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Статті в журналах з теми "Series Solution"

Автор: Grafiati

Опубліковано: 7 червня 2025

Оновлено: 2 серпня 2025

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1

P. Khabazi, Navid, Mohsen Aryan, Jalil Jamali, and Kayvan Sadeghy. "Sakiadis Flow of Harris Fluids: a Series-Solution." Nihon Reoroji Gakkaishi 42, no. 4 (2014): 245–53. http://dx.doi.org/10.1678/rheology.42.245.

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2

S, Aiswarya, and Gerly T G. "Finite Series Solution Arising from Three-Dimensional q-Difference Equation." Journal of Computational Mathematica 2, no. 2 (2018): 41–50. http://dx.doi.org/10.26524/cm38.

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3

Trefethen, FRS, Lloyd Nicholas. "Series solution of Laplace problems." ANZIAM Journal 60 (October 7, 2018): 1. http://dx.doi.org/10.21914/anziamj.v60i0.13077.

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4

TREFETHEN, LLOYD N. "SERIES SOLUTION OF LAPLACE PROBLEMS." ANZIAM Journal 60, no. 1 (2018): 1–26. http://dx.doi.org/10.1017/s1446181118000093.

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Анотація:

At the ANZIAM conference in Hobart in February 2018, there were several talks on the solution of Laplace problems in multiply connected domains by means of conformal mapping. It appears to be not widely known that such problems can also be solved by the elementary method of series expansions with coefficients determined by least-squares fitting on the boundary. (These are not convergent series; the coefficients depend on the degree of the approximation.) Here we give a tutorial introduction to this method, which converges at an exponential rate if the boundary data are sufficiently well-behave

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5

Adzic, Nevenka, and Z. Ovcin. "Orthogonal series approximation for boundary layers." Theoretical and Applied Mechanics 31, no. 3-4 (2004): 201–14. http://dx.doi.org/10.2298/tam0404201a.

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In this paper we shall consider a class of singularly perturbed problems described by the ordinary differential equation of second order with small parameter multiplying the highest derivative and the appropriate boundary conditions, which describes certain flow problems in fluid mechanics. The solution of such problems displays boundary layers where the solution changes its values very rapidly. The domain decomposition will be performed determining layer subintervals which are adapted to the possibility of spectral approximation. The division point for the boundary layer is determined using a

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6

Zhou, Wei, Chee-Meng Chew, and Geok-Soon Hong. "Design of series damper actuator." Robotica 27, no. 3 (2009): 379–87. http://dx.doi.org/10.1017/s0263574708004797.

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SUMMARYIn our previous work, a novel force control actuator, called series damper actuator (SDA), has been proposed. This paper proposes a general design procedure for the SDA system. From design requirements, several key parameters of the SDA plant can be determined. Based on these parameters, the selection or design of the series damper and the motor can be carried out. A case study is included to illustrate the effectiveness of the procedure. As there could be more than one feasible solutions from the procedure, the mechatronic design quotient (MDQ) method can be adopted to select the best

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7

Hollkamp, J. J., and S. M. Batill. "Structural Identification Using Order Overspecified Time-Series Models." Journal of Dynamic Systems, Measurement, and Control 114, no. 1 (1992): 27–33. http://dx.doi.org/10.1115/1.2896504.

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An identification method that uses order overspecified time-series models and a truncated singular value decomposition (SVD) solution is studied. The overspecified model reduces the effects of noise during the identification process, but produces extraneous modes. A backwards approach coupled with a minimum norm approximation, using a truncated SVD solution, enables the system modes to be distinguished from the extraneous modes of the model. Experimental data from a large flexible truss is used to study the effects of varying the truncation of the SVD solution and an order recursive algorithm

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8

Sales, Lázaro Lima de, Jonatas Arizilanio Silva, Eliângela Paulino Bento de Souza, Hidalyn Theodory Clemente Mattos de Souza, Antonio Diego Silva Farias, and Otávio Paulino Lavor. "Gaussian integral by Taylor series and applications." REMAT: Revista Eletrônica da Matemática 7, no. 2 (2021): e3001. http://dx.doi.org/10.35819/remat2021v7i2id4330.

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In this paper, we present a solution for a specific Gaussian integral. Introducing a parameter that depends on a n index, we found out a general solution inspired by the Taylor series of a simple function. We demonstrated that this parameter represents the expansion coefficients of this function, a very interesting and new result. We also introduced some Theorems that are proved by mathematical induction. As a test for the solution presented here, we investigated a non-extensive version for the particle number density in Tsallis framework, which enabled us to evaluate the functionality of the

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9

Schorr, Susan, Hans-Joachim Hoebler, and Michael Tovar. "A neutron diffraction study of the stannite-kesterite solid solution series." European Journal of Mineralogy 19, no. 1 (2007): 65–73. http://dx.doi.org/10.1127/0935-1221/2007/0019-0065.

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10

Shawagfeh, N., and D. Kaya. "Series solution to the Pochhammer-Chreeequation and comparison with exact solutions." Computers & Mathematics with Applications 47, no. 12 (2004): 1915–20. http://dx.doi.org/10.1016/j.camwa.2003.02.012.

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11

Bello, Nakone, Ibrahim Mutawakilu, and Mustafa Aminu. "Maclaurin Series Solution for Seventh Order Boundary Value Problems." International Journal of Science for Global Sustainability 9, no. 4 (2023): 70–77. http://dx.doi.org/10.57233/ijsgs.v9i4.555.

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Анотація:

Maclaurin series expansion which is taught at undergraduate levels and which has been previously used for the solution of initial value problem is successfully used in this paper for the solutions of seventh-order boundary value problems. The governing equation defined between boundary points 0 and 1is differentiated to get high order derivatives and the assumed solution is expanded in Maclaurin series then, the boundary conditions at x=1 are utilized to determined unknown coefficients. The method is simple and the numerical results displayed on the tables were found to be in good agreement wi

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12

Prathom, Kiattisak, and Asama Jampeepan. "Direct numerical solutions of the SIR and SEIR models via the Dirichlet series approach." PLOS ONE 18, no. 6 (2023): e0287556. http://dx.doi.org/10.1371/journal.pone.0287556.

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Анотація:

Compartment models are implemented to understand the dynamic of a system. To analyze the models, a numerical tool is required. This manuscript presents an alternative numerical tool for the SIR and SEIR models. The same idea could be applied to other compartment models. The result starts with transforming the SIR model to an equivalent differential equation. The Dirichlet series satisfying the differential equation leads to an alternative numerical method to obtain the model’s solutions. The derived Dirichlet solution not only matches the numerical solution obtained by the fourth-order Runge-K

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13

Boshier, F. A. T., and A. J. Mestel. "Extended series solutions and bifurcations of the Dean equations." Journal of Fluid Mechanics 739 (December 17, 2013): 179–95. http://dx.doi.org/10.1017/jfm.2013.614.

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AbstractSteady, incompressible flow down a slowly curving circular pipe is considered. Both real and complex solutions of the Dean equations are found by analytic continuation of a series expansion in the Dean number, $K$. Higher-order Hermite–Padé approximants are used and the results compared with direct computations using a spectral method. The two techniques agree for large, real $K$, indicating that previously reported asymptotic behaviour of the series solution is incorrect, and thus resolving a long-standing paradox. It is further found that a second solution branch, known to exist at h

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14

Chi, Yeung Lo. "Series solution of the telegraph equation." Applicable Analysis 56, no. 1-2 (1995): 547–58. http://dx.doi.org/10.1080/00036819508840309.

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15

Evers, Ingmar. "A Series Solution for Bermudan Options." Applied Mathematical Finance 12, no. 4 (2005): 337–49. http://dx.doi.org/10.1080/13504860500080263.

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16

Adomian, G., and R. Rach. "Noise terms in decomposition solution series." Computers & Mathematics with Applications 24, no. 11 (1992): 61–64. http://dx.doi.org/10.1016/0898-1221(92)90031-c.

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17

Thorne, James D. "Lambert’s Theorem—A Complete Series Solution." Journal of the Astronautical Sciences 52, no. 4 (2004): 441–54. http://dx.doi.org/10.1007/bf03546411.

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18

Craig Bettenhausen. "Botanical Solution boosts series A haul." C&EN Global Enterprise 101, no. 29 (2023): 13. http://dx.doi.org/10.1021/cen-10129-buscon14.

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19

Shrivastava, Omkar Lal, Kuldeep Narain, and Sumita Shrivastava. "FIVE SERIES EQUATIONS INVOLVING GENERALIZED BATEMAN k-FUNCTIONS." jnanabha 54, no. 01 (2024): 291–94. http://dx.doi.org/10.58250/jnanabha.2024.54135.

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Анотація:

In this paper, the solution of five series equations involving generalized Bateman k-functions is obtained by reducing them to Fredholm integral equation of the second kind. The solution presented in this paper is obtained by employing the techniques of Narain and Lal [11] involving generalized Bateman k-functions by reducing them to the solution of a Fredholm inregral equation of second kind with different bounday conditions. Thus we have seen that Bateman k-functions are having interesting properties to solve double, triple, quadruple and five series equations as special functions. These sol

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20

Shrivastava, Omkar Lal, Kuldeep Narain, and Sumita Shrivastava. "FIVE SERIES EQUATIONS INVOLVING GENERALIZED BATEMAN k-FUNCTIONS." jnanabha 53, no. 02 (2023): 232–35. http://dx.doi.org/10.58250/jnanabha.2023.53227.

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Анотація:

In this paper, the solution of ve series equations involving generalized Bateman k-functions is obtained by reducing them to Fredholm integral equation of the second kind. The solution presented in this paper is obtained by employing the techniques of Narain and Lal [12] involving generalized Bateman k-functions by reducing them to the solution of a Fredholm inregral equation of second kind with different bounday conditions. Thus we have seen that Bateman k-functions are having interesting properties to solve double, triple, quadruple and ve series equations as special functions. These solutio

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21

Pakdemirli, Mehmet. "Convergent piecewise series solutions of ordinary differential equations." Journal of Interdisciplinary Mathematics 27, no. 7 (2024): 1599–614. https://doi.org/10.47974/jim-2001.

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Анотація:

The series approximations of ordinary differential equations usually possess a limited validity range. Outside the convergence interval, the solution does not comply with the real solution. A technique is proposed in this work to expand the validity interval of the series approximations. Solutions are constructed in the vicinity of several points and are joined to each other with satisfaction of the continuity conditions. By employing the piecewise series solutions, the validity range of the approximations can be made to cover the whole domain of interest. In constructing a single analytical e

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22

Kukla, Stanisław, and Urszula Siedlecka. "On Solutions of the Initial Value Problem for the Three-Term Fractional Differential Equation with Caputo Derivatives." Symmetry 12, no. 8 (2020): 1355. http://dx.doi.org/10.3390/sym12081355.

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In this paper, two forms of an exact solution and an analytical–numerical solution of the three-term fractional differential equation with the Caputo derivatives are presented. The Prabhakar function and an asymptotic expansion are utilized to present the double series solution. Using properties of the Pochhammer symbol, a solution is obtained in the form of an infinite series of generalized hypergeometric functions. As an alternative for the series solutions of the fractional commensurate equation, a solution received by an analytical–numerical method based on the Laplace transform technique

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23

Sciarretta, C., V. Luceri, E. Pavlis, and G. Bianco. "The ILRS EOP Time Series." Artificial Satellites 45, no. 2 (2010): 41–48. http://dx.doi.org/10.2478/v10018-010-0004-9.

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The ILRS EOP Time Series Since 2004, ILRS has been providing on a weekly basis combined SSC/EOP solutions from 7-day arcs to support IERS for EOP computation and reference frame maintenance. At present, eight Analysis Centers and two Combination Centers contribute to the weekly official ILRS combined solution from which the main SLR contribution to the IERS EOP reference series is derived. The most recent work performed jointly by all ILRS Analysis Centers (AC) was the generation of a contribution to the next ITRF, extending the time-span coverage of the 7-day arcs combined solution to 1983-20

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24

Ramos, Juan I. "Piecewise Analytical Approximation Methods for Initial-Value Problems of Nonlinear Ordinary Differential Equations." Mathematics 13, no. 3 (2025): 333. https://doi.org/10.3390/math13030333.

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Анотація:

Piecewise analytical solutions to scalar, nonlinear, first-order, ordinary differential equations based on the second-order Taylor series expansion of their right-hand sides that result in Riccati’s equations are presented. Closed-form solutions are obtained if the dependence of the right-hand side on the independent variable is not considered; otherwise, the solution is given by convergent series. Discrete solutions also based on the second-order Taylor series expansion of the right-hand side and the discretization of the independent variable that result in algebraic quadratic equations are a

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25

Jayawardane, NS. "Prediction of unsaturated hydraulic conductivity changes of a loamy soil in different salt solutions by using the equivalent salt solutions concept." Soil Research 30, no. 5 (1992): 565. http://dx.doi.org/10.1071/sr9920565.

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Equivalent salt solutions series have been previously defined as solutions with combinations of sodium absorption ratio (SAR) and electrolyte concentration (E,) producing the same extent of clay swelling in a given soil. These equivalent salt solutions series values have yielded satisfactory predictions of changes in saturated hydraulic conductivity, with changes in salt solution composition and concentrations. In the present study, previously published data on changes in saturated and unsaturated hydraulic conductivities of Gilat soil in salt solutions of cationic ratio 0-50 (mmol dm-3)1/2 an

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26

Alenazy, Aneefah H. S., Abdelhalim Ebaid, Ebrahem A. Algehyne, and Hind K. Al-Jeaid. "Advanced Study on the Delay Differential Equation y′(t) = ay(t) + by(ct)." Mathematics 10, no. 22 (2022): 4302. http://dx.doi.org/10.3390/math10224302.

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Анотація:

Many real-world problems have been modeled via delay differential equations. The pantograph delay differential equation y′(t)=ay(t)+byct belongs to such a set of delay differential equations. To the authors’ knowledge, there are no standard methods to solve the delay differential equations, i.e., unlike the ordinary differential equations, for which numerous and standard methods are well-known. In this paper, the Adomian decomposition method is suggested to analyze the pantograph delay differential equation utilizing two different canonical forms. A power series solution is obtained through th

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27

KYRIAKOPOULOS, E. "NEW SERIES OF ASYMPTOTICALLY FLAT AXISYMMETRIC AND STATIONARY GRAVITATIONAL SOLUTIONS II." International Journal of Modern Physics A 05, no. 07 (1990): 1285–92. http://dx.doi.org/10.1142/s0217751x90000581.

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The second solution of a recently found series of asymptotically flat and stationary solutions of the Ernst equation is calculated explicitly as ratio of two sums of monomials. The solution depends on nine arbitrary real constants.

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28

Hernandez, Elena, Octavio Manero, Fernando Bautista, and Juan Paulo Garcia-Sandoval. "Analytic Matrix Method for Frequency Response Techniques Applied to Nonlinear Dynamical Systems II: Large Amplitude Oscillations." Mathematics 10, no. 15 (2022): 2700. http://dx.doi.org/10.3390/math10152700.

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Анотація:

This work is the second in a series of articles that deal with analytical solutions of nonlinear dynamical systems under oscillatory input that may exhibit harmonic frequencies. Frequency response techniques of nonlinear dynamical systems are usually analyzed with numerical methods, because in most cases analytical solutions such as the harmonic balance series solution turn out to be difficult, if not impossible, as they are based on an infinite series of trigonometric functions with harmonic frequencies. The key contribution of the analytic matrix methods reported in the present series of art

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29

Susan H. Mohammad. "Residual Power Series Method for Solving Linear Complex Differential Equations." Advances in Nonlinear Variational Inequalities 27, no. 2 (2024): 164–78. http://dx.doi.org/10.52783/anvi.v27.715.

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Анотація:

This work uses the residual power series method (RPSM), primarily based on the general Taylor series formula with a residual error function, to offer solutions to instances of linear complex differential equations (LCDEs) as test problems. After finding approximate solutions to those problems, the current approach's findings have been compared with the exact solutions as shown in the tables and figures, which show the suggested method's reliability, precision, and rapid convergence. So, the importance of this research lies in demonstrating the method's ability to find an approximate solution c

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30

Peter Alpha, Lukonde. "Pell-Lucas Series Solution for Fredholm Integral Equations of the Second Kind." International Journal of Science and Research (IJSR) 10, no. 6 (2021): 1376–83. http://dx.doi.org/10.21275/mr21617222711.

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31

Suhhiem, Mazin H., and Raad I. Khwayyit. "Semi Analytical Solution for Fuzzy Autonomous Differential Equations." International Journal of Analysis and Applications 20 (November 16, 2022): 61. http://dx.doi.org/10.28924/2291-8639-20-2022-61.

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Анотація:

In this work, we have used fuzzy Adomian decomposition method to find the fuzzy semi analytical solution of the fuzzy autonomous differential equations with fuzzy initial conditions. This method allows for the solution of the fuzzy initial value problems to be calculated in the form of an infinite fuzzy series in which the fuzzy components can be easily calculated. The fuzzy series solutions that we have obtained are accurate solutions and very close to the fuzzy exact analytical solutions. Some numerical results have been given to illustrate the efficiency of the used method.

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32

Hayat, T., Z. Abbas, and M. Sajid. "On the Analytic Solution of Magnetohydrodynamic Flow of a Second Grade Fluid Over a Shrinking Sheet." Journal of Applied Mechanics 74, no. 6 (2007): 1165–71. http://dx.doi.org/10.1115/1.2723820.

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In this study, we derive an analytical solution describing the magnetohydrodynamic boundary layer flow of a second grade fluid over a shrinking sheet. Both exact and series solutions have been determined. For the series solution, the governing nonlinear problem is solved using the homotopy analysis method. The convergence of the obtained solution is analyzed explicitly. Graphical results have been presented and discussed for the pertinent parameters.

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33

Bokhari, Ashfaque H., Ghulam Mohammad, M. T. Mustafa, and F. D. Zaman. "Adomian Decomposition Method for a Nonlinear Heat Equation with Temperature Dependent Thermal Properties." Mathematical Problems in Engineering 2009 (2009): 1–12. http://dx.doi.org/10.1155/2009/926086.

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Анотація:

The solutions of nonlinear heat equation with temperature dependent diffusivity are investigated using the modified Adomian decomposition method. Analysis of the method and examples are given to show that the Adomian series solution gives an excellent approximation to the exact solution. This accuracy can be increased by increasing the number of terms in the series expansion. The Adomian solutions are presented in some situations of interest.

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34

Urbanowicz, Kamil. "Infinite Series Based on Bessel Zeros." Applied Sciences 13, no. 23 (2023): 12932. http://dx.doi.org/10.3390/app132312932.

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Анотація:

An interesting series based on Bessel function roots (zeros) is discussed and numerically analyzed. The novel-derived simplified general solutions are based on Lommel polynomials. This kind of series can have a large practical use in many scientific areas, such as solid mechanics, fluid mechanics, thermodynamics, electronics, physics, etc. Some practical examples connected with fluid mechanics are provided in this paper. The errors in Afanasiev solutions are corrected. In addition, the main solution for the series analyzed by Baricz and Angel is presented.

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35

Barnes, Benedict, Martin Anokye, Mohammed Muniru Iddrisu, Bismark Gawu, and Emmanuel Afrifa. "The Analytic Solutions of the Fractional-Order Model for the Spatial Epidemiology of the COVID-19 Infection." Advances in Mathematical Physics 2023 (May 4, 2023): 1–19. http://dx.doi.org/10.1155/2023/5578900.

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Анотація:

This paper provides a mathematical fractional-order model that accounts for the mindset of patients in the transmission of COVID-19 disease, the continuous inflow of foreigners into the country, immunization of population subjects, and temporary loss of immunity by recovered individuals. The analytic solutions, which are given as series solutions, are derived using the fractional power series method (FPSM) and the residual power series method (RPSM). In comparison, the series solution for the number of susceptible members, using the FPSM, is proportional to the series solution, using the RPSM

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36

Manaa, Ssaad A., Fadhil H. Easif, and Jomaa J. Murad. "Residual Power Series Method for Solving Klein-Gordon Schrödinger Equation." Science Journal of University of Zakho 9, no. 2 (2021): 123–27. http://dx.doi.org/10.25271/sjuoz.2021.9.2.810.

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Анотація:

In this work, the Residual Power Series Method(RPSM) is used to find the approximate solutions of Klein Gordon Schrödinger (KGS) Equation. Furthermore, to show the accuracy and the efficiency of the presented method, we compare the obtained approximate solution of Klein Gordon Schrödinger equation by Residual Power Series Method(RPSM) numerically and graphically with the exact solution.

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37

Groza, Ghiocel, and Marilena Jianu. "Functions represented into fractional Taylor series." ITM Web of Conferences 29 (2019): 01017. http://dx.doi.org/10.1051/itmconf/20192901017.

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Анотація:

Fractional Taylor series are studied. Then solutions of fractional linear ordinary differential equations (FODE), with respect to Caputo derivative, are approximated by fractional Taylor series. The Cauchy-Kowalevski theorem is proved to show the existence and uniqueness of local solutions for FODE with Cauchy initial data. Sufficient conditions for the global existence of the solution and the estimate of error are given for the method using fractional Taylor series. Two illustrative numerical examples are given to demonstrate the validity and applicability of this method.

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38

Raj Tripathi, Bhadra. "Extension of Connecting Formulas on Hypergeometric Function." Turkish Journal of Computer and Mathematics Education (TURCOMAT) 16, no. 1 (2025): 123–34. https://doi.org/10.61841/turcomat.v16i1.15233.

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Анотація:

The Hypergeometric series is the extension of the geometric series and the Confluent Hypergeometric Function is the solution of the Hypergeometric Differential Equation. Kummer has developed six solutions for the differential equation and twenty connecting formulas. The connecting formula consist of a solution expressed as the combination of two other solutions. Further extension was done by Poudel et al. This research work has extended the nine connecting formulas obtained by Poudel et al. to obtain the other nine solutions.

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39

Kincanon, Eric. "A series solution for the GVψ0 term of the Born series". Applied Mathematics and Computation 66, № 2-3 (1994): 227–32. http://dx.doi.org/10.1016/0096-3003(94)90118-x.

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40

Bairwa, R. K., and Ajay Kumar. "Solution of the Quadratic Integral Equation by Homotopy Analysis Method." Annals of Pure and Applied Mathematics 25, no. 01 (2022): 17–40. http://dx.doi.org/10.22457/apam.v25n1a03858.

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Анотація:

In the present paper, we derive an approximate solution of the quadratic integral equation by using the homotopy analysis method (HAM). This approach provides a solution in the form of a rapidly converging series, and it includes an auxiliary parameter that controls the series solution's convergence. We compare the HAM solution with the exact solution graphically. Additionally, an absolute error comparison between the exact and HAM solutions is performed. The findings indicate that HAM is a very straightforward and attractive approach for computation.

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41

Yun, Beong In. "An Iteration Method Generating Analytical Solutions for Blasius Problem." Journal of Applied Mathematics 2011 (2011): 1–8. http://dx.doi.org/10.1155/2011/925649.

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Анотація:

We derive a new iteration method for finding solution of the generalized Blasius problem. This method results in the analytical series solutions which are consistent with the existing series solutions for some special cases.

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He, Song Lin, and Yan Huang. "The Rapid Series Method for Solving Differential Equation of the Odd Power Oscillator." Applied Mechanics and Materials 226-228 (November 2012): 138–41. http://dx.doi.org/10.4028/www.scientific.net/amm.226-228.138.

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The new rapid series method to solve the differential equation of the periodic vibration of the strongly odd power nonlinear oscillator has been put forward in this paper. By adding the exponentially decaying factor to each harmonic term of the Fourier series of the periodic solution, the high accurate solution can be obtained with a few harmonic terms. The number of truncated terms is determined by the requirement of accuracy. Comparing with other approximate methods, the calculation of rapid series method is very easy and the accurate degrees of solution can be control. By comparing the anal

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Bairagi, Mithun. "SERIES SOLUTION OF ORDINARY DIFFERENTIAL EQUATION USING A MODIFIED VERSION OF THE ADOMIAN DECOMPOSITION METHOD." South East Asian J. of Mathematics and Mathematical Sciences 20, no. 01 (2024): 137–54. http://dx.doi.org/10.56827/seajmms.2024.2001.11.

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We present a new modified version of the Adomian decomposition method for computing the series solutions of the nonlinear ordinary differential equations (ODEs). The recently proposed Adomian matrix algorithm is used in this method to compute the Adomian polynomials for scalar-valued nonlinear polynomial functions, which allows us to get the series solution of the ODEs numerically and makes it much faster than symbolic computation. This method can test the convergence of the series solution of the ODE by calculating the global squared residual error of the solution. Several types of nonlinear

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Gulsen, Selahattin, Mustafa Inc, and Harivan R. Nabi. "Approximate Solutions of Two-Dimensional Burgers' and Coupled Burgers' Equations by Residual Power Series Method." ITM Web of Conferences 22 (2018): 01044. http://dx.doi.org/10.1051/itmconf/20182201044.

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In this study, two-dimensional Burgers' and coupled Burgers' equations are examined by the residual power series method. This method provides series solutions which are rapidly convergent and their components are easily calculable by Mathematica. When the solution is polynomial, the method gives the exact solution using Taylor series expansion. The results display that the method is more efficient, applicable and accuracy and the graphical consequences clearly present the reliability of the method.

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Janavičius, A. J., P. Norgela, and D. Jurgaitis. "UNIQUENESS AND CONVERGENCE OF THE ANALYTICAL SOLUTION OF NONLINEAR DIFFUSION EQUATION." Mathematical Modelling and Analysis 6, no. 1 (2001): 77–84. http://dx.doi.org/10.3846/13926292.2001.9637147.

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We have discussed the problems of uniqueness of the physical solution of the nonlinear diffusion equation. Here are considered two different ways to express the solutions in the power series. In the first case we will use the power‐series expansion about the zero point. The accuracy of the obtained physical solution is evaluated. However, in this case we get an infinity of different solutions and the problem of the choice of the unique physical solution is considered using the expansion about the point of maximum penetration of the impurities. Then we get only two solutions which differ one fr

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Mousa, Zainab I., and Mazin H. Suhhiem. "Legendre operational differential matrix for solving ordinary differential equations." International Journal of ADVANCED AND APPLIED SCIENCES 11, no. 1 (2024): 201–6. http://dx.doi.org/10.21833/ijaas.2024.01.024.

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In this paper, we used the Legendre operational differential matrix method based on the Tau method to find the approximate analytical solutions to the initial value problems and boundary value problems of ordinary differential equations. This method allows the solution of the ordinary differential equation to be computed in the form of an infinite series in which the components can be easily calculated. We introduced a comparison between the approximate solution that we computed and the exact solution of the selected problem, as we found the absolute error. According to the numerical results,

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Macphee, Donald E., and Stephanie J. Barnett. "Solution properties of solids in the ettringite—thaumasite solid solution series." Cement and Concrete Research 34, no. 9 (2004): 1591–98. http://dx.doi.org/10.1016/j.cemconres.2004.02.022.

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Papkov, S. O. "Vibrations of a Cantilevered Thick Plate." PNRPU Mechanics Bulletin, no. 2 (December 15, 2021): 106–17. http://dx.doi.org/10.15593/perm.mech/2021.2.10.

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It has been for the first time that an analytical solution to the problem of free vibrations of a cantilevered thick orthotropic plate is presented. This problem is quite cumbersome for using the exact methods of the theory of elasticity; therefore, methods based on the variational approach were developed to solve it. The paper suggests using the superposition method to construct a general solution of the vibration equations of a plate in the series form of particular solutions obtained with the help of a variables separation. The particular solutions of one of the coordinates are built in the

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S.S., Handibag, and Bhosale J.H. "Solution Of Nonlinear Integro-Differential Equations by Using Laplace Decomposition Method and Series Solution Method." Indian Journal of Science and Technology 18, no. 12 (2025): 974–82. https://doi.org/10.17485/IJST/v18i12.3988.

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Abstract <strong>Objectives:</strong> To investigate two mutually beneficial methods: One is the Laplace Adomian decomposition method, and the second is the series solution method. This study intends to investigate the efficiency of both approaches in resolving nonlinear integral-differential equations. <strong>Methods:</strong> Each strategy’s main ideas and theoretical underpinnings are methodically described, along with each approach’s advantages and uses. By contrasting their performances, this work sheds light on the effectiveness and practicality of the series

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Xu, Liang, and Sha Xu. "Analytic Solution to a Virus Infection Model." Applied Mechanics and Materials 367 (August 2013): 503–7. http://dx.doi.org/10.4028/www.scientific.net/amm.367.503.

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A functional analytic method was developed by E.K.Ifantis in 1987 to prove that certain non-linear ordinary differential equations (ODEs) have a unique power series solution which converges absolutely in a specified disc of the complex plane. In this paper, we first applied this method to certain systems of two non-linear ordinary differential equations. We proved that the power series solutions can be determined by some recurrence relations which depend on the parameters of the equations and the initial conditions. Then, we found a method to extend the range of the converge bound. At last, we

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