1,992 results on '"Convergent series"'
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2. The ideal test for the divergence of a series.
- Author
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Filipów, Rafał, Kwela, Adam, and Tryba, Jacek
- Abstract
We generalize the classical Olivier’s theorem which says that for any convergent series ∑ n a n with positive nonincreasing real terms the sequence (n a n) tends to zero. Our results encompass many known generalizations of Olivier’s theorem and give some new instances. The generalizations are done in two directions: we either drop the monotonicity assumption completely or we relax it to the monotonicity on a large set of indices. In both cases, the convergence of (n a n) is replaced by ideal convergence. In the second part of the paper, we examine families of sequences for which the assertions of our generalizations of Olivier’s theorem fail. Here, we are interested in finding large linear and algebraic substructures in these families. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
3. Presentation of Khayyamian Thing in Materialistic Re-Configurations
- Author
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sajad mombeini
- Subjects
khayyam ,materialism ,convergent series ,khayyamian thing ,conversion and transmission ,Philosophy (General) ,B1-5802 - Abstract
Reflecting on existence as a linguistic practice has always been one of the sublime features of literature. This issue has had a significant appearance in Khayyam, so that Khayyam's work can be considered as a kind of problematic project of questioning existence. In the current research, it was tried to achieve new ontological perceptions of Khayyamian Thing with a materialistic approach and from a post-structuralist perspective. Here, identifying a convergent series of objects on one hand, and proposing the Conversion and Transmission idea as an underlying mechanism in Khayyam on the other hand, prepare strong analytical possibilities for new ontological formulations of Khayyamian Thing. Designing the Conversion and Transmission idea showed that the convergent series of objects are continuously connected with each other, and by producing new ratios, it displays a process of physical inner expansion. Finally, by theoretical analysis of these extended and expanding objects’ relations, four ontological ideas including inexhaustibility of existence, the object having memory, inner-expansion of desire (Oedipal economy of desire), and fetishism, as the result of the symptomatic encounter with Khayyam, are presented.
- Published
- 2022
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4. Approximate solution for the nonlinear fractional order mathematical model
- Author
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Kahkashan Mahreen, Qura Tul Ain, Gauhar Rahman, Bahaaeldin Abdalla, Kamal Shah, and Thabet Abdeljawad
- Subjects
adomian decomposition method ,convergent series ,laplace transformation ,caputo fractional derivative ,homotopy perturbation method ,invariant set ,Mathematics ,QA1-939 - Abstract
Health organizations are working to reduce the outbreak of infectious diseases with the help of several techniques so that exposure to infectious diseases can be minimized. Mathematics is also an important tool in the study of epidemiology. Mathematical modeling presents mathematical expressions and offers a clear view of how variables and interactions between variables affect the results. The objective of this work is to solve the mathematical model of MERS-CoV with the simplest, easiest and most proficient techniques considering the fractional Caputo derivative. To acquire the approximate solution, we apply the Adomian decomposition technique coupled with the Laplace transformation. Also, a convergence analysis of the method is conducted. For the comparison of the obtained results, we apply another semi-analytic technique called the homotopy perturbation method and compare the results. We also investigate the positivity and boundedness of the selected model. The dynamics and solution of the MERS-CoV compartmental mathematical fractional order model and its transmission between the human populace and the camels are investigated graphically for θ=0.5,0.7,0.9,1.0. It is seen that the recommended schemes are proficient and powerful for the given model considering the fractional Caputo derivative.
- Published
- 2022
- Full Text
- View/download PDF
5. Airy Functions Demystified — III: A Fresh Look at the Relation Between Airy and Bessel Functions.
- Author
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Ramkarthik, M. S. and Pereira, Elizabeth Louis
- Subjects
AIRY functions ,BESSEL functions ,ASYMPTOTIC expansions ,SPECIAL functions ,MATHEMATICS - Abstract
Airy and Bessel functions are one of the most popular and important special functions in various branches of physics, mathematics, and engineering. An observation to their behavior for the real argument suggest that they are related. This relation was studied earlier, but were accompanied by a number of assumptions, approximations, and sometimes even misconceptions. This motivated us to develop a fresh and transparent method to establish these relations. As the continuation of our study of the two papers published in resonance already, here we have used the general asymptotic series and the convergent series of these functions and thereby developed two new methods which throw light on the subtle interrelationships between these functions. Numerical evidences of our claims are provided for better clarity and understanding. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
6. Approximate solution for the nonlinear fractional order mathematical model.
- Author
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Mahreen, Kahkashan, Ain, Qura Tul, Rahman, Gauhar, Abdalla, Bahaaeldin, Shah, Kamal, and Abdeljawad, Thabet
- Subjects
NONLINEAR analysis ,APPROXIMATION theory ,CAPUTO fractional derivatives ,MATHEMATICAL bounds ,LAPLACE transformation ,SET theory - Abstract
Health organizations are working to reduce the outbreak of infectious diseases with the help of several techniques so that exposure to infectious diseases can be minimized. Mathematics is also an important tool in the study of epidemiology. Mathematical modeling presents mathematical expressions and offers a clear view of how variables and interactions between variables affect the results. The objective of this work is to solve the mathematical model of MERS-CoV with the simplest, easiest and most proficient techniques considering the fractional Caputo derivative. To acquire the approximate solution, we apply the Adomian decomposition technique coupled with the Laplace transformation. Also, a convergence analysis of the method is conducted. For the comparison of the obtained results, we apply another semi-analytic technique called the homotopy perturbation method and compare the results. We also investigate the positivity and boundedness of the selected model. The dynamics and solution of the MERS-CoV compartmental mathematical fractional order model and its transmission between the human populace and the camels are investigated graphically for ϑ = 0:5; 0:7; 0:9; 1:0. It is seen that the recommended schemes are proficient and powerful for the given model considering the fractional Caputo derivative. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
7. Minimality criteria for convergent power series over Zp and rational maps with good reduction on the projective line over Qp.
- Author
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Jeong, Sangtae, Ko, Dohyun, Kwon, Yongjae, and Kwon, Youngwoo
- Subjects
- *
POWER series - Abstract
In this paper, we first characterize the minimality criterion for a convergent power series f on Z p in terms of its coefficients for the cases p = 2 or 3. For an arbitrary prime p ≥ 5 , the minimality criterion of such a series can be obtained explicitly provided that the prescribed minimal conditions for the reduction of f modulo p are found. Second, we provide the minimality criterion for a rational map of at least degree 2 with good reduction on the projective line P 1 ( Q p) over Q p . This criterion enables us to obtain a complete description of minimal conditions for such a map on P 1 ( Q p) in terms of its coefficients for p = 2 or 3. For an arbitrary prime p ≥ 5 , we present a method of characterizing minimal rational maps ϕ of degree ≥ 2 on P 1 ( Q p) , provided that the prescribed conditions for the reduction of ϕ on P 1 (F p) to be transitive are known. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
8. Minimality criteria for convergent power series over Zp and rational maps with good reduction on the projective line over Qp.
- Author
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Jeong, Sangtae, Ko, Dohyun, Kwon, Yongjae, and Kwon, Youngwoo
- Subjects
POWER series - Abstract
In this paper, we first characterize the minimality criterion for a convergent power series f on Z p in terms of its coefficients for the cases p = 2 or 3. For an arbitrary prime p ≥ 5 , the minimality criterion of such a series can be obtained explicitly provided that the prescribed minimal conditions for the reduction of f modulo p are found. Second, we provide the minimality criterion for a rational map of at least degree 2 with good reduction on the projective line P 1 ( Q p) over Q p . This criterion enables us to obtain a complete description of minimal conditions for such a map on P 1 ( Q p) in terms of its coefficients for p = 2 or 3. For an arbitrary prime p ≥ 5 , we present a method of characterizing minimal rational maps ϕ of degree ≥ 2 on P 1 ( Q p) , provided that the prescribed conditions for the reduction of ϕ on P 1 (F p) to be transitive are known. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
9. Convergent subseries of divergent series.
- Author
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Balcerzak, Marek and Leonetti, Paolo
- Abstract
Let X be the set of positive real sequences x = (x n) such that the series ∑ n x n is divergent. For each x ∈ X , let I x be the collection of all A ⊆ N such that the subseries ∑ n ∈ A x n is convergent. Moreover, let A be the set of sequences x ∈ X such that lim n x n = 0 and I x ≠ I y for all sequences y = (y n) ∈ X with lim inf n y n + 1 / y n > 0 . We show that A is comeager and that contains uncountably many sequences x which generate pairwise nonisomorphic ideals I x . This answers, in particular, an open question recently posed by M. Filipczak and G. Horbaczewska. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
10. The limit of reciprocal sum of some subsequential Fibonacci numbers
- Author
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Ho-Hyeong Lee and Jong-Do Park
- Subjects
fibonacci number ,reciprocal sum ,floor function ,convergent series ,catalan's identity ,Mathematics ,QA1-939 - Abstract
This paper deals with the sum of reciprocal Fibonacci numbers. Let $ f_0 = 0 $, $ f_1 = 1 $ and $ f_{n+1} = f_n+f_{n-1} $ for any $ n\in\mathbb{N} $. In this paper, we prove new estimates on $ \sum\limits^\infty_{k = n}\frac{1}{f_{mk-\ell}} $, where $ m\in\mathbb{N} $ and $ 0\leq\ell\leq m-1 $. As a consequence of some inequalities, we prove $ \lim\limits_{n\rightarrow \infty}\left\{\left(\sum\limits^\infty_{k = n}\frac{1}{f_{mk-\ell}} \right)^{-1} -(f_{mn-\ell}-f_{m(n-1)-\ell})\right\} = 0. $ And we also compute the explicit value of $ \left\lfloor\left(\sum\limits^\infty_{k = n}\frac{1}{f_{mk-\ell}}\right)^{-1}\right\rfloor $. The interesting observation is that the value depends on $ m(n+1)+\ell $.
- Published
- 2021
- Full Text
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11. Small divisors in discrete local holomorphic dynamics
- Author
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Reppekus, Josias
- Published
- 2023
- Full Text
- View/download PDF
12. Approximate analytical solution of one dimensional nonlinear Burger's equation using Homotopy Perturbation method.
- Author
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Lal, Diwari and Yadav, Manoj
- Subjects
- *
HOMOTOPY theory , *PERTURBATION theory , *ANALYTICAL solutions , *BURGERS' equation , *TOPOLOGY - Abstract
As a solution to the nonlinear Burger's issue in a single dimension, the homotopy perturbation method (HPM) is what we recommend utilizing according to the findings of this research. We have reached a series solution of the equations in terms of convergent series with easily computable components thanks to the fact that the nonlinear elements of Burger's equations may be addressed by using the homotopy perturbation approach (HPM). HPM is closely associated with the concept of the sum of an infinite series term, which frequently and rapidly converges to the correct response. The intricate equation is simplified into a more understandable form by using the HPM. According to the data that was gathered, the method that was recommended performs better than the state-of-the-art solutions for similar PDEs, and it is also much easier to put into practice. [ABSTRACT FROM AUTHOR]
- Published
- 2022
13. Asymptotic behavior of reciprocal sum of two products of Fibonacci numbers
- Author
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Ho-Hyeong Lee and Jong-Do Park
- Subjects
Fibonacci number ,Reciprocal sum ,Catalan’s identity ,Convergent series ,Mathematics ,QA1-939 - Abstract
Abstract Let { f k } k = 1 ∞ $\{f_{k} \} _{k=1}^{\infty}$ be a Fibonacci sequence with f 1 = f 2 = 1 $f_{1}=f_{2}=1$ . In this paper, we find a simple form g n $g_{n}$ such that lim n → ∞ { ( ∑ k = n ∞ a k ) − 1 − g n } = 0 , $$\lim_{n\rightarrow\infty} \Biggl\{ \Biggl(\sum^{\infty}_{k=n}{a_{k}} \Biggr)^{-1}-g_{n} \Biggr\} =0, $$ where a k = 1 f k 2 $a_{k}=\frac{1}{f_{k}^{2}}$ , 1 f k f k + m $\frac{1}{f_{k}f_{k+m}}$ , or 1 f 3 k 2 $\frac{1}{f_{3k}^{2}}$ . For example, we show that lim n → ∞ { ( ∑ k = n ∞ 1 f 3 k 2 ) − 1 − ( f 3 n 2 − f 3 n − 3 2 + 4 9 ( − 1 ) n ) } = 0 . $$\lim_{n\rightarrow\infty} \Biggl\{ \Biggl(\sum^{\infty}_{k=n}{ \frac {1}{f_{3k}^{2}}} \Biggr)^{-1}- \biggl(f_{3n}^{2}-f_{3n-3}^{2}+ \frac {4}{9}(-1)^{n} \biggr) \Biggr\} =0. $$
- Published
- 2020
- Full Text
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14. Tail probability and divergent series.
- Author
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Chou, Yu-Lin
- Subjects
- *
PROBABILITY theory , *NUMBER theory , *INTEGRAL functions , *MEASURE theory , *REAL numbers , *FINITE, The - Abstract
From mostly a measure-theoretic consideration, we show that for every nonnegative, finite, and L1 function on a given finite measure space there is some nontrivial sequence of real numbers such that the series, obtained from summing over the term-by-term products of the reals and the summands of any divergent series with positive, vanishing summands such as the harmonic series, is convergent and no greater than the integral of the function. In terms of inequalities, the implications add additional information on mathematical expectation and the behavior of divergent series with positive, vanishing summands, and establish in a broad sense some new, unexpected connections between probability theory and, for instance, number theory. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
15. A SERIES EXPANSION FOR THE b(s) BROUNCKER-RAMANUJAN FUNCTION.
- Author
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Alegri, Mateus
- Subjects
- *
POWER series , *FUNCTIONAL equations - Abstract
Our basic aim is to provide a power series representation for b(s), 0 < s < 3, the well-known function satisfying b(s - 1)b(s + 1) = s². We will do this by using integer compositions of n. In the last section, some properties involving the coefficients of sn in the power series expansion of b(s) are given, as well an expression for 4/π. [ABSTRACT FROM AUTHOR]
- Published
- 2021
16. IDEAL EXTENSIONS OF OLIVIER'S THEOREM.
- Author
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Mišík, Ladislav and Tóth, János T.
- Subjects
- *
REAL numbers - Abstract
Let p; q be given positive numbers and a; b non-negative ones. In this paper we study and characterize the class S(a; b; p; q) of all admissible ideals I ⊄ 2N with the following property... for all sequences (an) of positive real numbers. In a series of corollaries we discuss special cases including, also, several previously published theorems on this topic. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
17. Solving a nonhomogeneous integral equation with the variable lower limit
- Author
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M.T. Kosmakova, D.M. Akhmanova, Zh.M. Tuleutaeva, and L.Zh. Kasymova
- Subjects
nonhomogeneous singular integral equation ,auxiliary equation ,Laplace transform ,convergent series ,Analysis ,QA299.6-433 ,Analytic mechanics ,QA801-939 ,Probabilities. Mathematical statistics ,QA273-280 - Abstract
An nonhomogeneous integral equation with a singular kernel is considered. A feature of the equation under study is the incompressibility of the integral operator. In the study of the equation, an auxiliary simpler equation is used with the right - hand side equal to 1. The incompressibility of the integral operator for the equation under study is shown. Using the relations for an independent variable, the equation is equivalently reduced to a certain simplified equation. With the help of replacements for independent variables, the equation is reduced to an integral equation with a difference kernel. By applying the Laplace transform, the obtained equation is reduced to an ordinary first - order differential equation (linear). Its solution is found. By using the inverse Laplace transform, a solution of the auxiliary integral equation is obtained in the form of a convergent series in some domain. The solution of the initial equation with an arbitrary right - hand side is written through the solution of the auxiliary equation.
- Published
- 2019
- Full Text
- View/download PDF
18. Asymptotic behavior of reciprocal sum of two products of Fibonacci numbers.
- Author
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Lee, Ho-Hyeong and Park, Jong-Do
- Subjects
- *
RECIPROCALS (Mathematics) , *FIBONACCI sequence , *MANUFACTURED products , *BEHAVIOR - Abstract
Let { f k } k = 1 ∞ be a Fibonacci sequence with f 1 = f 2 = 1 . In this paper, we find a simple form g n such that lim n → ∞ { (∑ k = n ∞ a k) − 1 − g n } = 0 , where a k = 1 f k 2 , 1 f k f k + m , or 1 f 3 k 2 . For example, we show that lim n → ∞ { (∑ k = n ∞ 1 f 3 k 2) − 1 − (f 3 n 2 − f 3 n − 3 2 + 4 9 (− 1) n) } = 0. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
19. Cauchy and Continuity
- Author
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Gray, Jeremy, Chaplain, M.A.J., Series editor, Erdmann, K., Series editor, MacIntyre, Angus, Series editor, Süli, Endre, Series editor, Tehranchi, M R, Series editor, Toland, J.F., Series editor, and Gray, Jeremy
- Published
- 2015
- Full Text
- View/download PDF
20. Uniform Convergence
- Author
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Gray, Jeremy, Chaplain, M.A.J., Series editor, Erdmann, K., Series editor, MacIntyre, Angus, Series editor, Süli, Endre, Series editor, Tehranchi, M R, Series editor, Toland, J.F., Series editor, and Gray, Jeremy
- Published
- 2015
- Full Text
- View/download PDF
21. Series
- Author
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Pedersen, Steen and Pedersen, Steen
- Published
- 2015
- Full Text
- View/download PDF
22. Sequences and Series
- Author
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Montesinos, Vicente, Zizler, Peter, Zizler, Václav, Montesinos, Vicente, Zizler, Peter, and Zizler, Václav
- Published
- 2015
- Full Text
- View/download PDF
23. Solutions for time-fractional coupled nonlinear Schrödinger equations arising in optical solitons
- Author
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Emamuzo N. Okposo, Newton I. Okposo, and P. Veeresha
- Subjects
symbols.namesake ,Nonlinear system ,State variable ,Homotopy ,symbols ,General Physics and Astronomy ,Applied mathematics ,Uniqueness ,Fixed point ,System of linear equations ,Convergent series ,Schrödinger equation ,Mathematics - Abstract
In this work, an efficient novel technique, namely, the q -homotopy analysis transform method ( q -HATM) is applied to obtain analytical solutions for a system of time-fractional coupled nonlinear Schrodinger (TF-CNLS) equations with the time-fractional derivative taken in the Caputo sense. This system of equations incorporate nonlocality behaviours which cannot be modeled under the framework of classical calculus. With numerous important applications in nonlinear optics, it describes interactions between waves of different frequencies or the same frequency but belonging to different polarizations. We first establish existence and uniqueness of solutions for the considered time-fractional problem via a fixed point argument. To demonstrate the effectiveness and efficiency of the q − HATM, two cases each of two time-fractional problems are considered. One important feature of the q − HATM is that it provides reliable algorithms which can be used to generate easily computable solutions for the considered problems in the form of rapidly convergent series. Numerical simulation are provided to capture the behaviour of the state variables for distinct values of the fractional order parameter. The results demonstrate that the general response expression obtained by the q − HATM contains the fractional order parameter which can be varied to obtain other responses. Particularly, as this parameter approaches unity, the responses obtained for the considered fractional equations approaches that of the corresponding classical equations.
- Published
- 2022
24. Analytical and approximate solutions of nonlinear Schrödinger equation with higher dimension in the anomalous dispersion regime
- Author
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Lanre Akinyemi, Mohamed S. Osman, and Mehmet Şenol
- Subjects
Physics ,Environmental Engineering ,Laplace transform ,Mathematical analysis ,Ocean Engineering ,Oceanography ,symbols.namesake ,Dimension (vector space) ,Riccati equation ,symbols ,Soliton ,Representation (mathematics) ,Nonlinear Schrödinger equation ,Analysis method ,Convergent series - Abstract
The generalized Riccati equation mapping method (GREMM) is used in this paper to obtain different types of soliton solutions for nonlinear Schrodinger equation with higher dimension that existed in the regimes of anomalous dispersion. Later, we use the q-homotopy analysis method combined with the Laplace transform (q-HATM) to obtain approximate solutions of the bright and dark optical solitons. The q-HATM illustrates the solutions as a rapid convergent series. In addition, to show the physical behavior of the solutions obtained by the proposed techniques, the graphical representation has been provided with some parameter values. The findings demonstrate that the proposed techniques are useful, efficient and reliable mathematical method for the extraction of soliton solutions.
- Published
- 2022
25. An efficient hybrid computational technique for the time dependent Lane-Emden equation of arbitrary order
- Author
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Amit Prakash, Manish Goyal, and Dumitru Baleanu
- Subjects
Nonlinear system ,Environmental Engineering ,Laplace transform ,Linearization ,Convergence (routing) ,Applied mathematics ,Ocean Engineering ,Uniqueness ,Lane–Emden equation ,Oceanography ,Convergent series ,Mathematics ,Fractional calculus - Abstract
The study of dynamic behaviour of nonlinear models that arise in ocean engineering play a vital role in our daily life. There are many examples of ocean water waves which are nonlinear in nature. In shallow water, the linearization of the equations imposes severe conditions on wave amplitude than it does in deep water, and the strong nonlinear effects are observed. In this paper, q-homotopy analysis Laplace transform scheme is used to inspect time dependent nonlinear Lane-Emden type equation of arbitrary order. It offers the solution in a fast converging series. The uniqueness and convergence analysis of the considered model is presented. The given examples confirm the competency as well as accuracy of the presented scheme. The behavior of obtained solution for distinct orders of fractional derivative is discussed through graphs. The auxiliary parameter ħ offers a suitable mode of handling the region of convergence. The outcomes reveal that the q-HATM is attractive, reliable, efficient and very effective.
- Published
- 2022
26. Some generalizations of Olivier's theorem
- Author
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Alain Faisant, Georges Grekos, and Ladislav Mišík
- Subjects
convergent series ,Olivier's theorem ,ideal ,$\mathcal{I}$-convergence ,$\mathcal{I}$-monotonicity ,Mathematics ,QA1-939 - Abstract
Let $\sum\limits_{n=1}^\infty a_n$ be a convergent series of positive real numbers. L. Olivier proved that if the sequence $(a_n)$ is non-increasing, then $\lim\limits_{n \to\infty} n a_n = 0$. In the present paper: (a) We formulate and prove a necessary and sufficient condition for having $\lim\limits_{n \to\infty} n a_n = 0$; Olivier's theorem is a consequence of our Theorem \ref{import}. (b) We prove properties analogous to Olivier's property when the usual convergence is replaced by the $\mathcal I$-convergence, that is a convergence according to an ideal $\mathcal I$ of subsets of $\mathbb N$. Again, Olivier's theorem is a consequence of our Theorem \ref{Iol}, when one takes as $\mathcal I$ the ideal of all finite subsets of $\mathbb N$.
- Published
- 2016
- Full Text
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27. Miscellanea
- Author
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Gander, Martin J., Kung, Joseph P. S., Series editor, Brezinski, Claude, editor, and Sameh, Ahmed, editor
- Published
- 2014
- Full Text
- View/download PDF
28. Rate of Convergence of Multi-Indexed Series
- Author
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Klesov, Oleg, Asmussen, Soren, Editor-in-chief, Glynn, Peter W., Editor-in-chief, Kurtz, Thomas G., Editor-in-chief, Le Jan, Yves, Editor-in-chief, Gani, Joe, Series editor, Hairer, Martin, Series editor, Jagers, Peter, Series editor, Karatzas, Ioannis, Series editor, Kelly, Frank P, Series editor, Kyprianou, Andreas E., Series editor, Øksendal, Bernt, Series editor, Papanicolaou, George, Series editor, Pardoux, Etienne, Series editor, Perkins, Edwin, Series editor, Soner, Halil Mete, Series editor, and Klesov, Oleg
- Published
- 2014
- Full Text
- View/download PDF
29. Numerical Series
- Author
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Choudary, A. D. R., Niculescu, Constantin P., Choudary, A. D. R., and Niculescu, Constantin P.
- Published
- 2014
- Full Text
- View/download PDF
30. On fractional calculus with general analytic kernels.
- Author
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Fernandez, Arran, Özarslan, Mehmet Ali, and Baleanu, Dumitru
- Subjects
- *
FRACTIONAL calculus , *FRACTIONAL differential equations - Abstract
Abstract Many possible definitions have been proposed for fractional derivatives and integrals, starting from the classical Riemann–Liouville formula and its generalisations and modifying it by replacing the power function kernel with other kernel functions. We demonstrate, under some assumptions, how all of these modifications can be considered as special cases of a single, unifying, model of fractional calculus. We provide a fundamental connection with classical fractional calculus by writing these general fractional operators in terms of the original Riemann–Liouville fractional integral operator. We also consider inversion properties of the new operators, prove analogues of the Leibniz and chain rules in this model of fractional calculus, and solve some fractional differential equations using the new operators. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
31. ГРАНИЧНА ВРЕДНОСТ НИЗОВА.
- Author
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Петровић, Марија
- Subjects
PROPERTY - Abstract
Copyright of MAT-KOL (Banja Luka), Matematicki Kolokvijum is the property of Scientific Society of Mathematicians Banja Luka and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
- Published
- 2019
- Full Text
- View/download PDF
32. Effective numerical technique for nonlinear Caputo-Fabrizio systems of fractional Volterra integro-differential equations in Hilbert space
- Author
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Fatima Youbi, Shaher Momani, Shatha Hasan, and Mohammed Al-Smadi
- Subjects
Differential equation ,Computer science ,Caputo-Fabrizio operator ,Numerical solution ,General Engineering ,Hilbert space ,Stability analysis ,Engineering (General). Civil engineering (General) ,Domain (mathematical analysis) ,Iterative reproducing kernel algorithm ,Fractional integro-differntial equations ,Nonlinear system ,symbols.namesake ,Kernel (statistics) ,symbols ,Applied mathematics ,Orthonormal basis ,TA1-2040 ,Fourier series ,Convergent series - Abstract
The point of this paper is to analyze and investigate the analytic-approximate solutions for fractional system of Volterra integro-differential equations in framework of Caputo-Fabrizio operator. The methodology relies on creating the reproducing kernel functions to gain analytical solutions in a uniform form of a rapidly convergent series in the Hilbert space. Using the Gram-Schmidt orthonomalization process, the orthonormal basis system is constructed in a dense compact domain to encompass the Fourier series expansion in view of reproducing kernel properties. Besides, convergence and error analysis of the proposed technique are discussed. For this purpose, several numerical examples are tested to demonstrate the great feasibility and efficiency of the present method and to support theoretical aspect as well. From a numerical point of view, the acquired solutions simulation indicates that the methodology used is sound, straightforward, and appropriate to deal with many physical issues in light of Caputo-Fabrizio derivatives.
- Published
- 2022
33. Application of Laplace residual power series method for approximate solutions of fractional IVP’s
- Author
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Mohammad Alaroud
- Subjects
Power series ,Coupling ,Laplace transform ,020209 energy ,General Engineering ,Fractional power series ,02 engineering and technology ,Engineering (General). Civil engineering (General) ,Residual ,01 natural sciences ,010305 fluids & plasmas ,Operator (computer programming) ,Fractional initial value problems ,0103 physical sciences ,0202 electrical engineering, electronic engineering, information engineering ,Caputo’s derivative operator ,Initial value problem ,Applied mathematics ,Limit (mathematics) ,TA1-2040 ,Laplace residual power series ,Convergent series ,Mathematics - Abstract
In this study, different systems of linear and non-linear fractional initial value problems are solved analytically utilizing an attractive novel technique so-called the Laplace residual power series approach, and which is based on the coupling of the residual power series approach with the Laplace transform operator to generate analytical and approximate solutions in fast convergent series forms by using the concept of the limit with less time and effort compared with the residual power series technique. To confirm the simplicity, performance, and viability of the proposed technique, three problems are tested and simulated. Analysis of the obtained results reveals that the aforesaid technique is straightforward, accurate, and suitable to investigate the solutions of the non-linear physical and engineering problems.
- Published
- 2022
34. Approximate Analytical Solutions of Space-Fractional Telegraph Equations by Sumudu Adomian Decomposition Method.
- Author
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Khan, Hasib, Tunç, Cemil, Khan, Rahmat Ali, Shirzoi, Akhtyar Gul, and Khan, Aziz
- Subjects
- *
DECOMPOSITION method , *ANALYTICAL solutions , *TELEGRAPH & telegraphy , *EQUATIONS , *CAPUTO fractional derivatives - Abstract
The main goal in this work is to establish a new and efficient analytical scheme for space fractional telegraph equation (FTE) by means of fractional Sumudu decomposition method (SDM). The fractional SDM gives us an approximate convergent series solution. The stability of the analytical scheme is also studied. The approximate solutions obtained by SDM show that the approach is easy to implement and computationally very much attractive. Further, some numerical examples are presented to illustrate the accuracy and stability for linear and nonlinear cases. [ABSTRACT FROM AUTHOR]
- Published
- 2018
35. Mathematical modeling of tsunami wave propagation at mid ocean and its amplification and run-up on shore
- Author
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Archana C. Varsoliwala and Twinkle Singh
- Subjects
Shore ,Elzaki Adomian Decomposition Method ,geography ,Environmental Engineering ,Partial differential equation ,Tsunami wave ,geography.geographical_feature_category ,Perturbation (astronomy) ,Geophysics ,Oceanography ,Tsunami wave propagation ,Physics::Geophysics ,Ocean engineering ,Shallow water equations ,Linearization ,Convergence analysis ,Adomian decomposition method ,TC1501-1800 ,Geology ,Convergent series ,Physics::Atmospheric and Oceanic Physics ,Run-up height - Abstract
The paper deals with the study of the mathematical model of tsunami wave propagation along a coastline of an ocean. The model is based on shallow-water assumption which is represented by a system of non-linear partial differential equations. In this study, we employ the Elzaki Adomian Decomposition Method (EADM) to successfully obtain the solution for the proposed model for different coastal slopes and ocean depths. How tsunami wave velocity and run-up height are affected by the coast slope and sea depth are demonstrated. The Adomian Decomposition Method together with Elzaki transform allows for solutions, without the need of any linearization or perturbation, in the form of rapidly converging series. The obtained numerical results for tsunami wave height and velocity are very close match to the real physical phenomenon of tsunami.
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- 2021
36. Square-Law Detection of Exponential Targets in Weibull-Distributed Ground Clutter
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Fernando Dario Almeida Garcia, Jose Candido Silveira Santos Filho, Henry Ramiro Carvajal Mora, and Gustavo Fraidenraich
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Cumulative distribution function ,Monte Carlo method ,Residue theorem ,Cauchy distribution ,Clutter ,Probability density function ,Statistical physics ,Electrical and Electronic Engineering ,Geotechnical Engineering and Engineering Geology ,Convergent series ,Weibull distribution ,Mathematics - Abstract
Modern radar systems use square-law detectors to search and track fluctuating targets embedded in Weibull-distributed ground clutter. However, the theoretical performance analysis of square-law detectors in the presence of Weibull clutter leads to cumbersome mathematical formulations. Some studies have circumvented this problem by using approximations or mathematical artifacts to simplify calculations. In this work, we derive a closed-form and exact expression for the probability of detection (PD) of a square-law detector in the presence of exponential targets and Weibull-distributed ground clutter, given in terms of the Fox H-function. Unlike previous studies, no approximations nor simplifying assumptions are made throughout our analysis. Furthermore, we derive a fast convergent series for the referred PD by exploiting the orthogonal selection of poles in Cauchy’s residue theorem. In passing, we also obtain closed-form solutions and series representations for the probability density function and the cumulative distribution function of the sum statistics that govern the output of a square-law detector. Numerical results and Monte Carlo simulations corroborate the validity of our expressions.
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- 2021
37. Macphail’s theorem revisited
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Janiely Silva and Daniel Pellegrino
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Combinatorics ,Mathematics::Functional Analysis ,Sequence ,Constructive proof ,Series (mathematics) ,General Mathematics ,Banach space ,Convergent series ,Mathematics - Abstract
In 1947, M.S. Macphail constructed a series in $$\ell _{1}$$ that converges unconditionally but does not converge absolutely. According to the literature, this result helped Dvoretzky and Rogers to finally answer a long standing problem of Banach space theory, by showing that in all infinite-dimensional Banach spaces, there exists an unconditionally summable sequence that fails to be absolutely summable. More precisely, the Dvoretzky–Rogers theorem asserts that in every infinite-dimensional Banach space E, there exists an unconditionally convergent series $$\sum x^{\left( j\right) }$$ such that $$\sum \Vert x^{(j)}\Vert ^{2-\varepsilon }=\infty $$ for all $$\varepsilon >0$$ . Their proof is non-constructive and Macphail’s result for $$E=\ell _{1}$$ provides a constructive proof just for $$\varepsilon \ge 1$$ . In this note, we revisit Macphail’s paper and present two alternative constructions that work for all $$\varepsilon >0.$$
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- 2021
38. A Novel Attractive Algorithm for Handling Systems of Fractional Partial Differential Equations
- Author
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Mohammad Alaroud and Yousef Al-Qudah
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Power series ,Partial differential equation ,Series (mathematics) ,Laplace transform ,General Mathematics ,Initial value problem ,Limit (mathematics) ,Algorithm ,Convergent series ,Mathematics ,Fractional calculus - Abstract
The purpose of this work is to provide and analyzed the approximate analytical solutions for certain systems of fractional initial value problems (FIVPs) under the time-Caputo fractional derivatives by means of a novel attractive algorithm, called the Laplace residual power series (LRPS) algorithm. It combines the Laplace transform operator and the RPS algorithm. The proposed algorithm produces the fractional series solutions in the Laplace space based upon basically on the limit concept and then transforming bake them to original spaces to get a rapidly convergent series approximate solution. To validate the efficiency, accuracy, and applicability of the proposed algorithm, two illustrative examples are performed. Obtained solutions are simulated graphically and numerically. The analysis of results reached shows that the proposed algorithm is applicable, effective, and very fast in determining the solutions for many fractional problems arising in the various areas of applied mathematics
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- 2021
39. Solitary Wave Solutions with Compact Support for The Nonlinear Dispersive K(m,n) Equations by Using Approximate Analytical Method
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Adem Kilicman, Amirah Azmi, and Che Haziqah Che Hussin
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Fluid Flow and Transfer Processes ,Differential transform method ,Sequence ,Nonlinear system ,Differential transformation ,Nonlinear physics ,Initial value problem ,Applied mathematics ,Term (logic) ,Convergent series ,Mathematics - Abstract
The study of solitons and compactons is important in nonlinear physics. In this paper we combined the Adomian polynomials with the multi-step approach to present a new technique called Multi-step Modified Reduced Differential Transform Method (MMRDTM). The proposed technique has the advantage of producing an analytical approximation in a fast converging sequence with a reduced number of calculated terms. The MMRDTM is presented with some modification of the Reduced Differential Transformation Method (RDTM) with multi-step approach and its nonlinear term is replaced by the Adomian polynomials. Therefore, the nonlinear initial value problem can easily be solved with less computational effort. Besides that, the multi-step approach produces a solution in fast converging series that converges the solution in a wide time area. Two examples are provided to demonstrate the capability and benefits of the proposed method for approximating the solution of NKdVEs with compactons. Graphical inputs are used to represent the solution and to demonstrate the precision and validity of the MMRDTM in graphic illustration. From the results, it was found that it is possible to obtain highly accurate results or exact solutions by using the MMRDTM.
- Published
- 2021
40. Chapter 3 Cauchy’s 'Modern Analysis'
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Bottazzini, Umberto, Gray, Jeremy, Bottazzini, Umberto, and Gray, Jeremy
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- 2013
- Full Text
- View/download PDF
41. Sequences and Series
- Author
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Mahmudov, E. and Mahmudov, Elimhan
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- 2013
- Full Text
- View/download PDF
42. Uniform Convergence of Sequences of Functions
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Ponnusamy, S. and Ponnusamy, S.
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- 2012
- Full Text
- View/download PDF
43. Flow of Eyring-Powell liquid due to oscillatory stretchable curved sheet with modified Fourier and Fick’s model
- Author
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Zaheer Abbas, Muhammad Imran, and M. Naveed
- Subjects
Physics ,Partial differential equation ,Applied Mathematics ,Mechanical Engineering ,Prandtl number ,Mathematical analysis ,symbols.namesake ,Nonlinear system ,Fourier transform ,Flow (mathematics) ,Mechanics of Materials ,symbols ,Convergent series ,Homotopy analysis method ,Dimensionless quantity - Abstract
This study deals with the features of the mass and heat transport mechanism by adopting a modified version of Fourier and Fick’s model known as the Cattaneo-Christov double diffusive theory. The time-dependent magnetohydrodynamic (MHD) flow of the Eyring-Powell liquid across an oscillatory stretchable curved sheet in the presence of Fourier and Fick’s model is investigated. The acquired set of flow equations is transformed into the form of nonlinear partial differential equations (PDEs) by applying appropriate similarity variables. A convergent series solution to the developed nonlinear equations is accomplished with the help of an analytical approach, i.e., the homotopy analysis method (HAM). The consequences of diverse parameters, including the dimensionless Eyring-Powell liquid parameter, the radius of curvature, the Schmidt/Prandtl numbers, the ratio of the oscillatory frequency of the sheet to its stretchable rate constant, the mass and thermal relaxation variables involved in the flow, and the heat and mass properties, are displayed through graphs and tables. It is noted from this study that the amplitude of the pressure distribution rises for the high parametric values of the Eyring-Powell parameter.
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- 2021
44. Fractional Order Airy’s Type Differential Equations of Its Models Using RDTM
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Girma Gemechu, Diriba Gemechu, Daba Meshesha Gusu, and Dechasa Wegi
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Partial differential equation ,Article Subject ,Series (mathematics) ,Differential equation ,General Mathematics ,General Engineering ,Type (model theory) ,Engineering (General). Civil engineering (General) ,Ordinary differential equation ,QA1-939 ,Order (group theory) ,Applied mathematics ,TA1-2040 ,MATLAB ,computer ,Mathematics ,Convergent series ,computer.programming_language - Abstract
In this paper, we propose a novel reduced differential transform method (RDTM) to compute analytical and semianalytical approximate solutions of fractional order Airy’s ordinary differential equations and fractional order Airy’s and Airy’s type partial differential equations subjected to certain initial conditions. The performance of the proposed method was analyzed and compared with a convergent series solution form with easily computable coefficients. The behavior of approximated series solutions at different values of fractional order α and its modeling in 2-dimensional and 3-dimensional spaces are compared with exact solutions using MATLAB graphical method analysis. Moreover, the physical and geometrical interpretations of the computed graphs are given in detail within 2- and 3-dimensional spaces. Accordingly, the obtained approximate solutions of fractional order Airy’s ordinary differential equations and fractional order Airy’s and Airy’s type partial differential equations subjected to certain initial conditions exactly fit with exact solutions. Hence, the proposed method reveals reliability, effectiveness, efficiency, and strengthening of computed mathematical results in order to easily solve fractional order Airy’s type differential equations.
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- 2021
45. Increasing the Efficiency of Projection Models of Strip Lines
- Author
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A. N. Kovalenko and A. D. Yarlykov
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Power series ,Radiation ,Basis (linear algebra) ,Diagonal ,Mathematical analysis ,Condensed Matter Physics ,Chebyshev filter ,Projection (linear algebra) ,Electronic, Optical and Magnetic Materials ,Algebraic equation ,Matrix (mathematics) ,Electrical and Electronic Engineering ,Convergent series ,Mathematics - Abstract
—The matrix coefficients of projection models of strip lines obtained using the Chebyshev basis are presented as a sum of infinite slowly and rapidly converging series. Slowly converging series are summed up and transformed into rapidly converging power series. The diagonal character of the matrix of coefficients is established, which makes it possible to obtain an asymptotic solution of an infinite system of linear algebraic equations. Numerical results are presented that confirm the efficiency of using the obtained representations for slowly converging series when constructing projection models of strip lines.
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- 2021
46. An application of variational iteration method for solving fuzzy time-fractional diffusion equations
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Saurabh Kumar and Vikas Gupta
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Discretization ,Iterative method ,Fuzzy logic ,symbols.namesake ,Transformation (function) ,Artificial Intelligence ,Simple (abstract algebra) ,Lagrange multiplier ,symbols ,Applied mathematics ,Diffusion (business) ,Software ,Convergent series ,Mathematics - Abstract
In this paper, an approximate solution based on the variational iteration method is given to solve the fuzzy time-fractional diffusion equations. The time-fractional derivative is taken in the Caputo sense. In the variational iteration method, the solution appears as a convergent series with easily predictable terms. In this approach, the correctional functional is constructed and the Lagrange multiplier is identified optimally via variational theory. Some examples are also given to illustrate the performance and applicability of the proposed method for the discussed class of fuzzy time-fractional diffusion equations. To demonstrate the efficiency of the variational iteration method, comparisons have been made with the numerical solution obtained by the implicit finite difference scheme that exists in the literature. The proposed iterative algorithm is quite simple to use and does not require any discretization, transformation, or restrictive assumptions. Also, tabulated results show that the proposed algorithm gives better accuracy than the implicit finite difference scheme.
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- 2021
47. A Second Look at the Second Ratio Test
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Christopher N. B. Hammond and Edward Omey
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Discrete mathematics ,Class (set theory) ,General Mathematics ,Ratio test ,Convergent series ,Mathematics - Abstract
This article examines a class of series convergence tests, known as the mth ratio tests, that were introduced by Sayel A. Ali in 2008. We pay particular attention to the case where m = 2. We also c...
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- 2021
48. Analytical expressions for electrodynamic parameters of the shielded microstrip line
- Author
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A. N. Kovalenko and A. D. Yarlykov
- Subjects
Physics ,Power series ,high accuracy ,Information theory ,projection method ,Mathematical analysis ,Basis function ,quasi-static approximation ,chebyshev basis ,Microstrip ,Conductor ,microstrip line with a narrow strip conductor ,Quasistatic approximation ,wave impedance ,Line (geometry) ,General Earth and Planetary Sciences ,Wave impedance ,deceleration coefficient ,Q350-390 ,Convergent series ,rapidly converging power series ,General Environmental Science - Abstract
On the basis of an electrodynamic model of a screened microstrip line, built on the basis of the projection method using the Chebyshev basis, which explicitly takes into account the edge features of the field, a mathematical model of a microstrip line with a strip conductor was developed. The line width does not exceed the height of the substrate. In this case, the current density on the strip conductor is approximated by only one basis function. Analytical expressions are presented in the form of a sum of slowly and rapidly converging series to determine the main electrodynamic parameters of the line – wave resistance and deceleration coefficient. Due to logarithmic features, slowly converging series are summed up and transformed into rapidly converging power series. In addition, limit expressions in the form of improper integrals are given for the main electrodynamic parameters of an open microstrip line in the quasi-static approximation. Due to the logarithmic features, these integrals are also converted to rapidly converging power series. As a result, simple approximate formulas were obtained. They allow calculating the deceleration coefficient and wave impedance of the line with an error not exceeding 1%, when the width of the strip conductor is less than twice the thickness of the substrate. The results of calculating the electrodynamic parameters obtained on the basis of the developed mathematical model and on the basis of the projection method with an accuracy of up to 5 significant digits are presented. These results make it possible to establish the limits of applicability of the quasi-static approximation and to determine the error in calculating the deceleration coefficient and wave resistance using the obtained analytical expressions. The error does not exceed 0.1%, if the width of the strip conductor is less than twice the thickness of the substrate in a wide range of changes in the substrate dielectric constant and frequency.
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- 2021
49. Natural Transform along with HPM Technique for Solving Fractional ADE
- Author
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A. Gupta, D. L. Suthar, G. Agarwal, and N. Pareek
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Article Subject ,Discretization ,Physics ,QC1-999 ,Applied Mathematics ,Homotopy ,Computation ,General Physics and Astronomy ,02 engineering and technology ,Function (mathematics) ,021001 nanoscience & nanotechnology ,01 natural sciences ,010305 fluids & plasmas ,Fractional calculus ,0103 physical sciences ,Time derivative ,Initial value problem ,Applied mathematics ,0210 nano-technology ,Convergent series ,Mathematics - Abstract
The authors of this paper solve the fractional space-time advection-dispersion equation (ADE). In the advection-dispersion process, the solute movement being nonlocal in nature and the velocity of fluid flow being nonuniform, it leads to form a heterogeneous system which approaches to model the same by means of a fractional ADE which generalizes the classical ADE, where the time derivative is substituted through the Caputo fractional derivative. For the study of such fractional models, various numerical techniques are used by the researchers but the nonlocality of the fractional derivative causes high computational expenses and complex calculations so the challenge is to use an efficient method which involves less computation and high accuracy in solving such models numerically. Here, in order to get the FADE solved in the form of convergent infinite series, a novel method NHPM (natural homotopy perturbation method) is applied which couples Natural transform along with the homotopy perturbation method. The homotopy peturbation method has been applied in mathematical physics to solve many initial value problems expressed in the form of PDEs. Also, the HPM has an advantage over the other methods that it does not require any discretization of the domains, is independent of any physical parameters, and only uses an embedding parameter p ∈ 0 , 1 . The HPM combined with the Natural transform leads to rapidly convergent series solutions with less computation. The efficacy of the used method is shown by working out some examples for time-fractional ADE with various initial conditions using the NHPM. The Mittag-Leffler function is used to solve the fractional space-time advection-dispersion problem, and the impact of changing the fractional parameter α on the solute concentration is shown for all the cases.
- Published
- 2021
50. A NOTE ON THE AXISYMMETRIC DIFFUSION EQUATION
- Author
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Alexander E Patkowski
- Subjects
Diffusion equation ,Series (mathematics) ,Mathematical analysis ,Residue theorem ,Rotational symmetry ,General Medicine ,symbols.namesake ,Mathematics (miscellaneous) ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,Mellin inversion theorem ,Bessel function ,Convergent series ,Mathematics - Abstract
We consider the explicit solution to the axisymmetric diffusion equation. We recast the solution in the form of a Mellin inversion formula, and outline a method to compute a formula for $u(r,t)$ as a series using the Cauchy residue theorem. As a consequence, we are able to represent the solution to the axisymmetric diffusion equation as a rapidly converging series.
- Published
- 2021
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