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Journal articles on the topic 'MATHEMATICAL EQUATION'

Author: Grafiati

Published: 11 September 2023

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1

Mosnegutu, Emilian, Mirela Panainte-Lehadus, Valentin Nedeff, Claudia Tomozei, Narcis Barsan, Dana Chitimus, and Marcin Jasinski. "Extraction of Mathematical Correlations Applied in the Aerodynamic Separation of Solid Particles." Processes 10, no. 7 (June 21, 2022): 1234. http://dx.doi.org/10.3390/pr10071234.

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This article describes the methodology used to identify the mathematical equation that describes the correlations between the input and output parameters of an experiment. As a technological process, aerodynamic separation was chosen to represent the behavior of a solid particle within an ascending vertical airflow. The experimental data were used to identify two parameters, namely the average linear velocity and the angular velocity. The Table Curve 3D program was used to develop a mathematical equation describing the dependence between the input parameters (the shape and size of the solid particle, as well as the velocity of the airflow) and the monitored parameters. A pyramid-type analysis (following a filtering system, a general equation was determined from a large number of equations that characterize an experimental set mathematically) was designed in order to determine a single mathematical equation that describes the correlation between the input variables and those obtained as accurately as possible. The determination of the mathematical equation started with the number of equations generated by the Table Curve 3D program; then, the equations with a correlation coefficient greater than 0.85 were chosen; and finally, the common equations were identified. Respecting the working methodology, one equation was identified, which has for the average linear velocity, a correlation coefficient r2 of between 0.88–0.99 and 0.86–0.99 for the angular velocity.

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Fauzi, Ahmad, Dwi Teguh Rahardjo, Utoro Romadhon, and Kunthi Ratna Kawuri. "Using Spreadsheet Modeling in Basic Physics Laboratory Practice for Physics Education Curriculum." International Journal of Science and Applied Science: Conference Series 2, no. 1 (December 10, 2017): 8. http://dx.doi.org/10.20961/ijsascs.v2i1.16666.

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<p class="Abstract">Physics is one of a branch of science which uses much of mathematical concept. Usually, the concept of physics is expressed in a mathematical equation; it will make physics easier to be understood. Therefore, the students need to understand about mathematical modelling to help them understand physics. Students who take fundamental physics and physics laboratory course required to understand the concept of feedback that is mathematically expressed in differential equations. However, most of the students have not been taught the concept of differential equations at early semester. Therefore, we are interested in reviewing the use of mathematical modelling with a spreadsheet in the case of feedback that is integrated with laboratory practice. The results of this study indicate that students gave positive perceptions and improve their ability in understanding the concept of feedback that is mathematically expressed in the differential equation.<strong></strong></p>

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Shinde, Rajwardhan, Onkar Dherange, Rahul Gavhane, Hemant Koul, and Nilam Patil. "HANDWRITTEN MATHEMATICAL EQUATION SOLVER." International Journal of Engineering Applied Sciences and Technology 6, no. 10 (February 1, 2022): 146–49. http://dx.doi.org/10.33564/ijeast.2022.v06i10.018.

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With recent developments in Artificial intelligence and deep learning every major field which is using computers for any type of work is trying to ease the work using deep learning methods. Deep learning is used in a wide range of fields due to its diverse range of applications like health, sports, robotics, education, etc. In deep learning, a Convolutional neural network (CNN) is being used in image classification, pattern recognition, Text classification, face recognition, live monitoring systems, handwriting recognition, Digit recognition, etc. In this paper, we propose a system for educational use where the recognition and solving process of mathematical equations will be done by machine. In this system for recognition of equations, we use a Convolutional neural network (CNN) model. The proposed system can recognize and solve mathematical equations with basic operations (-,+,/,*) of multiple digits as well as polynomial equations. The model is trained with Modified National Institute of Standards and Technology (MNIST) dataset as well as a manually prepared dataset of operator symbols (“-”,”+”, “/”, “*”, “(“, “)” ). Further, the system uses the RNN model to solve the recognized operations.

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ERTEKİN, Özlem. "Example of A Kinetic Mathematical Modeling in Food Engineering." ITM Web of Conferences 22 (2018): 01029. http://dx.doi.org/10.1051/itmconf/20182201029.

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Mathematical modeling of biochemical, chemical reaction processes facilitates understanding. The kinetics of these reaction processes can be analyzed mathematically and kinetics are presented as systems of differential equations. Mathematical model of a reaction kinetic is studied in this study. Bernoulli-Sub equation function method is used in this study. This example can be new model for food engineering applications.

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Elías-Zúñiga, Alex, and Oscar Martínez-Romero. "Equivalent Mathematical Representation of Second-Order Damped, Driven Nonlinear Oscillators." Mathematical Problems in Engineering 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/670845.

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The aim of this paper focuses on applying a nonlinearization method to transform forced, damped nonlinear equations of motion of oscillatory systems into the well-known forced, damped Duffing equation. The accuracy obtained from the derived equivalent equations of motion is evaluated by studying the amplitude-time, the phase portraits, and the continuous wavelet transform diagrams of the cubic-quintic Duffing equation, the generalized pendulum equation, the power-form elastic term oscillator, the Duffing equation with linear and cubic damped terms, and the pendulum equation with a cubic damped term.

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Kranysˇ, M. "Causal Theories of Evolution and Wave Propagation in Mathematical Physics." Applied Mechanics Reviews 42, no. 11 (November 1, 1989): 305–22. http://dx.doi.org/10.1115/1.3152415.

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There are still many phenomena, especially in continuum physics, that are described by means of parabolic partial differential equations whose solution are not compatible with the causality principle. Compatibility with this principle is required also by the theory of relativity. A general form of hyperbolic operators for the most frequently occurring linear governing equations in mathematical physics is written down. It is then easy to convert any given parabolic equation to the hyperbolic form without necessarily entering into the cause of the inadequacy of the governing equation. The method is verified on the well-known example of Timoshenko’s correction of the Bernoulli–Euler–Rayleigh beam equation for flexural motion. The “Love–Rayleigh” fourth-order differential equations for the longitudinal and torsional wave propagation in the rod is generalized with this method. The hyperbolic version (not to mention others) of the linear Korteweg–de Vries equation and of the “telegraph” equation governing electromagnetic wave propagation through relaxing material are given. Lagrangians of all the equations studied are listed. For all the reasons given we believe the hyperbolic governing equations to be physically and mathematically more realistic and adequate.

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7

Khan, Kamruzzaman, M. Ali Akbar, and Norhashidah Hj Mohd Ali. "The Modified Simple Equation Method for Exact and Solitary Wave Solutions of Nonlinear Evolution Equation: The GZK-BBM Equation and Right-Handed Noncommutative Burgers Equations." ISRN Mathematical Physics 2013 (February 25, 2013): 1–5. http://dx.doi.org/10.1155/2013/146704.

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The modified simple equation method is significant for finding the exact traveling wave solutions of nonlinear evolution equations (NLEEs) in mathematical physics. In this paper, we bring in the modified simple equation (MSE) method for solving NLEEs via the Generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony (GZK-BBM) equation and the right-handed noncommutative Burgers' (nc-Burgers) equations and achieve the exact solutions involving parameters. When the parameters are taken as special values, the solitary wave solutions are originated from the traveling wave solutions. It is established that the MSE method offers a further influential mathematical tool for constructing the exact solutions of NLEEs in mathematical physics.

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8

Lim, Kien, and Christopher Yakes. "Using Mathematical Equations to Communicate and Think About Karma." Journal of Humanistic Mathematics 11, no. 1 (January 2021): 300–317. http://dx.doi.org/10.5642/jhummath.202101.14.

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Two equations are presented in this article to communicate a particular understanding of karma. The first equation relates future experiences to past and present actions. Although the equation uses variables and mathematical symbols such as the integral sign and summation symbol, it reads more like a literal translation of an English sentence. Based on the key idea in the first equation, a second equation is then created to highlight the viability of using math to communicate concepts that are not readily quantifiable. Analyzing such equations can stimulate thinking, enhance understanding of spiritual concepts, raise issues, and uncover tensions between our ordinary conceptions of external reality and transcendental aspects of spirituality.

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Abu Doush, Iyad, and Sondos Al-Bdarneh. "Automatic Semantic Generation and Arabic Translation of Mathematical Expressions on the Web." International Journal of Web-Based Learning and Teaching Technologies 8, no. 1 (January 2013): 1–16. http://dx.doi.org/10.4018/jwltt.2013010101.

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Automatic processing of mathematical information on the web imposes some difficulties. This paper presents a novel technique for automatic generation of mathematical equations semantic and Arabic translation on the web. The proposed system facilitates unambiguous representation of mathematical equations by correlating equations to their known names. The ability to extract the equation meaning from its structure is vital when searching for mathematical equations. The general structure of the equation is recognized to identify the equation meaning. On the other hand, people who cannot understand Latin script notation of mathematical expressions have difficulty when they try to read them on the web as it is available mostly in Latin. Arabic mathematical expressions flow from right to left and they use specific symbols. The proposed system automatically translates the mathematical equations from Latin to Arabic. This translation can be combined with the text translation of mathematical web contents (generated by online tools) to be recognized by the people who understand only Arabic text. The proposed system is implemented using Java and it is evaluated using a set of web pages with MathML contents which is rendered in Mozilla web browser.

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Seadawy, Aly, Asghar Ali, and Noufe Aljahdaly. "The nonlinear integro-differential Ito dynamical equation via three modified mathematical methods and its analytical solutions." Open Physics 18, no. 1 (March 10, 2020): 24–32. http://dx.doi.org/10.1515/phys-2020-0004.

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AbstractIn this work, we construct traveling wave solutions of (1+1) - dimensional Ito integro-differential equation via three analytical modified mathematical methods. We have also compared our achieved results with other different articles. Portrayed of some 2D and 3D figures via Mathematica software demonstrates to understand the physical phenomena of the nonlinear wave model. These methods are powerful mathematical tools for obtaining exact solutions of nonlinear evolution equations and can be also applied to non-integrable equations as well as integrable ones. Hence worked-out results ascertained suggested that employed techniques best to deal NLEEs.

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NAKKEERAN, K. "MATHEMATICAL DESCRIPTION OF DIFFERENTIAL EQUATION SOLVING ELECTRICAL CIRCUITS." Journal of Circuits, Systems and Computers 18, no. 05 (August 2009): 985–91. http://dx.doi.org/10.1142/s0218126609005484.

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We show that the working principle of the differential equation solving analog electrical circuits is exactly the same as the Picard's method available for numerically solving the ordinary differential equations. The integrator circuit (low-pass filter) uses an initial condition and electrical input signal to generate the Maclaurin's series of a time varying function in recursion. This direct connection between the differential equation solving electrical circuits and Picard's method can be exploited to simplify the procedure of Picard's method to solve any order linear and nonlinear differential equations.

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Sulistyaningtyas, Annisa Dwi. "MODEL MATEMATIKA ALIRAN KONVEKSI BEBAS FLUIDA VISKOELASTIK YANG MELEWATI SILINDER ELIPTIK DENGAN PENGARUH MAGNETOHYDRODYNAMICS (MHD)." WAHANA 70, no. 1 (June 1, 2018): 1–6. http://dx.doi.org/10.36456/wahana.v70i1.1561.

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In fluid case, the mathematical model is the basic for translating a problem into a mathematical language using an equation or function. The governing equation is developed from continuity equation, momentum equation, and energy equation. Fluid characteristics are viscous and elastic result in boundary layer on the surface of elliptic cylinder. Dimensional equations are transformed into non-dimensional form and then classified into the similarity equations using boundary layer theory with influence of magnetic force. The results of this research is mathematical model of free convection flow in viscoelastic fluid passing through the elliptic cylinder with magnetohydrodinamics (MHD). For variation of magnetohydrodinamics (MHD) parameter, velocity and temperature increase when the parameter increases.

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Gepreel, Khaled A., Taher A. Nofal, and Ali A. Al-Thobaiti. "The Modified Rational Jacobi Elliptic Functions Method for Nonlinear Differential Difference Equations." Journal of Applied Mathematics 2012 (2012): 1–30. http://dx.doi.org/10.1155/2012/427479.

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We modified the rational Jacobi elliptic functions method to construct some new exact solutions for nonlinear differential difference equations in mathematical physics via the lattice equation, the discrete nonlinear Schrodinger equation with a saturable nonlinearity, the discrete nonlinear Klein-Gordon equation, and the quintic discrete nonlinear Schrodinger equation. Some new types of the Jacobi elliptic solutions are obtained for some nonlinear differential difference equations in mathematical physics. The proposed method is more effective and powerful to obtain the exact solutions for nonlinear differential difference equations.

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14

Dhunde, Ranjit R., and G. L. Waghmare. "Solutions of Some Linear Fractional Partial Differential Equations in Mathematical Physics." Journal of the Indian Mathematical Society 85, no. 3-4 (June 1, 2018): 313. http://dx.doi.org/10.18311/jims/2018/20144.

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In this article, we use double Laplace transform method to find solution of general linear fractional partial differential equation in terms of Mittag-Leffler function subject to the initial and boundary conditions. The efficiency of the method is illustrated by considering fractional wave and diffusion equations, Klein-Gordon equation, Burger’s equation, Fokker-Planck equation, KdV equation, and KdV-Burger’s equation of mathematical physics.

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Myrzakulova, Zh R., K. R. Yesmakhanova, and Zh S. Zhubayeva. "EQUIVALENCE OF THE HUNTER-SAXON EQUATION AND THE GENERALIZED HEISENBERG FERROMAGNET EQUATION." PHYSICO-MATHEMATICAL SERIES 2, no. 336 (April 15, 2021): 33–38. http://dx.doi.org/10.32014/2021.2518-1726.18.

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Integrable systems play an important role in modern mathematics, theoretical and mathematical physics. The display of integrable equations with exact solutions and some special solutions can provide important guarantees for the analysis of its various properties. The Hunter-Saxton equation belongs to the family of integrable systems. The extensive and interesting mathematical theory, underlying the Hunter-Saxton equation, creates active mathematical and physical research. The Hunter-Saxton equation (HSE) is a high-frequency limit of the famous Camassa-Holm equation. The physical interpretation of HSE is the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field. In this paper, we propose a matrix form of the Lax representation for HSE in 𝑠𝑢ሺ𝑛 ൅ 1ሻ/𝑠ሺ𝑢ሺ1ሻ ⊕ 𝑢ሺ𝑛ሻሻ - symmetric space for the case 𝑛 ൌ 2. Lax pairs, introduced in 1968 by Peter Lax, are a tool for finding conserved quantities of integrable evolutionary differential equations. The Lax representation expands the possibilities of the equation we are considering. For example, in this paper, we will use the matrix Lax representation for the HSE to establish the gauge equivalence of this equation with the generalized Heisenberg ferromagnet equation (GHFE). The famous Heisenberg Ferromagnet Equation (HFE) is one of the classical equations integrable through the inverse scattering transform. In this paper, we will consider its generalization. Andalso the connection between the decisions of the HSE and the GHFE will be presented.

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Kaplan, Melike, Arzu Akbulut, and Ahmet Bekir. "Exact Travelling Wave Solutions of the Nonlinear Evolution Equations by Auxiliary Equation Method." Zeitschrift für Naturforschung A 70, no. 11 (October 1, 2015): 969–74. http://dx.doi.org/10.1515/zna-2015-0122.

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AbstractThe auxiliary equation method presents wide applicability to handling nonlinear wave equations. In this article, we establish new exact travelling wave solutions of the nonlinear Zoomeron equation, coupled Higgs equation, and equal width wave equation. The travelling wave solutions are expressed by the hyperbolic functions, trigonometric functions, and rational functions. It is shown that the proposed method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering. Throughout the article, all calculations are made with the aid of the Maple packet program.

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Volobuev, Andrey N., Kaira A. Adyshirin-Zade, and Tatyana A. Antipova. "Genetic-mathematical modelling of the populations interaction." Physics of Wave Processes and Radio Systems 23, no. 4 (February 11, 2021): 116–22. http://dx.doi.org/10.18469/1810-3189.2020.23.4.116-122.

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Annotation The solution of the genetic-mathematical problem of interaction between the human population and the virus population in relation to the problem of the COVID-19 pandemic is presented. It is noted that the virus does not interact with the entire human body, as a set of complex organs, but with its individual cells. A mathematical model based on the HardyWeinberg law is used, consisting of two linear interdependent differential equations relatively to the frequency of an allele with different right sides. The equations reflect the time dynamics of human cells and virus populations during their interaction. In the equation for the virus population the right side is a constant value that characterizes the death of viruses due to the human immune system. In the equation for human cells population the right side depends linearly on the allele frequency of the virus population. The right side of this equation characterizes the death of a cell when a virus inserted in its DNA for its reproduction. Solutions of differential equations are found and the results of these solutions are analyzed. The duration of the pandemic was estimated using parameters of human liver cells and influenza virus.

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18

Abdelsalam, U. M., and M. G. M. Ghazal. "Analytical Wave Solutions for Foam and KdV-Burgers Equations Using Extended Homogeneous Balance Method." Mathematics 7, no. 8 (August 9, 2019): 729. http://dx.doi.org/10.3390/math7080729.

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In this paper, extended homogeneous balance method is presented with the aid of computer algebraic system Mathematica for deriving new exact traveling wave solutions for the foam drainage equation and the Kowerteg-de Vries–Burgers equation which have many applications in industrial applications and plasma physics. The method is effective to construct a series of analytical solutions including many types like periodical, rational, singular, shock, and soliton wave solutions for a wide class of nonlinear evolution equations in mathematical physics and engineering sciences.

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Zayed, E. M. E., and K. A. E. Alurrfi. "The Generalized Projective Riccati Equations Method for Solving Nonlinear Evolution Equations in Mathematical Physics." Abstract and Applied Analysis 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/259190.

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We apply the generalized projective Riccati equations method to find the exact traveling wave solutions of some nonlinear evolution equations with any-order nonlinear terms, namely, the nonlinear Pochhammer-Chree equation, the nonlinear Burgers equation and the generalized, nonlinear Zakharov-Kuznetsov equation. This method presents wider applicability for handling many other nonlinear evolution equations in mathematical physics.

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Gepreel, Khaled A., Taher A. Nofal, and Fawziah M. Alotaibi. "Exact Solutions for Nonlinear Differential Difference Equations in Mathematical Physics." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/756896.

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We modified the truncated expansion method to construct the exact solutions for some nonlinear differential difference equations in mathematical physics via the general lattice equation, the discrete nonlinear Schrodinger with a saturable nonlinearity, the quintic discrete nonlinear Schrodinger equation, and the relativistic Toda lattice system. Also, we put a rational solitary wave function method to find the rational solitary wave solutions for some nonlinear differential difference equations. The proposed methods are more effective and powerful to obtain the exact solutions for nonlinear difference differential equations.

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21

Latifu, Waidi Adebayo. "Mathematical Modeling of the Dynamics of Lubrication." Journal of Electronics,Computer Networking and Applied Mathematics, no. 26 (November 15, 2021): 13–19. http://dx.doi.org/10.55529/jecnam.26.13.19.

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A mathematical model is a model that is abstract and employs mathematical objects in order to explore and explain the real-life situation behaviour. The main thrust of the paper is to describe the dynamics of lubrication by assessing oil based on its viscosity .In this paper, a model is built using Reynolds equation together with the momentum and energy equations. Another model called Modulus equation is established by adopting experimental data .It is therefore shown that dependence of viscosity on temperature can be weakened when viscosity increases with growth of pressure.

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Yasunaga, Michihiro, and John D. Lafferty. "TopicEq: A Joint Topic and Mathematical Equation Model for Scientific Texts." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 7394–401. http://dx.doi.org/10.1609/aaai.v33i01.33017394.

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Scientific documents rely on both mathematics and text to communicate ideas. Inspired by the topical correspondence between mathematical equations and word contexts observed in scientific texts, we propose a novel topic model that jointly generates mathematical equations and their surrounding text (TopicEq). Using an extension of the correlated topic model, the context is generated from a mixture of latent topics, and the equation is generated by an RNN that depends on the latent topic activations. To experiment with this model, we create a corpus of 400K equation-context pairs extracted from a range of scientific articles from arXiv, and fit the model using a variational autoencoder approach. Experimental results show that this joint model significantly outperforms existing topic models and equation models for scientific texts. Moreover, we qualitatively show that the model effectively captures the relationship between topics and mathematics, enabling novel applications such as topic-aware equation generation, equation topic inference, and topic-aware alignment of mathematical symbols and words.

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23

Iqbal, Mujahid, Aly R. Seadawy, Dianchen Lu, and Xianwei Xia. "Construction of bright–dark solitons and ion-acoustic solitary wave solutions of dynamical system of nonlinear wave propagation." Modern Physics Letters A 34, no. 37 (December 6, 2019): 1950309. http://dx.doi.org/10.1142/s0217732319503097.

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The nonlinear (2 + 1)-dimensional Zakharov–Kuznetsov (ZK) equations deal with the nonlinear behavior of waves in collision-less plasma, which contains non-isothermal cold ions and electrons. Two-dimensional dust acoustic solitary waves (DASWs) in magnetized plasma, which consist of trapped electrons and ions are leading to (2 + 1)-dim (ZK) equation by using the perturbation technique. We found the solitary wave solutions of (2 + 1)-dimensional (ZK)-equation, generalized (ZK)-equation and generalized form of modified (ZK)-equation by implementing the modified mathematical method. As a result, we obtained the bright–dark solitons, traveling wave and solitary wave solutions. The physical structure of obtained solutions is represented in 2D and 3D, graphically with the help of Mathematica.

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Khachatryan, Kh A., A. Zh Narimanyan, and A. Kh Khachatryan. "On mathematical modelling of temporal spatial spread of epidemics." Mathematical Modelling of Natural Phenomena 15 (2020): 6. http://dx.doi.org/10.1051/mmnp/2019056.

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In the present work a generalized epidemic model containing a system of integral-differential equations is described. Using different transformations the system is reduced to a single nonlinear multidimensional integral equation. For the obtained equation the existence and uniqueness results are proved. Based on theoretical convergence results several application examples are presented with corresponding numerical results.

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Abdou, Mohamed A., and Essam M. Abulwafa. "Application of the Exp-Functionmethod to the Riccati Equation and New Exact Solutions with Three Arbitrary Functions of Quantum Zakharov Equations." Zeitschrift für Naturforschung A 63, no. 10-11 (November 1, 2008): 646–52. http://dx.doi.org/10.1515/zna-2008-10-1107.

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The Exp-function method with the aid of the symbolic computational system is used for constructing generalized solitary solutions of the generalized Riccati equation. Based on the Riccati equation and its generalized solitary solutions, new exact solutions with three arbitrary functions of quantum Zakharov equations are obtained. It is shown that the Exp-function method provides a straightforward and important mathematical tool for nonlinear evolution equations in mathematical physics.

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Al Nazer, Safaa, Carole Rosier, and Munkhgerel Tsegmid. "Mathematical analysis of a Dupuit-Richards model." Electronic Journal of Differential Equations 2022, no. 01-87 (January 17, 2022): 06. http://dx.doi.org/10.58997/ejde.2022.06.

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This article concerns an alternative model to the 3D-Richards equation to describe the flow of water in shallow aquifers. The model couples the two dominant types of flow existing in the aquifer. The first is described by the classic Richards problem in the upper capillary fringe. The second results from Dupuit's approximation after vertical integration of the conservation laws between the bottom of the aquifer and the saturation interface. The final model consists of a strongly coupled system of parabolic-type partial differential equations that are defined in a time-dependent domain. First, we show how taking the low compressibility of the fluid into account eliminates the nonlinearity in the time derivative of the Richards equation. Then, the general framework of parabolic equations is used in non-cylindrical domains to give a global in time existence result to this problem.

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Ahmed, N. U. "Mathematical problems in modeling artificial heart." Mathematical Problems in Engineering 1, no. 3 (1995): 245–54. http://dx.doi.org/10.1155/s1024123x95000159.

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In this paper we discuss some problems arising in mathematical modeling of artificial hearts. The hydrodynamics of blood flow in an artificial heart chamber is governed by the Navier-Stokes equation, coupled with an equation of hyperbolic type subject to moving boundary conditions. The flow is induced by the motion of a diaphragm (membrane) inside the heart chamber attached to a part of the boundary and driven by a compressor (pusher plate). On one side of the diaphragm is the blood and on the other side is the compressor fluid. For a complete mathematical model it is necessary to write the equation of motion of the diaphragm and all the dynamic couplings that exist between its position, velocity and the blood flow in the heart chamber. This gives rise to a system of coupled nonlinear partial differential equations; the Navier-Stokes equation being of parabolic type and the equation for the membrane being of hyperbolic type. The system is completed by introducing all the necessary static and dynamic boundary conditions. The ultimate objective is to control the flow pattern so as to minimize hemolysis (damage to red blood cells) by optimal choice of geometry, and by optimal control of the membrane for a given geometry. The other clinical problems, such as compatibility of the material used in the construction of the heart chamber, and the membrane, are not considered in this paper. Also the dynamics of the valve is not considered here, though it is also an important element in the overall design of an artificial heart. We hope to model the valve dynamics in later paper.

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Javeed, Shumaila, Khurram Saleem Alimgeer, Sidra Nawaz, Asif Waheed, Muhammad Suleman, Dumitru Baleanu, and M. Atif. "Soliton Solutions of Mathematical Physics Models Using the Exponential Function Technique." Symmetry 12, no. 1 (January 19, 2020): 176. http://dx.doi.org/10.3390/sym12010176.

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This paper is based on finding the exact solutions for Burger’s equation, Zakharov-Kuznetsov (ZK) equation and Kortewegde vries (KdV) equation by utilizing exponential function method that depends on the series of exponential functions. The exponential function method utilizes the homogeneous balancing principle to find the solutions of nonlinear equations. This method is simple, wide-reaching and helpful for finding the exact solution of nonlinear conformable PDEs.

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Gepreel, Khaled A., and A. R. Shehata. "Jacobi Elliptic Solutions for Nonlinear Differential Difference Equations in Mathematical Physics." Journal of Applied Mathematics 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/710375.

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We put a direct new method to construct the rational Jacobi elliptic solutions for nonlinear differential difference equations which may be called the rational Jacobi elliptic functions method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential difference equations in mathematical physics via the lattice equation and the discrete nonlinear Schrodinger equation with a saturable nonlinearity. The proposed method is more effective and powerful to obtain the exact solutions for nonlinear differential difference equations.

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30

Gekkieva, S. Kh, M. M. Karmokov, and M. A. Kerefov. "ON BOUNDARY VALUE PROBLEM FOR GENERALIZED ALLER EQUATION." Vestnik of Samara University. Natural Science Series 26, no. 2 (February 1, 2021): 7–14. http://dx.doi.org/10.18287/2541-7525-2020-26-2-7-14.

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The mathematical models of fluid filtration processes in porous media with a fractal structure and memory are based on differential equations of fractional order in both time and space variables. The dependence of the soil water content can significantly affect the moisture transport in capillary-porous media. The paper investigates the generalized Aller equation widely used in mathematical modeling of the processes related to water table dynamics in view of fractal structure. As a mathematical model of the Aller equation withRiemann Liouville fractional derivatives, a loaded fractional order equation is proposed, and a solution to the Goursat problem has been written out for this model in explicit form.

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Kurkin, A. S. "Mathematical research of the phase transformation kinetics of alloyed steel." Industrial laboratory. Diagnostics of materials 85, no. 12 (December 29, 2019): 25–32. http://dx.doi.org/10.26896/1028-6861-2019-85-12-25-32.

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Regulation of the process parameters allows obtaining the desired properties of the metal. Computer simulation of technological processes with allowance for structural and phase transformations of the metal forms the basis for the proper choice of those parameters. Methods of mathematical modeling are used to study the main diffusion and diffusion-free processes of transformations in alloyed steels during heating and cooling. A comparative analysis of the kinetic equations of phase transformations including the Kolmogorov – Avrami and Austin – Rickett equations which describe in different ways the time dependence of the diffusion transformation rate and attained degree of transformation has been carried out. It is shown that the Austin – Rickett equation is equivalent to the Kolmogorov – Avrami equation with a smooth decrease of the Avrami exponent during the transformation process. The advantages of the Kolmogorov – Avrami equation in modeling the kinetics of ferrite-pearlite and bainite transformations and validity of this equation for modeling the kinetics of martensite transformations during tempering are shown. The parameters for describing the tempering process of steel 35 at different temperatures are determined. The proposed model is compared with equations based on the Hollomon – Jaffe parameter. The diagrams of martensitic transformation of alloyed steels and disadvantages of the Koistinen – Marburger equation used to describe them are analyzed. The equations of the temperature dependence of the transformation degree, similar to the Kolmogorov – Avrami and Austin – Rickett equations, are derived. The equations contain the minimum set of the parameters that can be found from published data. An iterative algorithm for determining parameters of the equations is developed, providing the minimum standard deviation of the constructed dependence from the initial experimental data. The dependence of the accuracy of approximation on the temperature of the onset of transformation is presented. The complex character of the martensitic transformation development for some steels is revealed. The advantage of using equations of the Austin – Rickett type when constructing models from a limited amount of experimental data is shown. The results obtained make it possible to extend the approaches used in modeling diffusion processes of austenite decomposition to description of the processes of formation and decomposition of martensite in alloyed steels.

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Glagolev, Mikhail V., Aleksandr F. Sabrekov, and Vladimir M. Goncharov. "Delay differential equations as a tool for mathematical modelling of population dynamic." Environmental Dynamics and Global Climate Change 9, no. 2 (November 27, 2018): 40–63. http://dx.doi.org/10.17816/edgcc10483.

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The manuscript constitutes a lecture from a course “Mathematical modelling of biological processes”, adapted to the format of the journal paper. This course of lectures is held by one of authors in Ugra State University. Delay differential equations are widely used in different ecological and biological problems. It has to do with the fact that delay differential equations are able to take into account that different biological processes depend not only on the state of the system at the moment but on the state of the system in previous moments too. The most popular case of using delay differential equations in biology is modelling in population ecology (including the modelling of several interacting populations dynamic, for example, in predator-prey system). Also delay differential equations are considered in demography, immunology, epidemiology, molecular biology (to provide mathematical description of regulatory mechanisms in a cell functioning and division), physiology as well as for modelling certain important production processes (for example, in agriculture). In the beginning of the paper as introduction some basic concepts of differential difference equation theory (delay differential equations are specific type of differential difference equations) is considered and their classification is presented. Then it is discussed in more detail how such an important equations of population dynamic as Maltus equation and logistic (Verhulst-Pearl) equation are transformed into corresponsive delay differential equations – Goudriaan-Roermund and Hutchinson. Then several discussion questions on using of a delay differential equations in biological models are considered. It is noted that in a certain cases using of a delay differential equations lead to an incorrect behavior from the point of view of common sense. Namely solution of Goudriaan-Roermund equation with harvesting, stopped when all species were harvested, shows that spontaneous generation takes place in the system. This incorrect interpretation has to do with the fact that delay differential equations are used to simplify considered models (that are usually are systems of ordinary differential equations). Using of integro-differential equations could be more appropriate because in these equations background could be taken into account in a more natural way. It is shown that Hutchinson equation can be obtained by simplification of Volterra integral equation with a help of Diraq delta function. Finally, a few questions of analytical and numerical solution of delay differential equations are discussed.

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Gepreel, Khaled A., and Amr M. S. Mahdy. "Algebraic computational methods for solving three nonlinear vital models fractional in mathematical physics." Open Physics 19, no. 1 (January 1, 2021): 152–69. http://dx.doi.org/10.1515/phys-2021-0020.

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Abstract This research paper uses a direct algebraic computational scheme to construct the Jacobi elliptic solutions based on the conformal fractional derivatives for nonlinear partial fractional differential equations (NPFDEs). Three vital models in mathematical physics [the space-time fractional coupled Hirota Satsuma KdV equations, the space-time fractional symmetric regularized long wave (SRLW equation), and the space-time fractional coupled Sakharov–Kuznetsov (S–K) equations] are investigated through the direct algebraic method for more explanation of their novel characterizes. This approach is an easy and powerful way to find elliptical Jacobi solutions to NPFDEs. The hyperbolic function solutions and trigonometric functions where the modulus and, respectively, are degenerated by Jacobi elliptic solutions. In this style, we get many different kinds of traveling wave solutions such as rational wave traveling solutions, periodic, soliton solutions, and Jacobi elliptic solutions to nonlinear evolution equations in mathematical physics. With the suggested method, we were fit to find much explicit wave solutions of nonlinear integral differential equations next converting them into a differential equation. We do the 3D and 2D figures to define the kinds of outcome solutions. This style is moving, reliable, powerful, and easy for solving more difficult nonlinear physics mathematically.

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Hasan, Faeza, Mohamed Abdoon, Rania Saadeh, Mohammed Berir, and Ahmad Qazza. "A New Perspective on the Stochastic Fractional Order Materialized by the Exact Solutions of Allen-Cahn Equation." International Journal of Mathematical, Engineering and Management Sciences 8, no. 5 (October 1, 2023): 912–26. http://dx.doi.org/10.33889/ijmems.2023.8.5.052.

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Stochastic fractional differential equations are among the most significant and recent equations in physical mathematics. Consequently, several scholars have recently been interested in these equations to develop analytical approximations. In this study, we highlight the stochastic fractional space Allen-Cahn equation (SFACE) as a major application of this class. In addition, we utilize the simplest equation method (SEM) with a dual sense of Brownian motion to convert the presented equation into an ordinary differential equation (ODE) and apply an effective computational technique to obtain exact solutions. By carefully comparing the derived solutions with solutions from other articles, we prove the distinction of these solutions for their diversity and the discovery of new solutions for SFACE that appear in many scientific fields, such as mathematical biology, quantum mechanics, and plasma physics. The results introduced in this article were obtained by plotting several graphs and examining how noise affects exact solutions using Mathematica and MATLAB software packages.

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van Odyck, Daniel E. A., John B. Bell, Franck Monmont, and Nikolaos Nikiforakis. "The mathematical structure of multiphase thermal models of flow in porous media." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2102 (November 4, 2008): 523–49. http://dx.doi.org/10.1098/rspa.2008.0268.

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This paper is concerned with the formulation and numerical solution of equations for modelling multicomponent, two-phase, thermal fluid flow in porous media. The fluid model consists of individual chemical component (species) conservation equations, Darcy's law for volumetric flow rates and an energy equation in terms of enthalpy. The model is closed with an equation of state and phase equilibrium conditions that determine the distribution of the chemical components into phases. It is shown that, in the absence of diffusive forces, the flow equations can be split into a system of hyperbolic conservation laws for the species and enthalpy and a parabolic equation for pressure. This decomposition forms the basis of a sequential formulation where the pressure equation is solved implicitly and then the component and enthalpy conservation laws are solved explicitly. A numerical method based on this sequential formulation is presented and used to demonstrate some typical flow behaviour that occurs during fluid injection into a reservoir.

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Stanciu, Ioana, and Noureddine Ouerfelli. "Application Extended Vogel-Tammann-Fulcher Equation for soybean oil." Oriental Journal Of Chemistry 37, no. 6 (December 30, 2021): 1287–94. http://dx.doi.org/10.13005/ojc/370603.

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Mathematical models that describe the variation of soybean oil viscosity with temperature according to the recent WLF and VTF (or VFT) equations and traditionally by the Arrhenius equation. The Arrhenius equation shows that the viscosity of the oil is proportional to the absolute temperature and is determined by the activation energy parameter. In Arrhenius' equation the absolute temperature is replaced by T + b where both adjustable T and b are in ° C. The mathematical models described by the equations WLF and VTF, are equal to each other. All three equations are in the same model when used for experimental data of temperature-viscosity dependence, they give exactly the same very high regression coefficient.

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Tang, Bo, Xuemin Wang, Yingzhe Fan, and Junfeng Qu. "Exact Solutions for a Generalized KdV-MKdV Equation with Variable Coefficients." Mathematical Problems in Engineering 2016 (2016): 1–10. http://dx.doi.org/10.1155/2016/5274243.

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By using solutions of an ordinary differential equation, an auxiliary equation method is described to seek exact solutions of variable-coefficient KdV-MKdV equation. As a result, more new exact nontravelling solutions, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions, and combined Jacobi elliptic function solutions, for the KdV-MKdV equation are obtained. It is shown that the considered method provides a very effective, convenient, and powerful mathematical tool for solving many other nonlinear partial differential equations with variable coefficients in mathematical physics.

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HASSANABADI, S., A. A. RAJABI, and S. ZARRINKAMAR. "CORNELL AND KRATZER POTENTIALS WITHIN THE SEMIRELATIVISTIC TREATMENT." Modern Physics Letters A 27, no. 10 (March 28, 2012): 1250057. http://dx.doi.org/10.1142/s0217732312500575.

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We obtain approximate analytical solutions of the two-body Spinless Salpeter equation under both Cornell (Funnel) and Kratzer potentials and thereby provide a basis to study semirelativistic two-body systems which frequently appear in physics. Apart from the physical significance, the work is mathematically appealing as we have actually reported approximate analytical solutions of the corresponding Heun equations which appear as one of the most challenging differential equations of mathematical physics.

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Plecas, I., and S. Dimovic. "Mathematical modelling of transport phenomena in concrete matrix." Facta universitatis - series: Physics, Chemistry and Technology 9, no. 1 (2011): 21–27. http://dx.doi.org/10.2298/fupct1101021p.

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Two fundamental concerns must be addressed when attempting to isolate low-level waste in a disposal facility on land. The first concern is isolating the waste from water, or hydrologic isolation. The second is preventing movement of the radionuclides out of the disposal facility, or radionuclide migration. Particularly, we have investigated the latter modified scenario. To assess the safety for disposal of radioactive waste-concrete composition, the leakage of 60Co from a waste composite into a surrounding fluid has been studied. Leakage tests were carried out by original method, developed in Vinca Institute. Transport phenomena involved in the leaching of a radioactive material from a cement composite matrix are investigated using three methods based on theoretical equations. These are: the diffusion equation for a plane source an equation for diffusion coupled to a first-order equation, and an empirical method employing a polynomial equation. The results presented in this paper are from a 25-year mortar and concrete testing project that will influence the design choices for radioactive waste packaging for a future Serbian radioactive waste disposal center.

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Zaenal, Reza Muhamad, and Tio Heriyana. "Students' Mathematical Communication Skills in Solving Quadratic Equation Problems." Indo-MathEdu Intellectuals Journal 2, no. 2 (October 20, 2021): 92–105. http://dx.doi.org/10.54373/imeij.v2i2.24.

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Mathematics learning today only emphasizes the mastery of the material alone and more one-way communication with students. The buildup of information from teachers results in the learning style of students who tend to memorize. This research aims to find out the mathematical communication skills of students in solving quadratic equation problems. This type of research is qualitative descriptive. Determination of the subject using purposive sampling techniques with the criteria of the student's test results. The study subjects numbered three people consisting of students who obtained grades in high, medium, and low categories. Indicators of communication skills in this researcher are (1) explain and make mathematical statements that have been studied, (2) state everyday events in a language or mathematical symbol, and (3) provide an explanation of mathematical ideas, concepts, or situations with their own language in the form of writing mathematically. The instruments used are researchers, tests and interviews. Data analysis techniques are using quantitataif data analysis with data reduction stages, data presentation, and conclusion withdrawal. The results of the data analysis showed that the KSS subject met 3 indicators of mathematical communication skills. AHB subjects meet 2 of the 3 indicators, namely the first and third indicators. WL subjects meet reached 1 of 3 indicators of mathematical communication skills, namely the first indicators.

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FASANO, ANTONIO, DIETMAR HÖMBERG, and LUCIA PANIZZI. "A MATHEMATICAL MODEL FOR CASE HARDENING OF STEEL." Mathematical Models and Methods in Applied Sciences 19, no. 11 (November 2009): 2101–26. http://dx.doi.org/10.1142/s0218202509004054.

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A mathematical model for the case hardening of steel is presented. Carbon is dissolved in the surface layer of a low-carbon steel part at a temperature sufficient to render the steel austenitic, followed by quenching to form a martensitic microstructure. The model consists of a nonlinear evolution equation for the temperature, coupled with a nonlinear evolution equation for the carbon concentration, both coupled with two ordinary differential equations to describe the evolution of phase fractions. We investigate questions of existence and uniqueness of a solution and finally present some numerical simulations.

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Zhang, Sheng, Bo Xu, and Ao Xue Peng. "A Generalized Auxiliary Equation Method for the Quadratic Nonlinear KG Equation." Applied Mechanics and Materials 394 (September 2013): 571–76. http://dx.doi.org/10.4028/www.scientific.net/amm.394.571.

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A generalized auxiliary equation method with symbolic computation is used to construct more general exact solutions of the quadratic nonlinear Klein-Gordon (KG) equation. As a result, new and more general solutions are obtained. It is shown that the generalized auxiliary equation method provides a more powerful mathematical tool for solving nonlinear partial differential equations arising in the fields of nonlinear sciences.

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Boyle, P. P., W. Tian, and Fred Guan. "The Riccati Equation in Mathematical Finance." Journal of Symbolic Computation 33, no. 3 (March 2002): 343–55. http://dx.doi.org/10.1006/jsco.2001.0508.

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Okino, T., T. Shimozaki, R. Fukuda, and Hiroki Cho. "Analytical Solutions of the Boltzmann Transformation Equation." Defect and Diffusion Forum 322 (March 2012): 11–31. http://dx.doi.org/10.4028/www.scientific.net/ddf.322.11.

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The so-called continuity equation derived by Fick is one of the most fundamental and extremely important equations in physics and/or in materials science. As is well known, this partial differential equation is also called the diffusion equation or the heat conduction equation and is applicable to physical phenomena of the conservation system. Incorporating the parabolic law relevant to a random movement into it, Boltzmann obtained the ordinary differential equation (B-equation). Matano then applied the B-equation to the analysis of the nonlinear problem for the interdiffusion experiment. The empirical Boltzmann-Matano (B-M) method has been successful in the metallurgical field. However, the nonlinear B-equation was not mathematically solved for a long time. Recently, the analytical solutions of the B-equation were obtained in accordance with the results of the B-M method. In the present study, an applicable limitation of the B-equation to the interdiffusion problems is investigated from a mathematical point of view.

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Teskeredzic, Armin, and Rejhana Blazevic. "Transient Radiator Room Heating—Mathematical Model and Solution Algorithm." Buildings 8, no. 11 (November 18, 2018): 163. http://dx.doi.org/10.3390/buildings8110163.

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A mathematical model and robust numerical solution algorithm for radiator heating of an arbitrary room is presented in this paper. Three separate and coupled transient thermal energy equations are solved. A modified transient heat conduction equation is used for solving the heat transfer at multi-layer outer walls and room assembly. Heat exchange between the inner walls and the observed room are represented with their own transport equation and the transient thermal energy equation is solved for radiators as well. Explicit coupling of equations and linearization of source terms result in a simple, accurate, and stabile solution algorithm. Verification of the developed methodology is demonstrated on three carefully selected test cases for which an analytical solution can be found. The obtained results show that even for the small temperature differences between inner walls and room air, the corresponding heat flux can be larger than the transmission heat flux through outer walls or windows. The benefits of the current approach are stressed, while the plans for the further development and application of the methodology are highlighted at the end.

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Na'imah, Farohatin, Yuni Yulida, and Muhammad Ahsar Karim. "PEMBENTUKAN PERSAMAAN VAN DER POL DAN SOLUSI MENGGUNAKAN METODE MULTIPLE SCALE." JURNAL MATEMATIKA MURNI DAN TERAPAN EPSILON 14, no. 2 (March 2, 2021): 104. http://dx.doi.org/10.20527/epsilon.v14i2.958.

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Mathematical modeling is one of applied mathematics that explains everyday life in mathematical equations, one example is Van der Pol equation. The Van der Pol equation is an ordinary differential equation derived from the Resistor, Inductor, and Capacitor (RLC) circuit problem. The Van der Pol equation is a nonlinear ordinary differential equations that has a perturbation term. Perturbation is a problem in the system, denoted by ε which has a small value 0<E<1. The presence of perturbation tribe result in difficulty in solving the equation using anlytical methode. One method that can solve the Van der Pol equation is a multiple scale method. The purpose of this study is to explain the constructions process of Van der Pol equation, analyze dynamic equations around equilibrium, and determine the solution of Van der Pol equation uses a multiple scale method. From this study it was found that the Van der Pol equation system has one equilibrium. Through stability analysis, the Van der Pol equation system will be stable if E= 0 and -~<E<=-2. The solution of the Van der Pol equation with the multiple scale method is Keywords: Van der Pol equation, equilibrium, stability, multiple scale.

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Kadirov, Yorkin, Abdukhalil Samadov, and Olmos Goziev. "Equation of dynamics of greenhouse microclimate parameters." E3S Web of Conferences 390 (2023): 04019. http://dx.doi.org/10.1051/e3sconf/202339004019.

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When developing and researching management tasks, a climate model is needed. There are different ways to mathematically describe or predict parameters for controlling a greenhouse complex. Most authors represent the ambient air temperature by a Fourier series or its segment, as well as by trigonometric expressions. This approach makes it possible to use the obtained representation for solving specific thermal and physical problems, as well as for calculating heat and moisture resistance. A mathematical model of greenhouse microclimate dynamics has been developed. The mathematical model describes parameters such as air temperature, relative humidity and soil humidity. The developed model in the form of a differential equation is calculated numerically using the Maple 17.

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Dukhnovsky, Sergey. "A self-similar solution for the two-dimensional Broadwell system via the Bateman equation." Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3(100) (June 2023): 30–40. http://dx.doi.org/10.56415/basm.y2022.i3.p30.

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A self-similar solution of the Broadwell system is found. Here the solution is sought using a reduction that transforms the given system into a system of differential equations. Further, the solution is constructed using the Painlev\'e series. Here the system already passes the Painlev\'e test and it is possible to find the solution if the equations in resonance satisfy the solution of the two-dimensional Bateman equation. Exact solution of the Bateman equation is established, allowing to find new explicit solution for the original system. In the process of calculations, we use the Wolfram Mathematica program. The proof of these results is carried out at a rigorous mathematical level.

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Evseev, Dmitriy, Yuriy Sarychev, and Sergey Bespal'ko. "MATHEMATICAL MODEL OF THE CAR SHOCK ABSORBER BASED ON VISCOUS FRICTION." Transport engineering 2022, no. 01-02 (February 21, 2022): 89–95. http://dx.doi.org/10.30987/2782-5957-2022-01-02-89-95.

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The work objective is to develop a mathematical model of the force characteristics of shock absorbers based on viscous friction, including elastomeric ones. For this purpose, the following tasks are solved in the work: the existing approaches to the mathematical modelling of hydraulic shock absorbers are analysed, a mathematical model based on hydraulic equations (the Darcy–Weisbach equation) is proposed, the developed mathematical model is verified and the results are compared with the results obtained based on the existing approach. Research methods: the equation of fluid flow through holes with hydraulic friction; D'Alembert's principle for composing the equation of car swaying motions; the Euler method for numerical integration of the differential equation. The novelty of the work consists in the fact that a mathematical model of the power characteristics of shock absorbers with viscous friction is proposed, based on the quadratic dependence of the reaction on the deformation rate. The results are the study of car swaying motions based on the traditional and proposed approaches. The proposed mathematical models can be used to develop shock absorbers with improved characteristics when designing passenger cars.

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Wang, Hong Lei, and Chun Huan Xiang. "Riccati Equation Solutions to Higher Order Korteweg-de Vries Equation." Advanced Materials Research 926-930 (May 2014): 3240–44. http://dx.doi.org/10.4028/www.scientific.net/amr.926-930.3240.

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The traveling wave solutions to the heigher order Korteweg-de Vries equation is obtained by using Riccati equation. The method is straightforward and concise, the applications are promising to obtain traveling wave solutions of various partial differential equations. It is shown that the Riccati equation method, with the symbolic computation, provide an effective and powerful mathematical tools for solving such systems. The numerical simulation of the solutions are given for completeness.

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