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Journal articles on the topic 'Hybrid differential equations'

Author: Grafiati

Published: 4 June 2025

Last updated: 23 June 2025

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1

Sadhasivam, V., and M. Deepa. "Oscillation criteria for fractional impulsive hybrid partial differential equations." Issues of Analysis 26, no. 2 (2019): 73–91. http://dx.doi.org/10.15393/j3.art.2019.5910.

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2

Dhage, Bapurao C. "Differential inequalities for hybrid fractional differential equations." Journal of Mathematical Inequalities, no. 3 (2013): 453–59. http://dx.doi.org/10.7153/jmi-07-40.

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3

Harir, A., S. Melliani, and L. S. Chadli. "Fuzzy fractional hybrid differential equations." Carpathian Mathematical Publications 14, no. 2 (2022): 332–44. http://dx.doi.org/10.15330/cmp.14.2.332-344.

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This article is related to present and solve the theory of fractional hybrid differential equations with fuzzy initial values involving the fuzzy Riemann-Liouville fractional differential operators of order $0 < q < 1$. For the concerned presentation, we study the existence and uniqueness of a fuzzy solution are brought in detail basing on the concept of generalized division of fuzzy numbers. We have developed and investigated a fuzzy solution of a fuzzy fractional hybrid differential equation. At the end we have given an example is provided to illustrate the theory.

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4

Hilal, Khalid, and Ahmed Kajouni. "Existence of the Solution for System of Coupled Hybrid Differential Equations with Fractional Order and Nonlocal Conditions." International Journal of Differential Equations 2016 (2016): 1–9. http://dx.doi.org/10.1155/2016/4726526.

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This paper is motivated by some papers treating the fractional hybrid differential equations with nonlocal conditions and the system of coupled hybrid fractional differential equations; an existence theorem for fractional hybrid differential equations involving Caputo differential operators of order1<α≤2is proved under mixed Lipschitz and Carathéodory conditions. The existence and uniqueness result is elaborated for the system of coupled hybrid fractional differential equations.

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5

Khan, Subuhi, Mumtaz Riyasat, and Shahid Ahmad Wani. "On some classes of differential equations and associated integral equations for the Laguerre–Appell polynomials." Advances in Pure and Applied Mathematics 9, no. 3 (2018): 185–94. http://dx.doi.org/10.1515/apam-2017-0079.

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Abstract The article aims to explore some new classes of differential equations and associated integral equations for some hybrid families of Laguerre polynomials. The recurrence relations and differential, integro-differential and partial differential equations for the hybrid Laguerre–Appell polynomials are derived via the factorization method. An analogous study of these results for the hybrid Laguerre–Bernoulli, Euler and Genocchi polynomials is presented. Further, the Volterra integral equations for the hybrid Laguerre–Appell polynomials and for their corresponding members are also explored.

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6

Benallou, Mohamed, Hamid Beddani, and Moustafa Beddani. "Existence of solution for a tripled system of fractional hybrid differential equations with laplacie involving Caputo derivatives." STUDIES IN ENGINEERING AND EXACT SCIENCES 5, no. 2 (2024): e11971. https://doi.org/10.54021/seesv5n2-734.

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In this paper, we study the existence of solutions for a tripled system of Fractional hybrid differential equations with nonlocal integro multi point boundary conditions by using the Laplacian operator of degree p and the Caputo derivatives, we know that the differential equations with the Laplacian operator appeared for the first time when Leibenson was attempting to derive an accurate formula to model turbulent flow in the porous medium, in this work we study the case of the Fractional hybrid differential equations who are the quadratic perturbations of nonlinear differential equations. Dhage and Lakshmikantham [12] discussed the hybrid differential equation They established the existence, uniqueness results, and some fundamental differential inequalities for hybrid differential equations initiating the study of the theory of such systems and proved to utilize the theory of inequalities, its existence of extremal solutions, and comparison results. Hilal and Kajouni [19] have studied boundary fractional hybrid differential equations involving Caputo differential operators, so In this article, we are interested in the existence result of the solution of hybrid nonlinear differential equations. obtained by the hybrid fixed point theorem for a sum of three operators due to Dhage. An illustrative example is presented at the end to show the applicability of the results. To the best of our knowledge, this is the first time where such problem is considered.

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7

Ivaz, K., A. Khastan, and Juan J. Nieto. "A Numerical Method for Fuzzy Differential Equations and Hybrid Fuzzy Differential Equations." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/735128.

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Numerical algorithms for solving first-order fuzzy differential equations and hybrid fuzzy differential equations have been investigated. Sufficient conditions for stability and convergence of the proposed algorithms are given, and their applicability is illustrated with some examples.

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8

Herzallah, Mohamed A. E., and Dumitru Baleanu. "On Fractional Order Hybrid Differential Equations." Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/389386.

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We develop the theory of fractional hybrid differential equations with linear and nonlinear perturbations involving the Caputo fractional derivative of order0<α<1. Using some fixed point theorems we prove the existence of mild solutions for two types of hybrid equations. Examples are given to illustrate the obtained results.

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9

Dhage, Bapurao C., and V. Lakshmikantham. "Basic results on hybrid differential equations." Nonlinear Analysis: Hybrid Systems 4, no. 3 (2010): 414–24. http://dx.doi.org/10.1016/j.nahs.2009.10.005.

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10

Zhao, Yige, Shurong Sun, Zhenlai Han, and Qiuping Li. "Theory of fractional hybrid differential equations." Computers & Mathematics with Applications 62, no. 3 (2011): 1312–24. http://dx.doi.org/10.1016/j.camwa.2011.03.041.

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11

Deng, Feiqi, Qi Luo, and Xuerong Mao. "Stochastic stabilization of hybrid differential equations." Automatica 48, no. 9 (2012): 2321–28. http://dx.doi.org/10.1016/j.automatica.2012.06.044.

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12

Yin, Gang, and Jifeng Zhang. "Hybrid singular systems of differential equations." Science in China Series F Information Sciences 45, no. 4 (2002): 241–58. http://dx.doi.org/10.1360/02yf9022.

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13

Dhage, Bapurao Chandrabahan. "Approximation methods in the theory of hybrid differential equations with linear perturbations of second type." Tamkang Journal of Mathematics 45, no. 1 (2014): 39–61. http://dx.doi.org/10.5556/j.tkjm.45.2014.1328.

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In this paper, some existence theorems for the extremal solutions are proved for an initial value problem of nonlinear hybrid differential equations via constructive methods. The monotone iterative techniques for initial value problems of first order hybrid differential equations are developed and it is shown that the sequences of successive iterations defined in a certain way converge to the minimal and maximal solutions of the hybrid differential equations.

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14

Hannabou, Mohamed, and Hilal Khalid. "Investigation of a Mild Solution to Coupled Systems of Impulsive Hybrid Fractional Differential Equations." International Journal of Differential Equations 2019 (December 10, 2019): 1–9. http://dx.doi.org/10.1155/2019/2618982.

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The study of coupled systems of hybrid fractional differential equations requires the attention of scientists for the exploration of their different important aspects. Our aim in this paper is to study the existence and uniqueness of the solution for impulsive hybrid fractional differential equations. The novelty of this work is the study of a coupled system of impulsive hybrid fractional differential equations with initial and boundary hybrid conditions. We used the classical fixed-point theorems such as the Banach fixed-point theorem and Leray–Schauder alternative fixed-point theorem for existence results. We also give an example of the main results.

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15

S. Sekar and A. Sakthivel. "Numerical investigation of the hybrid fuzzy differential equations using He's homotopy perturbation method." Malaya Journal of Matematik 5, no. 02 (2017): 475–82. http://dx.doi.org/10.26637/mjm502/026.

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This paper presents an efficient method namely He's Homotopy Perturbation Method (HHPM) is introduced for solving hybrid fuzzy differential equations based on Seikkala derivative with initial value problem [2]. The proposed method is tested on hybrid fuzzy differential equations. The discrete solutions obtained through He's Homotopy Perturbation Method are compared with Leapfrog method [13]. The applicability of the He's Homotopy Perturbation Method is more suitable to solve the hybrid fuzzy differential equations. Error graphs are presented to highlight the efficiency of the He's Homotopy Perturbation Method.

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16

Geer, James F., and Carl M. Andersen. "Hybrid Pade´-Galerkin Technique for Differential Equations." Applied Mechanics Reviews 46, no. 11S (1993): S255—S265. http://dx.doi.org/10.1115/1.3122644.

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A three-step hybrid analysis technique, which successively uses the regular perturbation expansion method, the Pade´ expansion method, and then a Galerkin approximation, is presented and applied to some model boundary value problems. In the first step of the method, the regular perturbation method is used to construct an approximation to the solution in the form of a finite power series in a small parameter ε associated with the problem. In the second step of the method, the series approximation obtained in step one is used to construct a Pade´ approximation in the form of a rational function in the parameter ε. In the third step, the various powers of ε which appear in the Pade´ approximation are replaced by new (unknown) parameters {δj}. These new parameters are determined by requiring that the residual formed by substituting the new approximation into the governing differential equation is orthogonal to each of the perturbation coordinate functions used in step one. The technique is applied to model problems involving ordinary or partial differential equations. In general, the technique appears to provide good approximations to the solution even when the perturbation and Pade´ approximations fail to do so. The method is discussed and topics for future investigations are indicated.

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17

S, SALUNKHE. "Approximation Method for Hybrid Functional Differential Equations." International Journal of Mathematics Trends and Technology 65, no. 6 (2019): 21–25. http://dx.doi.org/10.14445/22315373/ijmtt-v65i6p504.

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18

Ruan, Dehao, Liping Xu, and Jiaowan Luo. "Stability of hybrid stochastic functional differential equations." Applied Mathematics and Computation 346 (April 2019): 832–41. http://dx.doi.org/10.1016/j.amc.2018.03.064.

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19

Dhage, Bapurao C., Shyam B. Dhage, and Sotiris K. Ntouyas. "Approximating solutions of nonlinear hybrid differential equations." Applied Mathematics Letters 34 (August 2014): 76–80. http://dx.doi.org/10.1016/j.aml.2014.04.002.

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20

Hale, Jack K., and Wenzhang Huang. "Variation of constants for hybrid systems of functional differential equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 125, no. 1 (1995): 1–12. http://dx.doi.org/10.1017/s0308210500030729.

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The objective is to derive a variation of constants formula for systems of functional differential equations (or delay differential equations) coupled with functional equations (or difference equations). The difficulties arise because of the constraints imposed by the functional equations.

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21

Razzaghi, Mohsen. "Hybrid approximations for fractional calculus." ITM Web of Conferences 29 (2019): 01001. http://dx.doi.org/10.1051/itmconf/20192901001.

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In this paper, a numerical method for solving the fractional differential equations is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting ofblock-pulse functions and Taylor polynomials are presented. The Riemann-Liouville fractional integral operator for hybrid functions is given. This operator is then utilized to reduce the solution of the initial value problems for fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

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22

Noroozi, Hossein, Alireza Ansari, and Mohammad Shafi Dahaghin. "Existence Results for the Distributed Order Fractional Hybrid Differential Equations." Abstract and Applied Analysis 2012 (2012): 1–16. http://dx.doi.org/10.1155/2012/163648.

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We introduce the distributed order fractional hybrid differential equations (DOFHDEs) involving the Riemann-Liouville differential operator of order0<q<1with respect to a nonnegative density function. Furthermore, an existence theorem for the fractional hybrid differential equations of distributed order is proved via a fixed point theorem in the Banach algebras under the mixed Lipschitz and Caratheodory conditions.

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23

Nahid, Tabinda, and Subuhi Khan. "Differential equations for certain hybrid special matrix polynomials." Boletim da Sociedade Paranaense de Matemática 41 (December 24, 2022): 1–10. http://dx.doi.org/10.5269/bspm.52758.

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The main aim of this article is to find the matrix recurrence relation and shift operators for the Gould-Hopper-Laguerre-Appell matrix polynomials. The matrix differential, matrix integro-differential and matrix partial differential equations are derived for these polynomials via factorization method. Certain examples are constructed in order to illustrate the applications of the results.

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24

Dhandapani, Prasantha Bharathi, Jayakumar Thippan, Carlos Martin-Barreiro, Víctor Leiva, and Christophe Chesneau. "Numerical Solutions of a Differential System Considering a Pure Hybrid Fuzzy Neutral Delay Theory." Electronics 11, no. 9 (2022): 1478. http://dx.doi.org/10.3390/electronics11091478.

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In this paper, we propose and derive a new system called pure hybrid fuzzy neutral delay differential equations. We apply the classical fourth-order Runge–Kutta method (RK-4) to solve the proposed system of ordinary differential equations. First, we define the RK-4 method for hybrid fuzzy neutral delay differential equations and then establish the efficiency of this method by utilizing it to solve a particular type of fuzzy neutral delay differential equation. We provide a numerical example to verify the theoretical results. In addition, we compare the RK-4 and Euler solutions with the exact solutions. An error analysis is conducted to assess how much deviation from exactness is found in the two numerical methods. We arrive at the same conclusion for our hybrid fuzzy neutral delay differential system since the RK-4 method outperforms the classical Euler method.

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25

Razzaghi, M. "A numerical scheme for problems in fractional calculus." ITM Web of Conferences 20 (2018): 02001. http://dx.doi.org/10.1051/itmconf/20182002001.

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In this paper, a new numerical method for solving the fractional differential equations with boundary value problems is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The Riemann-Liouville fractional integral operator for hybrid functions is given. This operator is then utilized to reduce the solution of the boundary value problems for fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

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26

Awadalla, Muath, and Nazim I. Mahmudov. "On System of Mixed Fractional Hybrid Differential Equations." Journal of Function Spaces 2022 (June 1, 2022): 1–11. http://dx.doi.org/10.1155/2022/1258823.

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In this article, we find the necessary conditions for the existence and uniqueness of solutions to a system of hybrid equations that contain mixed fractional derivatives (Caputo and Riemann-Liouville). We also verify the stability of these solutions using the Ulam-Hyers (U-H) technique. Finally, this study ends with applied examples that show how to proceed and verify the conditions of our theoretical results.

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27

Wu, Hao, Junhao Hu, and Chenggui Yuan. "Stability of hybrid pantograph stochastic functional differential equations." Systems & Control Letters 160 (February 2022): 105105. http://dx.doi.org/10.1016/j.sysconle.2021.105105.

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28

An, Truong Vinh, Ngo Van Hoa, and Nguyen Anh Tuan. "Impulsive hybrid interval-valued functional integro-differential equations." Journal of Intelligent & Fuzzy Systems 32, no. 1 (2017): 529–41. http://dx.doi.org/10.3233/jifs-152405.

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29

Duane, Simon, and John B. Kogut. "Hybrid Stochastic Differential Equations Applied to Quantum Chromodynamics." Physical Review Letters 55, no. 25 (1985): 2774–77. http://dx.doi.org/10.1103/physrevlett.55.2774.

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30

Dassios, Ioannis, Angel Vaca, and Federico Milano. "On hybrid dynamical systems of differential–difference equations." Chaos, Solitons & Fractals 187 (October 2024): 115431. http://dx.doi.org/10.1016/j.chaos.2024.115431.

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31

Ibrahim, Iman H., and Fatma M. Yousry. "Hybrid special class for solving differential-algebraic equations." Numerical Algorithms 69, no. 2 (2014): 301–20. http://dx.doi.org/10.1007/s11075-014-9897-x.

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32

Fard, Omid Solaymani, and Tayebeh Aliabdoli Bidgoli. "Solving hybrid fuzzy differential equations by Chebyshev wavelet." SeMA Journal 72, no. 1 (2015): 61–82. http://dx.doi.org/10.1007/s40324-015-0049-6.

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33

Salgado, G. H. O., and L. A. Aguirre. "A hybrid algorithm for Caputo fractional differential equations." Communications in Nonlinear Science and Numerical Simulation 33 (April 2016): 133–40. http://dx.doi.org/10.1016/j.cnsns.2015.08.024.

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34

Adebisi, O., OA Ajala, AA Bepo, and M. Taiwo. "Thermal Performance of Hybrid Nanoparticles on Radiative Flow with Slip Boundary Condition Under the Influence of Riga Plate." International Journal of Advanced Multidisciplinary Research and Studies 5, no. 1 (2025): 1270–79. https://doi.org/10.62225/2583049x.2025.5.1.3793.

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The rapid development of modern nanotechnology in industries and medical area have brought about the idea of mixing more than one nanoparticle in a base fluid which is called hybrid nanofluid. Hybrid nanofluid enhances the thermophysical properties of flow better compare to nanofluid. The main object of this study is to examine the thermal performance of hybrid nanoparticles on radiative flow with slip boundary condition under the influence of Riga Plate. This lead to a mathematical flow model in terms of highly non-linear partial differential equations (PDEs). The partial differential equations and their boundary conditions were reduced to ordinary differential equations (ODEs) using a suitable similarity variable. The resulting non-linear system of equations is then solved using Chebychev Collocation Method with the aid of Mathematica 11.0 software. It is found that the heat transfer rate of the hybrid nanofluid is higher as compared to the nanofluid. The imposed magnetic field of high strength is a better tool to control the motion of hybrid nanofluids inside the boundary layer. Thermal radiations and slip parameter are observed to be beneficial for thermal enhancement for both hybrid and nanofluids.

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35

Derbazi, Choukri, Hadda Hammouche, Abdelkrim Salim, and Mouffak Benchohra. "Measure of noncompactness and fractional hybrid differential equations with hybrid conditions." Differential Equations & Applications, no. 2 (2022): 145–61. http://dx.doi.org/10.7153/dea-2022-14-09.

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36

Ahmad, Sufia Zulfa, Fudziah Ismail, and Norazak Senu. "Solving Oscillatory Delay Differential Equations Using Block Hybrid Methods." Journal of Mathematics 2018 (October 1, 2018): 1–7. http://dx.doi.org/10.1155/2018/2960237.

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A set of order condition for block explicit hybrid method up to order five is presented and, based on the order conditions, two-point block explicit hybrid method of order five for the approximation of special second order delay differential equations is derived. The method is then trigonometrically fitted and used to integrate second-order delay differential equations with oscillatory solutions. The efficiency curves based on the log of maximum errors versus the CPU time taken to do the integration are plotted, which clearly demonstrated the superiority of the trigonometrically fitted block hybrid method.

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37

Rashid, Umair, Azhar Iqbal, and Abdullah Alsharif. "Numerical Study of (Au-Cu)/Water and (Au-Cu)/Ethylene Glycol Hybrid Nanofluids Flow and Heat Transfer over a Stretching Porous Plate." Energies 14, no. 24 (2021): 8341. http://dx.doi.org/10.3390/en14248341.

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The purpose of the study is to investigate the (Au-Cu)/Water and (Au-Cu)/Ethylene glycol hybrid nanofluids flow and heat transfer through a linear stretching porous plate with the effects of thermal radiation, ohmic heating, and viscous dissipation. Similarity transformations technique is used to transform a governing system of partial differential equations into ordinary differential equations. The NDSolve Mathematica program is used to solve the nonlinear ordinary differential equations. Furthermore, the results are compared with the results of homotopy analysis method. The impacts of relevant physical parameters on velocity, temperature, and the Nusselt number are represented in graphical form. The key points indicate that the temperature of (Au-Cu)/water and (Au-Cu)/Ethylene glycol hybrid nanofluids is increased with the effects of Eckert number and magnetic field. The (Au-Cu)/Ethylene glycol hybrid nanofluid also has a greater rate of heat transfer than (Au-Cu)/Water hybrid nanofluid.

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38

Zirra, Donald J., Paul I. Dalatu, and Johnson Ishaku. "Development of Hybrid Block Methods for Oscillatory Solutions of Second-Order Differential Equations." International Journal of Development Mathematics (IJDM) 2, no. 1 (2025): 032–46. https://doi.org/10.62054/ijdm/0201.03.

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This study explores the application of hybrid block method in solving oscillatory second-order initial value problems (IVPs), a category of differential equations relevant in modeling various real-life phenomena, such as mechanical and electrical oscillations. Traditional numerical methods often face challenges in accuracy and stability when applied to oscillatory problems, prompting a need for advanced methods like hybrid block method. This research developed a hybrid block method that offers improved error control, stability and convergence for oscillatory differential equations. Error analysis is conducted to assess the method's effectiveness, comparing it with existing approaches to highlight its robustness in handling oscillatory behavior. The proposed method demonstrates an expanded stability region, making it suitable for complex, real-world applications that require high precision. The study's findings emphasize the potential of block hybrid methods in advancing numerical solutions for differential equations, providing valuable insights for further research and application in science and engineering

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39

You, Jin, and Zhenlai Han. "Analysis of fractional hybrid differential equations with impulses in partially ordered Banach algebras." Nonlinear Analysis: Modelling and Control 26, no. 6 (2021): 1071–86. http://dx.doi.org/10.15388/namc.2021.26.24939.

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In this paper, we investigate a class of fractional hybrid differential equations with impulses, which can be seen as nonlinear differential equations with a quadratic perturbation of second type and a linear perturbation in partially ordered Banach algebras. We deduce the existence and approximation of a mild solution for the initial value problems of this system by applying Dhage iteration principles and related hybrid fixed point theorems. Compared with previous works, we generalize the results to fractional order and extend some existing conclusions for the first time. Meantime, we take into consideration the effect of impulses. Our results indicate the influence of fractional order for nonlinear hybrid differential equations and improve some known results, which have wider applications as well. A numerical example is included to illustrate the effectiveness of the proposed results.

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40

Shatnawi, Taqi A. M., Nadeem Abbas, and Wasfi Shatanawi. "Comparative study of Casson hybrid nanofluid models with induced magnetic radiative flow over a vertical permeable exponentially stretching sheet." AIMS Mathematics 7, no. 12 (2022): 20545–64. http://dx.doi.org/10.3934/math.20221126.

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<abstract> <p>In this paper, the steady flow of an incompressible hybrid Casson nanofluid over a vertical permeable exponential stretching sheet is considered. The influence of the induced magnetic field is investigated. The influence of heat production and nonlinear radiation on slip effects is studied. Typically, three hybrid nanofluidic models are presented in this paper, namely: Xue, Yamada-Ota, and Tiwari Das. A study of a single-walled carbon nanotube and a multi-walled carbon nanotube with base fluid water is also provided. The governing equations are developed under flow assumptions in the form of partial differential equations by using boundary layer approximations. Using the appropriate transformations, partial differential equations are converted into ordinary differential equations. The ordinary differential equations are solved by the fifth-order Runge-Kutta-Fehlberg approach. Impacts concerning physical parameters are revealed by graphs and numerical values through tables. Temperature profile increases as concentration of solid nanoparticles increases. Because the thermal conductivity of the fluid is enhanced due to an increment in solid nanoparticles, which enhanced the temperature of the magneto-Casson hybrid nanofluid. The skin friction achieved higher values in the Yamada-Ota model of hybrid nanofluid as compared to the Xue model and Tiwari Das model. The results of this study show the Yamada-Ota model achieved a higher heat transfer rate than the Xue and Tiwari Das models of hybrid nanofluid.</p> </abstract>

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41

Sidheshwar, Sangram Bellale* Ganesh Babruwan Dapke. "HYHYBRID FIXED POINT THEOREM FOR NONLINEAR DIFFERENTIAL EQUATIONSBRID FIXED POINT THEOREM FOR NONLINEAR DIFFERENTIAL EQUATIONS." INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY 6, no. 1 (2017): 272–77. https://doi.org/10.5281/zenodo.246805.

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For Hybrid fixed point theorem for nonincreasing mapping in partially ordered complete metric space to prove existence as well as initial value problem of nonlinear first order ordinary differential equations.

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42

Benyoub, Mohammed, and Özyurt Gülyaz. "On extremal solutions of weighted fractional hybrid differential equations." Filomat 38, no. 6 (2024): 2091–107. http://dx.doi.org/10.2298/fil2406091b.

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This research studies the existence of a solution for an initial value problem of nonlinear fractional hybrid differential equations involving Riemann-Liouville derivative in weighted space of continuous functions. An existence theorem for this equations is proved under mixed Lipschitz and Carath?odory conditions.

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43

Drăgan, Vasile, Ivan Ganchev Ivanov, and Ioan-Lucian Popa. "A Game — Theoretic Model for a Stochastic Linear Quadratic Tracking Problem." Axioms 12, no. 1 (2023): 76. http://dx.doi.org/10.3390/axioms12010076.

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In this paper, we solve a stochastic linear quadratic tracking problem. The controlled dynamical system is modeled by a system of linear Itô differential equations subject to jump Markov perturbations. We consider the case when there are two decision-makers and each of them wants to minimize the deviation of a preferential output of the controlled dynamical system from a given reference signal. We assume that the two decision-makers do not cooperate. Under these conditions, we state the considered tracking problem as a problem of finding a Nash equilibrium strategy for a stochastic differential game. Explicit formulae of a Nash equilibrium strategy are provided. To this end, we use the solutions of two given terminal value problems (TVPs). The first TVP is associated with a hybrid system formed by two backward nonlinear differential equations coupled by two algebraic nonlinear equations. The second TVP is associated with a hybrid system formed by two backward linear differential equations coupled by two algebraic linear equations.

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44

Swalmeh, Mohammed Zaki. "Numerical Solutions of Hybrid Nanofluids Flow Via Free Convection Over a Solid Sphere." Journal of Advanced Research in Fluid Mechanics and Thermal Sciences 83, no. 1 (2021): 34–45. http://dx.doi.org/10.37934/arfmts.83.1.3445.

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The purpose of the existing study is to examine how heat transfer enables consolidated by variations in the basic advantages of fluids in the existence of free convection with the assistance of suspended hybrid nanofluids. Iron-graphene oxide suspended in water as a hybrid nanofluid flow on a solid sphere is also considered in this work. The partial differential equations are gotten, for this problem, by transforming the mathematical governing equations using similarity equations (stream function). These partial differential equations are solved numerically by Keller-Box method and programmed by MATLAB program. the acquired numerical results are in excellent agreement with the preceding literature results. Graphical results of the influence of the hybrid nanofluid parameters on some physical quantities regarded to examine the behavior of hybrid nanofluid flow were attained, and they proved that hybrid nanofluid flow represents a more essential role in the operation of heat transfer than a regular nanofluid flow.

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45

Yang, Hua, Feng Jiang, and Jun Hao Hu. "Numerical Method of Hybrid Stochastic Functional Differential Equations with the Local Lipschitz Coefficients." Advanced Materials Research 267 (June 2011): 422–26. http://dx.doi.org/10.4028/www.scientific.net/amr.267.422.

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Recently, hybrid stochastic differential equations have received a great deal of attention. It is surprising that there are not any numerical schemes established for the hybrid stochastic functional differential equations. In this paper, the Euler—Maruyama method is developed, and the main aim is to show that the numerical solutions will converge to the true solutions under the local Lipschitz condition. The result obtained generalizes the earlier results.

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46

Marzban, Hamid Reza, and Sayyed Mohammad Hoseini. "Solution of Nonlinear Volterra-Fredholm Integrodifferential Equations via Hybrid of Block-Pulse Functions and Lagrange Interpolating Polynomials." Advances in Numerical Analysis 2012 (December 10, 2012): 1–14. http://dx.doi.org/10.1155/2012/868279.

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An efficient hybrid method is developed to approximate the solution of the high-order nonlinear Volterra-Fredholm integro-differential equations. The properties of hybrid functions consisting of block-pulse functions and Lagrange interpolating polynomials are first presented. These properties are then used to reduce the solution of the nonlinear Volterra-Fredholm integro-differential equations to the solution of algebraic equations whose solution is much more easier than the original one. The validity and applicability of the proposed method are demonstrated through illustrative examples. The method is simple, easy to implement and yields very accurate results.

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47

Arguchintsev, Alexander, and Vasilisa Poplevko. "An Optimal Control Problem by a Hybrid System of Hyperbolic and Ordinary Differential Equations." Games 12, no. 1 (2021): 23. http://dx.doi.org/10.3390/g12010023.

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This paper deals with an optimal control problem for a linear system of first-order hyperbolic equations with a function on the right-hand side determined from controlled bilinear ordinary differential equations. These ordinary differential equations are linear with respect to state functions with controlled coefficients. Such problems arise in the simulation of some processes of chemical technology and population dynamics. Normally, general optimal control methods are used for these problems because of bilinear ordinary differential equations. In this paper, the problem is reduced to an optimal control problem for a system of ordinary differential equations. The reduction is based on non-classic exact increment formulas for the cost-functional. This treatment allows to use a number of efficient optimal control methods for the problem. An example illustrates the approach.

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48

Alshammari, Saleh, Mohammad Alshammari, and Mohammed S. Abdo. "Nonlocal Hybrid Integro-Differential Equations Involving Atangana–Baleanu Fractional Operators." Journal of Mathematics 2023 (June 17, 2023): 1–11. http://dx.doi.org/10.1155/2023/5891342.

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In this study, we develop a theory for the nonlocal hybrid boundary value problem for the fractional integro-differential equations featuring Atangana–Baleanu derivatives. The corresponding hybrid fractional integral equation is presented. Then, we establish the existence results using Dhage’s hybrid fixed point theorem for a sum of three operators. We also offer additional exceptional cases and similar outcomes. In order to demonstrate and verify the results, we provide an example as an application.

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Ahmad, Naeem, Raziya Sabri, Mohammad Faisal Khan, Mohammad Shadab, and Anju Gupta. "Relevance of Factorization Method to Differential and Integral Equations Associated with Hybrid Class of Polynomials." Fractal and Fractional 6, no. 1 (2021): 5. http://dx.doi.org/10.3390/fractalfract6010005.

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This article has a motive to derive a new class of differential equations and associated integral equations for some hybrid families of Laguerre–Gould–Hopper-based Sheffer polynomials. We derive recurrence relations, differential equation, integro-differential equation, and integral equation for the Laguerre–Gould–Hopper-based Sheffer polynomials by using the factorization method.

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50

Ali Hussain, Eman, and Yahya Mourad Abdul – Abbass. "On Fuzzy differential equation." Journal of Al-Qadisiyah for computer science and mathematics 11, no. 2 (2019): 1–9. http://dx.doi.org/10.29304/jqcm.2019.11.2.540.

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In this paper, we introduce a hybrid method to use fuzzy differential equation, and Genetic Turing Machine developed for solving nth order fuzzy differential equation under Seikkala differentiability concept [14]. The Errors between the exact solutions and the approximate solutions were computed by fitness function and the Genetic Turing Machine results are obtained. After comparing the approximate solution obtained by the GTM method with approximate to the exact solution, the approximate results by Genetic Turing Machine demonstrate the efficiency of hybrid methods for solving fuzzy differential equations (FDE).

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