Journal articles: 'Complex analytical functions'

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Klunnik, A. A. "Interpolation of generalized analytical functions of complex variable." Journal of Soviet Mathematics 58, no. 5 (February 1992): 391–95. http://dx.doi.org/10.1007/bf01100061.

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Sagindykov, Bimurat. "Analytical functions of generalized complex variables and some appllications." INTERNATIONAL JOURNAL OF RESEARCH IN EDUCATION METHODOLOGY 5, no. 1 (April 30, 2014): 569–75. http://dx.doi.org/10.24297/ijrem.v5i1.3922.

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The object of this work is to use functions of a generalized complex variable to solve the problems of fluid dynamics and elasticity theory. In this paper, we obtain Cauchy-Riemann conditions, generalized Laplace equation and the generalized Poisson formula for such functions.

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Yin, Yilin, Zhenghong Yang, and Meilun Shi. "Analytical Expression of Complex Modulus for Viscoelastic Material." International Journal of Applied Mechanics 12, no. 05 (June 2020): 2050048. http://dx.doi.org/10.1142/s1758825120500489.

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Transfer functions in the linear dynamic system theory are applied to characterize dynamic mechanical properties of viscoelastic materials. Correlation between transfer functions and typical rheological models and fractional derivative ones are briefly introduced. The transfer function of a rheological model may be expressed in terms of multiplication of factored polynomials. The frequency–response data are presented in the form of a Bode plot of magnitude, from which a transfer function can be established. The characteristic times can be conveniently identified via the corner frequencies of asymptotes of the magnitude curve. Dynamic frequency sweep results for a typical viscoelastic solid are presented to illustrate the use of the Bode diagram method for parameter identification.

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McKinney, Nellie, and Josefa Guerrero-Millan. "Implementing an Algorithm for Locating Complex Zeros of Analytical Functions in the Complex Plane." Arsenal: The Undergraduate Research Journal of Augusta University 4, no. 1 (May 18, 2021): 24. http://dx.doi.org/10.21633/issn.2380.5064/s.2021.04.01.24.

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Hu, X., H. Wang, and D. S. Guo. "Phased Bessel functions." Canadian Journal of Physics 86, no. 7 (July 1, 2008): 863–70. http://dx.doi.org/10.1139/p08-009.

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In the study of photon-state transitions, we found a natural extension of the first kind of Bessel functions that extends both the range and domain of the Bessel functions from the real number field to the complex number field. We term the extended Bessel functions as phased Bessel functions. This extension is completely different from the traditional “analytical extension”. The new complex Bessel functions satisfy addition, subtraction, and recurrence theorems in a complex range and a complex domain. These theorems provide short cuts in calculations. The single-phased Bessel functions are generalized to multiple-phased Bessel functions to describe various photon-state transitions.PACS Nos.: 02.30.Gp, 32.80.Rm, 42.50.Hz

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Šauli, Vladimir. "Analytical solution for the correlator with Gribov propagators." Open Physics 14, no. 1 (January 1, 2016): 570–78. http://dx.doi.org/10.1515/phys-2016-0065.

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AbstractPropagators approximated by meromorphic functions with complex conjugated poles are widely used to model the infrared behavior of QCD Green’s functions. In this paper, analytical solutions for two point correlators made out of functions with complex conjugated poles or branch points have been obtained in the Minkowski space for the first time. As a special case the Gribov propagator has been considered as well. The result is different from the naive analytical continuation of the correlator primarily defined in the Euclidean space. It is free of ultraviolet divergences and instead of Lehmann it rather satisfies Gribov integral representation.

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Rao, S., and C. R. Houska. "X-ray diffraction profiles described by refined analytical functions." Acta Crystallographica Section A Foundations of Crystallography 42, no. 1 (January 1, 1986): 14–19. http://dx.doi.org/10.1107/s010876738609997x.

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The line profiles from a sample containing small spherical particles, non-uniform strain and instrumental broadening can be described exactly by using error functions with complex arguments. Consequently, the development by Houska & Smith [J. Appl. Phys. (1981). 52, 748-754] has been revised in terms of these functions. This calculation has been extended, by the use of error functions with complex arguments, to include a more general distribution of particle size or column heights than that obtained from a single sphere. The latter extension is applied to profiles obtained from a partially stabilized zirconia wear debris. It is found, in this example, that a column-height variation coefficient that is greater than that from a single sphere gives a somewhat better fit of the experimental line profiles. We find that if the single-sphere model is used to fit the profiles the particle size and root-mean-square strain differ by about 12 and 5% respectively.

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Reiter, Jerome P., Elaine L. Zanutto, and Larry W. Hunter. "Analytical Modeling in Complex Surveys of Work Practices." ILR Review 59, no. 1 (October 2005): 82–100. http://dx.doi.org/10.1177/001979390505900105.

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Quantitative industrial relations research frequently relies on data collected from large surveys of establishments that use complex sampling designs, such as stratified and unequal probability sampling. The authors analyze two complex surveys of establishments, the National Organizations Survey and the National Survey of Establishments. They discuss design-based (survey-weighted) and model-based (unweighted) strategies for analyzing these data. They show that the choice of strategy can affect inferences about parameters, and hence conclusions drawn from analyses. They discuss the advantages of model-based approaches that include independent variables corresponding to design features, such as functions of size measures or indicator variables for strata or clusters, relative to purely design-based approaches.

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AHN, JOONSEON, and TAISOOK HAN. "AN ANALYTICAL METHOD FOR PARALLELIZATION OF RECURSIVE FUNCTIONS." Parallel Processing Letters 10, no. 01 (March 2000): 87–98. http://dx.doi.org/10.1142/s012962640000010x.

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Programming with parallel skeletons is an attractive framework because it encourages programmers to develop efficient and portable parallel programs. However, extracting parallelism from sequential specifications and constructing efficient parallel programs using the skeletons are still difficult tasks. In this paper, we propose an analytical approach to transforming recursive functions on general recursive data structures into compositions of parallel skeletons. Using static slicing, we have defined a classification of subexpressions based on their data-parallelism. Then, skeleton-based parallel programs are generated from the classification. To extend the scope of parallelization, we have adopted more general parallel skeletons which do not require the associativity of argument functions. In this way, our analytical method can parallelize recursive functions with complex data flows.

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AHN, JOONSEON, and TAISOOK HAN. "AN ANALYTICAL METHOD FOR PARALLELIZATION OF RECURSIVE FUNCTIONS." Parallel Processing Letters 10, no. 04 (December 2000): 359–70. http://dx.doi.org/10.1142/s0129626400000330.

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Programming with parallel skeletons is an attractive framework because it encourages programmers to develop efficient and portable parallel programs. However, extracting parallelism from sequential specifications and constructing efficient parallel programs using the skeletons are still difficult tasks. In this paper, we propose an analytical approach to transforming recursive functions on general recursive data structures into compositions of parallel skeletons. Using static slicing, we have defined a classification of subexpressions based on their data-parallelism. Then, skeleton-based parallel programs are generated from the classification. To extend the scope of parallelization, we have adopted more general parallel skeletons which do not require the associativity of argument functions. In this way, our analytical method can parallelize recursive functions with complex data flows.

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Hwu, Chyanbin, Chung-Lei Hsu, and Wei-Ren Chen. "Corrective evaluation of multi-valued complex functions for anisotropic elasticity." Mathematics and Mechanics of Solids 22, no. 10 (September 11, 2017): 2040–62. http://dx.doi.org/10.1177/1081286517728542.

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It is well known that the Stroh formalism is an elegant and powerful complex variable method for anisotropic elasticity. Through this formalism, several analytical solutions for the problems of anisotropic elasticity have been presented in the literature. To evaluate the analytical solutions, some problems may occur on the numerical evaluation of multi-valued complex functions and their related singular integrals. In this paper, to get a correct single-valued solution, proper branch cuts are suggested for several different multi-valued complex functions, such as the logarithmic function, inverse trigonometric function, power function, mapping function, Plemelj function, and the logarithmic function with mapped variables. To get the correct numerical integration for weakly and strongly singular integrals with multi-valued complex variables, based upon the concept of finite part integrals, formulae employing the Gaussian quadrature rules of standard, logarithmic, and inverse type are derived. According to the branch cuts selected for different complex functions and the integration formulae for singular integrals, some remarks on the computer programming are provided. Verification of the remarks is then made by typical examples of anisotropic elasticity such as holes, cracks, punches, and singular integrals used in boundary element formulation.

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Shilin, A. P. "A hypersingular integro-differential equation of the Euler type." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 56, no. 1 (April 6, 2020): 17–29. http://dx.doi.org/10.29235/1561-2430-2020-56-1-17-29.

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In this paper, we study an integro-differential equation on a closed curve located on the complex plane. The integrals included in the equation are understood as a finite part by Hadamard. The coefficients of the equation have a particular structure. The analytical continuation method is applied. The equation is reduced to a boundary value linear conjugation problem for analytic functions and linear Euler differential equations in the domains of the complex plane. Solutions of the Euler equations, which are unambiguous analytical functions, are sought. The conditions of solvability of the initial equation are given explicitly. The solution of the initial equation obtained under these conditions is also given explicitly. Examples are considered.

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MATSUI, Toshiro, and Kiyoshi MATSUMOTO. "Physiological functions of food components. Analytical evaluation of complex functional food matrices. (Review)." BUNSEKI KAGAKU 49, no. 7 (2000): 477–91. http://dx.doi.org/10.2116/bunsekikagaku.49.477.

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Cveticanin, L. "Approximate analytical solutions to a class of non-linear equations with complex functions." Journal of Sound and Vibration 157, no. 2 (September 1992): 289–302. http://dx.doi.org/10.1016/0022-460x(92)90682-n.

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Wilson, Cathal, Rossella Venditti, Laura R. Rega, Antonino Colanzi, Giovanni D'Angelo, and M. Antonietta De Matteis. "The Golgi apparatus: an organelle with multiple complex functions." Biochemical Journal 433, no. 1 (December 15, 2010): 1–9. http://dx.doi.org/10.1042/bj20101058.

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Remarkable advances have been made during the last few decades in defining the organizational principles of the secretory pathway. The Golgi complex in particular has attracted special attention due to its central position in the pathway, as well as for its fascinating and complex structure. Analytical studies of this organelle have produced significant advances in our understanding of its function, although some aspects still seem to elude our comprehension. In more recent years a level of complexity surrounding this organelle has emerged with the discovery that the Golgi complex is involved in cellular processes other than the ‘classical’ trafficking and biosynthetic pathways. The resulting picture is that the Golgi complex can be considered as a cellular headquarters where cargo sorting/processing, basic metabolism, signalling and cell-fate decisional processes converge.

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Ghanim, Firas, Khalifa Al-Shaqsi, Maslina Darus, and Hiba Fawzi Al-Janaby. "Subordination Properties of Meromorphic Kummer Function Correlated with Hurwitz–Lerch Zeta-Function." Mathematics 9, no. 2 (January 19, 2021): 192. http://dx.doi.org/10.3390/math9020192.

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Recently, Special Function Theory (SPFT) and Operator Theory (OPT) have acquired a lot of concern due to their considerable applications in disciplines of pure and applied mathematics. The Hurwitz-Lerch Zeta type functions, as a part of Special Function Theory (SPFT), are significant in developing and providing further new studies. In complex domain, the convolution tool is a salutary technique for systematic analytical characterization of geometric functions. The analytic functions in the punctured unit disk are the so-called meromorphic functions. In this present analysis, a new convolution complex operator defined on meromorphic functions related with the Hurwitz-Lerch Zeta type functions and Kummer functions is considered. Certain sufficient stipulations are stated for several formulas of this defining operator to attain subordination. Indeed, these outcomes are an extension of known outcomes of starlikeness, convexity, and close to convexity.

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Nijimbere, Victor. "Analytical and asymptotic evaluations of Dawson’s integral and related functions in mathematical physics." Journal of Applied Analysis 25, no. 2 (December 1, 2019): 179–88. http://dx.doi.org/10.1515/jaa-2019-0019.

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Abstract Dawson’s integral and related functions in mathematical physics that include the complex error function (Faddeeva’s integral), Fried–Conte (plasma dispersion) function, Jackson function, Fresnel function and Gordeyev’s integral are analytically evaluated in terms of the confluent hypergeometric function. And hence, the asymptotic expansions of these functions on the complex plane {\mathbb{C}} are derived by using the asymptotic expansion of the confluent hypergeometric function.

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Wang, Yun Tao, Nian Chun Lü, and Cheng Jin. "Analytical Solutions of Mode III Moving Crack under Concentrated Force Boundary Condition." Advanced Materials Research 214 (February 2011): 271–75. http://dx.doi.org/10.4028/www.scientific.net/amr.214.271.

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By application of the theory of complex functions, the problem for mode Ⅲ dynamic crack under concentrated force were researched. The Riemann-Hilbert problem is formulated according to relationships of self-similar functions. Analytical solutions of stresses, displacements and dynamic stress intensity factors were obtained by the measures of self-similar functions and corresponding differential and integral operation.

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Brassel, B. P., S. D. Maharaj, and G. Govender. "Analytical Models for Gravitating Radiating Systems." Advances in Mathematical Physics 2015 (2015): 1–7. http://dx.doi.org/10.1155/2015/274251.

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We analyse the gravitational behaviour of a relativistic heat conducting fluid in a shear-free spherically symmetric spacetime. We show that the isotropy of pressure is a consistency condition which realises a second order nonlinear ordinary differential equation with variable coefficients in the gravitational potentials. Several new classes of solutions are found to the governing equation by imposing various forms on one of the potentials. Interestingly, a complex transformation leads to an exact solution with only real metric functions. All solutions are written in terms of elementary functions. We demonstrate graphically that the fluid pressure, energy density, and heat flux are well behaved for the model, and the model is consistent with a core-envelope framework.

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Eslami, Seyed Hashem. "Crack Growth Threshold Criterion Based on Calculation of Dynamic Stress Intensity Factors in the Mixed Mode (Complex Functions Theory Approach)." Advanced Materials Research 891-892 (March 2014): 333–38. http://dx.doi.org/10.4028/www.scientific.net/amr.891-892.333.

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In this paper, an analytical method is presented for crack growth threshold criterion based on calculation of dynamic stress intensity factors (DSIF) in the mixed mode. The basis of the method is comparison of DSIF and fracture toughness. The analytical method is grounded on solving two-dimensional wave equations in the frequency domain, using complex functions and mapping technique. Using this method, the J-integrals are calculated and based on relation between the J-integrals and DSIF, these factors are derived. The proposed criterion for crack growth threshold is defined as a comparing calculated DSIF (KI, KII) with fracture toughness (Kc), using complex functions and mapping technique. The mapping technique helps to convert the complicated boundaries geometry into a simple one (unit circle). Finally, since using complex functions theory is an analytical method, the results have a high precision compared with numerical methods.

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Bondarenko, Natalia Pavlovna. "Inverse Sturm-Liouville problem with analytical functions in the boundary condition." Open Mathematics 18, no. 1 (June 10, 2020): 512–28. http://dx.doi.org/10.1515/math-2020-0188.

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Abstract The inverse spectral problem is studied for the Sturm-Liouville operator with a complex-valued potential and arbitrary entire functions in one of the boundary conditions. We obtain necessary and sufficient conditions for uniqueness and develop a constructive algorithm for the inverse problem solution. The main results are applied to the Hochstadt-Lieberman half-inverse problem. As an auxiliary proposition, we prove local solvability and stability for the inverse Sturm-Liouville problem by the Cauchy data in the non-self-adjoint case.

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Assier, R. C., and A. V. Shanin. "Diffraction by a quarter–plane. Analytical continuation of spectral functions." Quarterly Journal of Mechanics and Applied Mathematics 72, no. 1 (December 31, 2018): 51–86. http://dx.doi.org/10.1093/qjmam/hby021.

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Summary The problem of diffraction by a Dirichlet quarter-plane (a flat cone) in a 3D space is studied. The Wiener–Hopf equation for this case is derived and involves two unknown (spectral) functions depending on two complex variables. The aim of the present work is to construct an analytical continuation of these functions onto a well-described Riemann manifold and to study their behaviour and singularities on this manifold. In order to do so, integral formulae for analytical continuation of the spectral functions are derived and used. It is shown that the Wiener–Hopf problem can be reformulated using the concept of additive crossing of branch lines introduced in the article. Both the integral formulae and the additive crossing reformulation are novel and represent the main results of this work.

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Trybucka, Edyta. "Extremal Problems of Some Family of Holomorphic Functions of Several Complex Variables." Symmetry 11, no. 10 (October 16, 2019): 1304. http://dx.doi.org/10.3390/sym11101304.

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Many authors, e.g., Bavrin, Jakubowski, Liczberski, Pfaltzgraff, Sitarski, Suffridge, and Stankiewicz, have discussed some families of holomorphic functions of several complex variables described by some geometrical or analytical conditions. We consider a family of holomorphic functions of several complex variables described in n-circular domain of the space C n . We investigate relations between this family and some of type of Bavrin’s families. We give estimates of G-balance of k-homogeneous polynomial, a distortion type theorem and a sufficient condition for functions belonging to this family. Furthermore, we present some examples of functions from the considered class.

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JAMISON, JAMES, and RAENA KING. "UNBOUNDED HERMITIAN OPERATORS ON KOLASKI SPACES." Glasgow Mathematical Journal 56, no. 3 (August 30, 2013): 507–17. http://dx.doi.org/10.1017/s0017089513000426.

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AbstractWe investigate strongly continuous one-parameter (C0) groups of isometries acting on certain spaces of analytical functions which were introduced by Kolaski (C. J. Kolaski, Isometries of some smooth normed spaces of analytic functions, Complex Var. Theory Appl. 10(2–3) (1988), 115–122). We characterize the generators of these groups of isometries and also the spectrum of the generators. We provide an example on the Bloch space of an unbounded hermitian operator with non-compact resolvent.

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Shilin, A. P. "On the solution of one integro-differential equation with singular and hypersingular integrals." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 56, no. 3 (October 18, 2020): 298–309. http://dx.doi.org/10.29235/1561-2430-2020-56-3-298-309.

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A linear integro-differential equation of the first order given on a closed curve located on the complex plane is studied. The coefficients of the equation have a special structure. The equation contains a singular integral, which can be understood as the main value by Cauchy, and a hypersingular integral which can be understood as the end part by Hadamard. The analytical continuation method is applied. The equation is reduced to a sequential solution of the Riemann boundary value problem and two linear differential equations. The Riemann problem is solved in the class of analytic functions with special points. Differential equations are solved in the class of analytical functions on the complex plane. The conditions for the solvability of the original equation are explicitly given. The solution of the equation when these conditions are fulfilled is also given explicitly. Examples are considered. A non-obvious special case is analyzed.

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Wang, Yun Tao, Nian Chun Lü, Chao Ying Wang, and Jin Cheng. "Dislocation Distribution Functions of the Edges of Mode I Crack under Several Boundary Conditions." Advanced Materials Research 619 (December 2012): 98–102. http://dx.doi.org/10.4028/www.scientific.net/amr.619.98.

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Dislocation distribution functions of mode I dynamic crack subjected to two loads were studied by the methods of the theory of complex variable functions. By this way, the problems researched can be translated into Riemann-Hilbert problems and Keldysh-Sedov mixed boundary value problems. Analytical solutions of stresses, displacements and dynamic stress intensity factors were obtained by the measures of self-similar functions and corresponding differential and integral operation. The analytical solutions attained relate to the crack propagation velocity and time, but the solutions have nothing to the other parameters. In terms of the relationship between dislocation distribution functions and displacements, analytical solutions of dislocation distribution functions were gained, and variation rules of dislocation distribution functions were depicted.

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Wang, Yun Tao, and Nian Chun Lü. "Dislocation Distribution Model of Mode I Dynamic Crack." Advanced Materials Research 214 (February 2011): 235–39. http://dx.doi.org/10.4028/www.scientific.net/amr.214.235.

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Dislocation distribution functions of mode I dynamic crack subjected to two loads were studied by the methods of the theory of complex variable functions. By this way, the problems researched can be translated into Riemann-Hilbert problems and Keldysh-Sedov mixed boundary value problems. Analytical solutions of stresses, displacements and dynamic stress intensity factors were obtained by the measures of self-similar functions and corresponding differential and integral operation. The analytical solutions attained relate to the crack propagation velocity and time, but the solutions have nothing to the other parameters. In terms of the relationship between dislocation distribution functions and displacements, analytical solutions of dislocation distribution functions were gained, and variation rules of dislocation distribution functions were depicted.

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Shilin, Andrei P. "Solution of one hypersingular integro-differential equation defined by determinants." Journal of the Belarusian State University. Mathematics and Informatics, no. 2 (August 5, 2021): 17–28. http://dx.doi.org/10.33581/2520-6508-2021-2-17-28.

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The paper provides an exact analytical solution to a hypersingular inregro-differential equation of arbitrary order. The equation is defined on a closed curve in the complex plane. A characteristic feature of the equation is that if is written using determinants. From the view of the traditional classification of the equations, it should be classified as linear equations with vatiable coefficients of a special form. The method of analytical continuation id applied. The equation is reduced to a boundary value problem of linear conjugation for analytic functions with some additional conditions. If this problem is solvable, if is required to solve two more linear differential equations in the class of analytic functions. The conditions of solvability are indicated explicitly. When these conditions are met, the solution can also be written explicitly. An example is given.

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Miller, G. R. "A Green’s Function Solution for Plane Anisotropic Contact Problems." Journal of Applied Mechanics 53, no. 2 (June 1, 1986): 386–89. http://dx.doi.org/10.1115/1.3171769.

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An analytical expression for the stresses in contacting anisotropic cylinders containing a singularity is derived, including the effects of sliding friction. The solution is obtained using complex variables, and is expressed in terms of piece-wise analytic functions. Concentrated force and edge dislocation type singularities are both considered, providing a solution useful for various boundary integral applications.

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Marter-Kenyon, Jessica. "Origins and functions of climate-related relocation: An analytical review." Anthropocene Review 7, no. 2 (July 7, 2020): 159–88. http://dx.doi.org/10.1177/2053019620915633.

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Planned, state-led population relocation is advanced as an adaptation to climate change. Concerned that climate hazards will threaten settlement viability and provoke widespread, unplanned migration, global discourse overwhelmingly characterizes relocation as a voluntary, “last resort” effort to resettle and rebuild communities in safer areas. Over the past decade, scholars have investigated where and why climate-related relocation materializes and how it functions as an adaptation (or otherwise). This article systematically reviews the scientific literature, concluding that climate-related relocation is a more diverse and complex process than recognized within dominant research efforts and policy narratives. While climate-related relocation is sometimes a function of environmental migration pressures and adaptation imperatives, recent critical scholarship shows that climate-related relocation processes are embedded in historical responses to environment and development problems and unfold through political negotiation, discourse, and the social construction of risk and response. In practice, “adaptive relocation” frequently involves population redistribution (villagization and sedentarization) as well as resettlement, is often proactive and involuntary, and risks (re)producing maladaptive outcomes. Based on this analysis, I argue for an expanded research and policy agenda centered around a pluralistic conceptual framework that respects the diversity of relocation efforts undertaken as adaptation.

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Chao, C. K., and C. J. Tan. "On the General Solutions for Annular Problems With a Point Heat Source." Journal of Applied Mechanics 67, no. 3 (January 29, 1999): 511–18. http://dx.doi.org/10.1115/1.1312804.

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A general analytical solution for the annular problem with a point heat source is provided in this paper. Based upon the method of analytical continuation and the technique of Fourier series expansions, the series solutions of the temperature and stress functions are expressed in complex explicit form. Single-valuedness of complex functions in the doubly connected region has been examined for both the stress-free and displacement-free boundary conditions. The dilatation stress in the annulus due to the application of a point heat source is discussed and shown in graphic form. [S0021-8936(00)02803-8]

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Yang, B. "Distributed Transfer Function Analysis of Complex Distributed Parameter Systems." Journal of Applied Mechanics 61, no. 1 (March 1, 1994): 84–92. http://dx.doi.org/10.1115/1.2901426.

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This paper presents a new analytical and numerical method for modeling and synthesis of complex distributed parameter systems that are multiple continua combined with lumped parameter systems. In the analysis, the complex distributed parameter system is first divided into a number of subsystems; the distributed transfer functions of each subsystem are determined in exact and closed form by a state space technique. The complex distributed parameter system is then assembled by imposing displacement compatibility and force balance at the nodes where the subsystems are interconnected. With the distributed transfer functions and the transfer functions of the constraints and lumped parameter systems, exact, closed-form formulation is obtained for various dynamics and vibration problems. The method does not require a knowledge of system eigensolutions, and is valid for non-self-adjoint systems with inhomogeneous boundary conditions. In addition, the proposed method is convenient in computer coding and suitable for computerized symbolic manipulation.

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Algorabi, Maha Saeed, and M. A. Sharaf. "Fourier Series Expansions of Powers of the Trigonometric Sine and Cosine Functions." JOURNAL OF ADVANCES IN MATHEMATICS 12, no. 5 (June 30, 2016): 6248–53. http://dx.doi.org/10.24297/jam.v12i5.246.

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In this paper, Fourier series expansions of powers of sine and cosine functions are established for any possible power real or complex or positive integer. Recurrence relations are established to facilities the computations of the coefficients of expansions formulae. Numerical applications for real and complex powers are also included , the accuracy of the computed values are at least of order . While the applications for positive integer powers are given as exact analytical expressions.

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Wang, Yun Tao, Nian Chun Lü, Jin Kang Zheng, and Jin Cheng. "Dislocation Distribution Function of the Mode III Dynamic Crack Subjected to Moving Unit Step Load from a Point." Advanced Materials Research 314-316 (August 2011): 872–76. http://dx.doi.org/10.4028/www.scientific.net/amr.314-316.872.

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Dislocation distribution functions of the edges of mode III propagation crack subjected to Moving unit step load from a point was studied by the methods of the theory of complex variable functions.By the methods, the problems researched can be facilely transformed into Riemann-Hilbert problems and Keldysh-Sedov mixed boundary value problems. Analytical solutions of stresses, displacements, dynamic stress intensity factor and dislocation distribution function were obtained by the methods of the theory of self-similar functions.In terms of the relationship between dislocation distribution functions and displacements, analytical solutions of dislocation distribution functions were attained.

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Goloskokov, Dmitry Р., and Аlexander V. Matrosov. "Bending of a ribbed plate under complex loading." Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes 17, no. 2 (2021): 120–30. http://dx.doi.org/10.21638/11701/spbu10.2021.202.

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The article considers the problem of a rectangular plate, supported by a cross system of stiffening ribs, bending. In addition to the transverse load, the plate is subjected to forces in its plane, transmitted through the ribs. Аn analytical solution to the boundary value problem for the resolving differential equation with respect to the normal deflection of the plate, describing the deformation of a rectangular plate supported by stiffeners, is obtained. The solution is presented in the form of series in combinations of regular and special discontinuous functions, which converge quickly and lead to a simple computational algorithm. The influence of the ribs is taken into account in the equation in the form of additional terms containing factors with a delta function. This approach allows us to get rid of a number of assumptions regarding the interaction of the plate with its reinforcing elements. The use of the apparatus of generalized functions when modeling objects of this type simplifies the boundary conditions (there are no conditions for conjugation of various structural elements), but at the same time the differential equations become more complicated. The problem is reduced to the so-called partially degenerate equations. Development of analytical methods that allow obtaining exact solutions of differential equations of this type, and their introduction into computational practice, is one of the urgent tasks of the mechanics of objects with disturbed regularity.

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Shi, Jun-jie, and Shao-hua Pan. "Envelope wave functions and subband energies in superlattices with complex bases: Analytical solutions and numerical examples." Physical Review B 48, no. 11 (September 15, 1993): 8136–44. http://dx.doi.org/10.1103/physrevb.48.8136.

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Lü, N. C., Y. H. Cheng, X. G. Li, and J. Cheng. "Dynamic Problem Concerning Mode I Semi-Infinite Crack Propagation." Journal of Mechanics 29, no. 2 (January 29, 2013): 309–18. http://dx.doi.org/10.1017/jmech.2013.1.

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AbstractBy the approaches of the theory of complex functions, dynamic problem concerning mode I semi-infinite crack propagation was studied. Analytical solutions are very easily obtained by application of the measures of self-similar functions. The problems researched can be changed into Riemann-Hilbert problems and analytical solutions to semi-infinite crack under the condition of impulse loads and increasing loads Pt located at the origin of the coordinates, respectively, are attained.

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Kotova, E. V., A. V. Eremin, V. A. Kudinov, V. K. Tkachev, and A. E. Kuznetsova. "Method of additional unknown functions in heat conductivity problems with variable physical properties of the medium." Vestnik IGEU, no. 2 (2019): 59–70. http://dx.doi.org/10.17588/2072-2672.2019.2.059-070.

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Finding analytical solutions to the problems of thermal conductivity with variable physical properties of the medium by classical analytical methods is very complicated mathematically. The known expressions repre-senting complex infinite series including two types of Bessel functions and gamma-functions are, in fact, numerical as they require a numerical solution to complex transcendental equations with eigenvalues of the boundary problem. Such solutions can hardly be used in engineering applications, especially in cases when a solution to a certain problem is only an intermediate stage in other problems (such as thermoelasticity and control problems, inverse problems, etc.) which can be solved effectively only by finding analytical solutions to the initial problems. Therefore, an urgent problem now is to develop new methods of obtaining analytical solutions to the abovementioned problems, at least approximate ones. The study employed methods of additional boundary conditions and additional unknown functions in the integral method of heat balance. High-precision approximate analytical solutions to the transient heat conduction problem with nonhomogeneous physical properties of the medium for an infinite plate under symmetric boundary conditions of the first type have been obtained. The initial problem for partial differential equations is reduced to two problems in which ordinary differential equations are integrated. Additional boundary conditions are defined in such a way that their fulfillment in accordance with the new method is equivalent to the result of solving the initial partial differential equation at the boundary points and at the temperature perturbation front (for the first stage of the process). By combining methods with finite and infinite heat propagation rate we have been able to obtain high-precision analytical solutions for the whole time range of the unsteady process including its small and ultra small values. The solutions look like simple algebraical polynomials not including special functions (Bessel, Legendre, gamma-functions and others). Since it is not necessary to directly integrate the initial equations by the space variable and to reduce them to ordinary differential equations with additional unknown functions, the considered method can be used for solving complex boundary problems in which differential equations do not allow distinguishing between the variables (into nonlinear, with linear boundary conditions and heat sources, etc.).

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Lü, Nian Chun, Yun Hong Cheng, and Yun Tao Wang. "Dynamic Expansion Problems on Symmetrical Mode III Interface Crack." Advanced Materials Research 211-212 (February 2011): 1012–15. http://dx.doi.org/10.4028/www.scientific.net/amr.211-212.1012.

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By means of the complex variable functions, dynamic expension problems on symmetrical mode Ⅲ interface crack were researched. The problems considered can be very facilely transformed into Riemann-Hilbert problem by the measures of self-similar functions, and the general expressions of analytical solutions for the edges of mode Ⅲ symmetrical interface crack subjected to motive variable loadings Px2/t3 and Pt4/x3 were obtained by means of self-similar functions, respectively. After those solutions were utilized by superposition theorem, the solutions of arbitrary complex problems could be readily attained.

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Lü, Nian Chun, Yun Hong Cheng, and Yun Tao Wang. "Two Dynamic Propagation Problems of Mode III Interface Crack." Advanced Materials Research 214 (February 2011): 477–81. http://dx.doi.org/10.4028/www.scientific.net/amr.214.477.

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By the approaches of complex variable functions, two dynamic propagation problems of mode Ⅲ interface crack were researched. The problems considered can be very facilely changed into Riemann-Hilbert problem by means of self-similar functions, and analytical solutions of the stresses, displacements, dynamic stress intensity factors for the edges of mode Ⅲ symmetrical dynamix interface crack subjected to moving increasing loads Pt2/x2 and Px3/t2, respectively, were obtained by the methods of self-similar functions. After those solutions were utilized by superposition theorem, the solutions of arbitrary complex problems can be readily attained.

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Sharma, Sunil Kumar, Waseem A. Khan, and Cheon Seoung Ryoo. "A Parametric Kind of Fubini Polynomials of a Complex Variable." Mathematics 8, no. 4 (April 22, 2020): 643. http://dx.doi.org/10.3390/math8040643.

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In this paper, we propose a parametric kind of Fubini polynomials by defining the two specific generating functions. We also investigate some analytical properties (for example, summation formulae, differential formulae and relationships with other well-known polynomials and numbers) for our introduced polynomials in a systematic way. Furthermore, we consider some relationships for parametric kind of Fubini polynomials associated with Bernoulli, Euler, and Genocchi polynomials and Stirling numbers of the second kind.

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Basarab, Mikhail, Alain Giani, and Philippe Combette. "Thermal Accelerometer Simulation by the R‑Functions Method." Applied Sciences 10, no. 23 (November 25, 2020): 8373. http://dx.doi.org/10.3390/app10238373.

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As well as many modern devices, thermal accelerometers (TAs) need a sophisticated mathematical simulation to find the ways for their performance optimization. In the paper, a novel approach for solving computational fluid dynamics (CFD) problems in the TA’s cavity is proposed (MQ-RFM), which is based on the combined use of Rvachev’s R-functions method (RFM) and the Galerkin technique with multiquadric (MQ) radial basis functions (RBFs). The semi-analytical RFM takes an intermediate position between traditional analytical approaches and numerical methods, such as the finite-element method (FEM), belonging to the family of the so-called meshless techniques which became popular in the last decades in solving various CFD problems in complex-shaped cavities. Mathematical simulation of TA by using the MQ-RFM was carried out with the purpose to simulate the temperature response of the device and to study and improve its performance. The results of numerical experiments were compared with well-known analytical and numerical benchmark solutions for the circular annulus geometry and it demonstrated the effectiveness of the MQ-RFM for solving the convective heat-transfer problem in the TA’s cavity. The use of solution structures allows one to take a relatively small number of expansion terms to achieve an appropriate accuracy of the approximate solution satisfying at the same time the given boundary conditions exactly. The application of the MQ-RFM gives the possibility to obtain semi-analytical solutions to the diffusion-convection problems and to identify the main thermal characteristics of the TA, that allows one to improve the device performance.

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Kazakov, Kirill E. "Wear of elastic tube with nonuniform coating by rigid bush." E3S Web of Conferences 162 (2020): 02002. http://dx.doi.org/10.1051/e3sconf/202016202002.

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This article is devoted to the statement and construction of analytical solution of the wearcontact problem for a rigid bush and elastic pipe with a coating in the case when the coating is nonuniform. The presence of nonuniformity leads us to the necessity of constructing a solution in a special form over special functions, since standard methods does not allow us to effectively take into account the complex properties of the coating. Analytical representation for contact stresses under the bush is presented in series with separate factor, which connect with complex properties of coating. This allows provide effective calculation even if these properties are described by rapidly changing or discontinuous functions. It is also shown that contact stresses will be negligible over time.

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Lv, Nian Chun, Yun Hong Cheng, and Yun Tao Wang. "Dynamic Propagation Problems on Symmetrical Mode Ⅲ Interface Crack." Advanced Materials Research 197-198 (February 2011): 1728–31. http://dx.doi.org/10.4028/www.scientific.net/amr.197-198.1728.

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By the theory of complex functions, dynamic propagation problems on symmetrical mode Ⅲ interface crack were researched. The problems considered can be very facilely transformed into Riemann-Hilbert problem by the methods of self-similar functions, and the general expressions of analytical solutions for the surfaces of mode Ⅲ symmetrical interface crack subjected to motive variable loadings Px2/t2 and Pt3/x2 were obtained, respectively. After those solutions were utilized by superposition theorem, the solutions of arbitrary complex problems could be attained.

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Lü, Nian Chun, Yun Hong Cheng, and Yun Tao Wang. "Symmetrical Dynamic Propagation Problems of Mode III Interface Crack." Applied Mechanics and Materials 300-301 (February 2013): 806–9. http://dx.doi.org/10.4028/www.scientific.net/amm.300-301.806.

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By the theory of complex functions, symmetrical dynamic propagation problems of mode Ⅲ interface crack were investigated. The problems considered can be very easily translated into Riemann-Hilbert problem by the methods of self-similar functions, and the universal expressions of analytical solutions for the surfaces of symmetrical mode Ⅲ interface crack subjected to moving alterable loadings Pt3/x3 and Px4/t3 were obtained, respectively. After those solutions were utilized by superposition theorem, the solutions of arbitrary complex problems could be acquired.

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Singh, Sandeep, and Kuldeep Sharma. "Riemann–Hilbert approach-based analytical solutions for strip saturated two unequal collinear cracks in piezoelectric media." Strength, Fracture and Complexity 13, no. 4 (April 27, 2021): 177–95. http://dx.doi.org/10.3233/sfc-200261.

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The objective of the work is to derive analytical solutions based on the Riemann–Hilbert (R–H) approach for semipermeable strip saturated two unequal collinear cracks in arbitrary polarized piezoelectric media. We particularly consider the influence of far field electromechanical loadings, poling direction and different crack-face boundary conditions. The problem is mathematically formulated into a set of non-homogeneous R–H problems in terms of complex potential functions (related to field components) using complex variable and extended Stroh formalism approach. After solving these equations, explicit solutions are obtained for the involved unknown complex potential functions and hence, the stress and electric displacement components at any point within the domain. Furthermore, after employing standard limiting conditions, explicit expressions for some conventional fracture parameters such as saturated zone lengths (in terms of nonlinear equations), local stress intensity factors and crack opening displacement are obtained. Numerical studies are presented for the PZT-4H material to analyze the effects of prescribed electromechanical loadings, inter-cracks distance, crack-face conditions and poling direction on the defined fracture parameters.

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SUN, JIANSHE. "ANALYTICAL APPROXIMATE SOLUTIONS OF (N + 1)-DIMENSIONAL FRACTAL HARRY DYM EQUATIONS." Fractals 26, no. 06 (December 2018): 1850094. http://dx.doi.org/10.1142/s0218348x18500949.

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The new fractal models of the [Formula: see text]-dimensional and [Formula: see text]-dimensional nonlinear local fractional Harry Dym equation (HDE) on Cantor sets are derived and the analytical approximate solutions of the above two new models are obtained by coupling the fractional complex transform via local fractional derivative (LFD) and local fractional reduced differential transform method (LFRDTM). Fractional complex transform for functions of [Formula: see text]-dimensional variables is generalized and the theorems of [Formula: see text]-dimensional LFRDTM are supplementary extended. The travelling wave solutions of the fractal HDE show that the proposed LFRDTM is effective and simple for obtaining approximate solutions of nonlinear local fractional partial differential equations.

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Lu, Nian Chun, Fang Wang, and Chen Shi. "Symmetrical Dynamic Expansion Issues of Mode III Interface Crack." Advanced Materials Research 773 (September 2013): 649–53. http://dx.doi.org/10.4028/www.scientific.net/amr.773.649.

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By means of complex variable functions, symmetrical dynamic extension issues of mode III interface crack were researched. The problems considered can be very facilely transformed into Riemann-Hilbert problem by the approaches of self-similar functions, and the general expressions of analytical solutions of the stresses, displacements and dynamic stress intensity factors for the edges of symmetrical mode III interface crack subjected to motive variable loadingsPt4/x4andPx5/t4were attained, respectively. After those solutions were utilized by superposition theorem, the solutions of arbitrary complex problems could be acquired.

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Lü, N. C., X. G. Li, Y. H. Cheng, and J. Cheng. "Asymmetrical Dynamic Propagation Problem on the Edges of Mode III Interface Crack Subjected to Superimpose Loads." Journal of Mechanics 29, no. 2 (January 29, 2013): 319–26. http://dx.doi.org/10.1017/jmech.2013.2.

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AbstractBy application of the theory of complex variable functions, asymmetrical dynamic propagation problem on the edges of mode III interface crack subjected to superimpose loads was studied. Analytical solutions of the stresses displacements and dynamic stress intensity factors are obtained by means of self- similar functions. The problems researched can be facilely transformed into Riemann-Hilbert problems and their closed solutions are attained rather simple according to this measure. After those solutions are utilized by superposition theorem, the solutions of arbitrarily complex problems can be gained.

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Kartashov, E. M., and I. V. Antonova. "HYPERBOLIC MODEL OF NON -STATIONARY THERMAL CONDUCTIVITY." Fine Chemical Technologies 11, no. 2 (April 28, 2016): 74–80. http://dx.doi.org/10.32362/2410-6593-2016-11-2-74-80.

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The article presents fundamentally new results on the analytical theory of thermal conductivity for hyperbolic transport models. The questions of the correct formulation of boundary value problems are considered. A technique for finding exact analytical solutions of a rather complex class of boundary value problems based on the method of Green's functions and operational calculus is developed.

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Journal articles on the topic 'Complex analytical functions'

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