ldots, x_n)=d\mu_1(x_1)\cdots d\mu_n(x_n)\) be a
product measure which is not necessarily doubling in
\(\mathbb{R}^n\) (only assuming \(d\mu_i\) is doubling on \(\mathbb{R}\)
for \(i=2, \ldots, n\)), and \(M_{d\mu}^n\) be the strong maximal function defined by
\[ M_{d\mu}^n f(x)=\sup_{x\in R\in \mathcal{R}}\frac{1}{\mu(R)}\int_{R}|f(y)|d\mu(y),\]
where \(\mathcal{R}\) is the collection of rectangles with sides
parallel to the coordinate axes in \(\mathbb{R}^n\), and \(\omega,\nu\) are two nonnegative functions. We
give a sufficient condition on \(\omega,\nu\) for which the operator \(M_{d\mu}^n\) is bounded from \(L(1+(\log^{+})^{n-1})(\nu d\mu)\) to \(L^{1,\infty}(\omega d\mu)\). By interpolation,
\(M^{n}_{d\mu}\) is bounded from \(L^{p}(\nu d\mu)\) to \(L^{p}(\omega d\mu)\), \(1<p<\infty\).
1271
1281
Wei
Ding
School of Sciences
Nantong University
P. R. China
dingwei@ntu.edu.cn
Fefferman-Stein inequality
strong maximal function
nondoubling measure
\(A^\infty\) weights
reverse Holder's inequality
Article.7.pdf
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An accurate numerical method for solving the generalized time-fractional diffusion equation
An accurate numerical method for solving the generalized time-fractional diffusion equation
en
en
In this paper, a formulation for the fractional Legendre functions is
constructed to solve a class of time-fractional diffusion equation. The
fractional derivative is described in the Caputo sense. The method is based
on the collection Legendre. Analysis for the presented method is given and
numerical results are presented.
1282
1293
Muhammed
Syam
Department of Mathematical Sciences
United Arab Emirates University
UAE
m.syam@uaeu.ac.ae
Ibrahim
Al-Subaihi
Department of Mathematics
Taibah University
Saudi Arabia
Fractional-order Legendre function
collocation method
generalized time-fractional diffusion equation
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