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Journal articles on the topic 'Differential Equation Method de Wormald'

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Published: 25 May 2024

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1

Zou, Li, Zhen Wang, and Zhi Zong. "Generalized differential transform method to differential-difference equation." Physics Letters A 373, no. 45 (November 2009): 4142–51. http://dx.doi.org/10.1016/j.physleta.2009.09.036.

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2

Li, Meng-Rong, Tzong-Hann Shieh, C. Jack Yue, Pin Lee, and Yu-Tso Li. "Parabola Method in Ordinary Differential Equation." Taiwanese Journal of Mathematics 15, no. 4 (August 2011): 1841–57. http://dx.doi.org/10.11650/twjm/1500406383.

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3

Ali Hussain, Eman, and Yahya Mourad Abdul – Abbass. "On Fuzzy differential equation." Journal of Al-Qadisiyah for computer science and mathematics 11, no. 2 (August 21, 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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4

Chang, Ick-Soon, and Sheon-Young Kang. "Fredholm integral equation method for the integro-differential Schrödinger equation." Computers & Mathematics with Applications 56, no. 10 (November 2008): 2676–85. http://dx.doi.org/10.1016/j.camwa.2008.05.027.

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5

Jain, Pankaj, Chandrani Basu, and Vivek Panwar. "Reduced $pq$-Differential Transform Method and Applications." Journal of Inequalities and Special Functions 13, no. 1 (March 30, 2022): 24–40. http://dx.doi.org/10.54379/jiasf-2022-1-3.

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In this paper, Reduced Differential Transform method in the framework of (p, q)-calculus, denoted by Rp,qDT , has been introduced and applied in solving a variety of differential equations such as diffusion equation, 2Dwave equation, K-dV equation, Burgers equations and Ito system. While the diffusion equation has been studied for the special case p = 1, i.e., in the framework of q-calculus, the other equations have not been studied even in q-calculus.

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6

Abe, Kenji, Akira Ishida, Tsuguhiro Watanabe, Yasumasa Kanada, and Kyoji Nishikawa. "HIDM-New Numerical Method for Differential Equation." Kakuyūgō kenkyū 57, no. 2 (1987): 85–95. http://dx.doi.org/10.1585/jspf1958.57.85.

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7

Chen, Xi, and Ying Dai. "Differential transform method for solving Richards’ equation." Applied Mathematics and Mechanics 37, no. 2 (February 2016): 169–80. http://dx.doi.org/10.1007/s10483-016-2023-8.

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8

Youness, Ebrahim A., Abd El-Monem A. Megahed, Elsayed E. Eladdad, and Hanem F. A. Madkour. "Min-max differential game with partial differential equation." AIMS Mathematics 7, no. 8 (2022): 13777–89. http://dx.doi.org/10.3934/math.2022759.

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<abstract><p>In this paper, we are concerned with a min-max differential game with Cauchy initial value problem (CIVP) as the state trajectory for the differential game, we studied the analytical solution and the approximate solution by using Picard method (PM) of this problem. We obtained the equivalent integral equation to the CIVP. Also, we suggested a method for solving this problem. The existence, uniqueness of the solution and the uniform convergence are discussed for the two methods.</p></abstract>

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9

Khalili Golmankhaneh, Alireza, and Carlo Cattani. "Fractal Logistic Equation." Fractal and Fractional 3, no. 3 (July 11, 2019): 41. http://dx.doi.org/10.3390/fractalfract3030041.

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In this paper, we give difference equations on fractal sets and their corresponding fractal differential equations. An analogue of the classical Euler method in fractal calculus is defined. This fractal Euler method presets a numerical method for solving fractal differential equations and finding approximate analytical solutions. Fractal differential equations are solved by using the fractal Euler method. Furthermore, fractal logistic equations and functions are given, which are useful in modeling growth of elements in sciences including biology and economics.

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10

Tuluce Demiray, Seyma, Yusuf Pandir, and Hasan Bulut. "Generalized Kudryashov Method for Time-Fractional Differential Equations." Abstract and Applied Analysis 2014 (2014): 1–13. http://dx.doi.org/10.1155/2014/901540.

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In this study, the generalized Kudryashov method (GKM) is handled to find exact solutions of time-fractional Burgers equation, time-fractional Cahn-Hilliard equation, and time-fractional generalized third-order KdV equation. These time-fractional equations can be turned into another nonlinear ordinary differantial equation by travelling wave transformation. Then, GKM has been implemented to attain exact solutions of time-fractional Burgers equation, time-fractional Cahn-Hilliard equation, and time-fractional generalized third-order KdV equation. Also, some new hyperbolic function solutions have been obtained by using this method. It can be said that this method is a generalized form of the classical Kudryashov method.

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11

Abdulghafoor J. Salim and Ali F. Ali. "Studying the Ito ̃ ’s formula for some stochastic differential equation: (Quotient stochastic differential equation)." Tikrit Journal of Pure Science 26, no. 3 (July 10, 2021): 108–12. http://dx.doi.org/10.25130/tjps.v26i3.150.

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The aim of this paper is to study It ’s formula for some stochastic differential equation such as quotient stochastic differential equation, by using the function F (t, x (t)) which satisfies the product Ito’s formula, then we find some calculus relation for the quotient stochastic differential equation and we generalize the method for all m supported by some examples to explain the method.

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12

CARIÑENA, J. F., G. MARMO, and J. NASARRE. "THE NONLINEAR SUPERPOSITION PRINCIPLE AND THE WEI-NORMAN METHOD." International Journal of Modern Physics A 13, no. 21 (August 20, 1998): 3601–27. http://dx.doi.org/10.1142/s0217751x98001694.

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Group theoretical methods are used to study some properties of the Riccati equation, which is the only differential equation admitting a nonlinear superposition principle. The Wei–Norman method is applied to obtain the associated differential equation in the group SL(2, ℝ). The superposition principle for first order differential equation systems and Lie–Scheffers theorem are also analyzed from this group theoretical perspective. Finally, the theory is applied in the solution of second order differential equations like time independent Schrödinger equation.

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13

Saeed, Umer, and Mujeeb ur Rehman. "Hermite Wavelet Method for Fractional Delay Differential Equations." Journal of Difference Equations 2014 (July 2, 2014): 1–8. http://dx.doi.org/10.1155/2014/359093.

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We proposed a method by utilizing method of steps and Hermite wavelet method, for solving the fractional delay differential equations. This technique first converts the fractional delay differential equation to a fractional nondelay differential equation and then applies the Hermite wavelet method on the obtained fractional nondelay differential equation to find the solution. Several numerical examples are solved to show the applicability of the proposed method.

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14

Rajchel, Kazimierz, and Jerzy Szczęsny. "New method to solve certain differential equations." Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica 15, no. 1 (December 1, 2016): 107–11. http://dx.doi.org/10.1515/aupcsm-2016-0009.

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AbstractA new method to solve stationary one-dimensional Schroedinger equation is investigated. Solutions are described by means of representation of circles with multiple winding number. The results are demonstrated using the well-known analytical solutions of the Schroedinger equation.

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15

R. Rizkalla, Raafat, Seham SH. Tantawy, and Mahmoud H.Taha. "Applications on Differential Transform method for solving Singularly Perturbed Volterra integral equation, Volterra integral equation and integro-differential equation." International Journal of Mathematics Trends and Technology 23, no. 1 (July 25, 2015): 42–53. http://dx.doi.org/10.14445/22315373/ijmtt-v23p507.

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16

Ab. Aziz, Amiruddin, Noor Noor Syazana Ngarisan, and Nur Afriza Baki. "Solution of Finite Difference Method and Differential Quadrature Method in Burgers Equation." Journal of Ocean, Mechanical and Aerospace -science and engineering- (JOMAse) 63, no. 3 (November 30, 2019): 1–4. http://dx.doi.org/10.36842/jomase.v63i3.97.

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The Finite Difference Method and Differential Quadrature Method are used to solve the partial differential equation in Burgers equation. The different number of nodes is used in these methods to investigate the accuracy. The solutions of these methods are compared in terms of accuracy of the numerical solution. C language program have been developed based on the method in order to solve the Burgers equation. The results of this study are compared in terms of convergence as well as accuracy of the numerical solution. Generally, from the numerical results show that the Differential Quadrature Method is better than the Finite Different Method in terms of accuracy and convergence.

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17

Vitanov, Nikolay K., Zlatinka I. Dimitrova, and Kaloyan N. Vitanov. "Simple Equations Method (SEsM): Algorithm, Connection with Hirota Method, Inverse Scattering Transform Method, and Several Other Methods." Entropy 23, no. 1 (December 23, 2020): 10. http://dx.doi.org/10.3390/e23010010.

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The goal of this article is to discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations and to show that several well-known methods for obtaining exact solutions of such equations are connected to SEsM. In more detail, we show that the Hirota method is connected to a particular case of SEsM for a specific form of the function from Step 2 of SEsM and for simple equations of the kinds of differential equations for exponential functions. We illustrate this particular case of SEsM by obtaining the three- soliton solution of the Korteweg-de Vries equation, two-soliton solution of the nonlinear Schrödinger equation, and the soliton solution of the Ishimori equation for the spin dynamics of ferromagnetic materials. Then we show that a particular case of SEsM can be used in order to reproduce the methodology of the inverse scattering transform method for the case of the Burgers equation and Korteweg-de Vries equation. This particular case is connected to use of a specific case of Step 2 of SEsM. This step is connected to: (i) representation of the solution of the solved nonlinear partial differential equation as expansion as power series containing powers of a “small” parameter ϵ; (ii) solving the differential equations arising from this representation by means of Fourier series, and (iii) transition from the obtained solution for small values of ϵ to solution for arbitrary finite values of ϵ. Finally, we show that the much-used homogeneous balance method, extended homogeneous balance method, auxiliary equation method, Jacobi elliptic function expansion method, F-expansion method, modified simple equation method, trial function method and first integral method are connected to particular cases of SEsM.

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18

Yang, Xiao-Feng, and Yi Wei. "Bilinear Equation of the Nonlinear Partial Differential Equation and Its Application." Journal of Function Spaces 2020 (April 28, 2020): 1–14. http://dx.doi.org/10.1155/2020/4912159.

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The homogeneous balance of undetermined coefficient method is firstly proposed to derive a more general bilinear equation of the nonlinear partial differential equation (NLPDE). By applying perturbation method, subsidiary ordinary differential equation (sub-ODE) method, and compatible condition to bilinear equation, more exact solutions of NLPDE are obtained. The KdV equation, Burgers equation, Boussinesq equation, and Sawada-Kotera equation are chosen to illustrate the validity of our method. We find that the underlying relation among the G′/G-expansion method, Hirota’s method, and HB method is a bilinear equation. The proposed method is also a standard and computable method, which can be generalized to deal with other types of NLPDE.

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19

Peng, Hai-Jun, Sheng Zhang, Zhi-Gang Wu, and Biao-Song Chen. "Precise Integration Method for Solving Noncooperative LQ Differential Game." Mathematical Problems in Engineering 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/713725.

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The key of solving the noncooperative linear quadratic (LQ) differential game is to solve the coupled matrix Riccati differential equation. The precise integration method based on the adaptive choosing of the two parameters is expanded from the traditional symmetric Riccati differential equation to the coupled asymmetric Riccati differential equation in this paper. The proposed expanded precise integration method can overcome the difficulty of the singularity point and the ill-conditioned matrix in the solving of coupled asymmetric Riccati differential equation. The numerical examples show that the expanded precise integration method gives more stable and accurate numerical results than the “direct integration method” and the “linear transformation method”.

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20

Zhang, Yinnan, and Weian Zheng. "Discretizing a backward stochastic differential equation." International Journal of Mathematics and Mathematical Sciences 32, no. 2 (2002): 103–16. http://dx.doi.org/10.1155/s0161171202110234.

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We show a simple method to discretize Pardoux-Peng's nonlinear backward stochastic differential equation. This discretization scheme also gives a numerical method to solve a class of semi-linear PDEs.

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21

Ebaid, Abdelhalim. "ON A NEW DIFFERENTIAL TRANSFORMATION METHOD FOR SOLVING NONLINEAR DIFFERENTIAL EQUATIONS." Asian-European Journal of Mathematics 06, no. 04 (December 2013): 1350057. http://dx.doi.org/10.1142/s1793557113500575.

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The main difficulty in solving nonlinear differential equations by the differential transformation method (DTM) is how to treat complex nonlinear terms. This method can be easily applied to simple nonlinearities, e.g. polynomials, however obstacles exist for treating complex nonlinearities. In the latter case, a technique has been recently proposed to overcome this difficulty, which is based on obtaining a differential equation satisfied by this nonlinear term and then applying the DTM to this obtained differential equation. Accordingly, if a differential equation has n-nonlinear terms, then this technique must be separately repeated for each nonlinear term, i.e. n-times, consequently a system of n-recursive relations is required. This significantly increases the computational budget. We instead propose a general symbolic formula to treat any analytic nonlinearity. The new formula can be easily applied when compared with the only other available technique. We also show that this formula has the same mathematical structure as the Adomian polynomials but with constants instead of variable components. Several nonlinear ordinary differential equations are solved to demonstrate the reliability and efficiency of the improved DTM method, which increases its applicability.

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22

Viswam, Sonia. "THE ADOMIAN DECOMPOSITION METHOD AND THE DIFFERENTIAL EQUATION." International Journal of Engineering Applied Sciences and Technology 5, no. 4 (August 1, 2020): 173–77. http://dx.doi.org/10.33564/ijeast.2020.v05i04.023.

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23

Tate, Shivaji. "A New Iterative Method for Riccati Differential Equation." International Journal for Research in Applied Science and Engineering Technology V, no. XI (November 23, 2017): 2722–25. http://dx.doi.org/10.22214/ijraset.2017.11375.

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24

Abd Ali, Hadeel G. "Modify Lyapunov-Schmidt Method for Nonhomogeneous Differential Equation." BASRA JOURNAL OF SCIENCE 40, no. 2 (September 1, 2022): 306–20. http://dx.doi.org/10.29072/basjs.20220204.

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The Lyapunov-Schmidt reduction for nonhomogeneous problems is modified when the dimension of the null space is two. The novel method was utilized to approximate the solutions of the nonlinear wave equation. This equation's related key function has been identified

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25

Sharma, Devender, Puneet Sharma, and Ritu Sharma. "Non-Blind Deblurring Using Partial Differential Equation Method." International Journal of Computer Applications Technology and Research 2, no. 3 (May 10, 2013): 232–36. http://dx.doi.org/10.7753/ijcatr0203.1005.

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26

Pritchard, Jocelyun I., and Howard M. Adelman. "Differential Equation Based Method for Accurate Modal Approximations." AIAA Journal 29, no. 3 (March 1991): 484–86. http://dx.doi.org/10.2514/3.10609.

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27

CAI, Q. D. "CONTINUOUS NEWTON METHOD FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATION." Modern Physics Letters B 24, no. 13 (May 30, 2010): 1303–6. http://dx.doi.org/10.1142/s0217984910023487.

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Newton method is a widely used iteration method in solving nonlinear algebraic equations. In this method, a linear algebraic equations need to be solved in every step. The coefficient matrix of the algebraic equations is so-called Jacobian matrix, which needs to be determined at every step. For a complex non-linear system, usually no explicit form of Jacobian matrix can be found. Several methods are introduced to obtain an approximated matrix, which are classified as Jacobian-free method. The finite difference method is used to approximate the derivatives in Jacobian matrix, and a small parameter is needed in this process. Some problems may arise because of the interaction of this parameter and round-off errors. In the present work, we show that this kind of Newton method may encounter difficulties in solving non-linear partial differential equation (PDE) on fine mesh. To avoid this problem, the continuous Newton method is presented, which is a modification of classical Newton method for non-linear PDE.

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28

Shu, C., and H. Xue. "Solution of Helmholtz equation by differential quadrature method." Computer Methods in Applied Mechanics and Engineering 175, no. 1-2 (June 1999): 203–12. http://dx.doi.org/10.1016/s0045-7825(98)00370-3.

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29

Cveticanin, L. "Homotopy–perturbation method for pure nonlinear differential equation." Chaos, Solitons & Fractals 30, no. 5 (December 2006): 1221–30. http://dx.doi.org/10.1016/j.chaos.2005.08.180.

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30

Allahviranloo, T., and S. Salahshour. "Euler method for solving hybrid fuzzy differential equation." Soft Computing 15, no. 7 (October 1, 2010): 1247–53. http://dx.doi.org/10.1007/s00500-010-0659-y.

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31

Saeed, Umer. "CAS Picard method for fractional nonlinear differential equation." Applied Mathematics and Computation 307 (August 2017): 102–12. http://dx.doi.org/10.1016/j.amc.2017.02.044.

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32

Lin, Bin. "REDUCED DIFFERENTIAL TRANSFORM METHOD FOR NONLINEAR SCHRÖDINGER EQUATION." Far East Journal of Dynamical Systems 26, no. 2 (July 14, 2015): 91–98. http://dx.doi.org/10.17654/fjdsjun2015_091_098.

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33

Mredula, K. P., and D. C. Vakaskar. "Collocation Method Using Wavelet for Partial Differential Equation." Indian Journal of Industrial and Applied Mathematics 8, no. 1 (2017): 14. http://dx.doi.org/10.5958/1945-919x.2017.00002.0.

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34

Lei, Wu, Zong Feng-De, and Zhang Jie-Fang. "Adomian Decomposition Method for Nonlinear Differential-Difference Equation." Communications in Theoretical Physics 48, no. 6 (December 2007): 983–86. http://dx.doi.org/10.1088/0253-6102/48/6/004.

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35

Wang, Qi. "Extended rational expansion method for differential-difference equation." Applied Mathematics and Computation 219, no. 17 (May 2013): 8965–72. http://dx.doi.org/10.1016/j.amc.2013.03.097.

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36

Wang, Xiao, Yufu Ning, Tauqir A. Moughal, and Xiumei Chen. "Adams–Simpson method for solving uncertain differential equation." Applied Mathematics and Computation 271 (November 2015): 209–19. http://dx.doi.org/10.1016/j.amc.2015.09.009.

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37

Gao, Qiang, Shu-jun Tan, Wan-xie Zhong, and Hong-wu Zhang. "Improved precise integration method for differential Riccati equation." Applied Mathematics and Mechanics 34, no. 1 (December 13, 2012): 1–14. http://dx.doi.org/10.1007/s10483-013-1648-8.

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38

Tan, Yue, and Saeid Abbasbandy. "Homotopy analysis method for quadratic Riccati differential equation." Communications in Nonlinear Science and Numerical Simulation 13, no. 3 (June 2008): 539–46. http://dx.doi.org/10.1016/j.cnsns.2006.06.006.

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39

Aminikhah, Hossein, and Milad Hemmatnezhad. "An efficient method for quadratic Riccati differential equation." Communications in Nonlinear Science and Numerical Simulation 15, no. 4 (April 2010): 835–39. http://dx.doi.org/10.1016/j.cnsns.2009.05.009.

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40

Kudryashov, Nikolai A., and Nadejda B. Loguinova. "Extended simplest equation method for nonlinear differential equations." Applied Mathematics and Computation 205, no. 1 (November 2008): 396–402. http://dx.doi.org/10.1016/j.amc.2008.08.019.

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41

El-Tawil, Magdy A., Ahmed A. Bahnasawi, and Ahmed Abdel-Naby. "Solving Riccati differential equation using Adomian's decomposition method." Applied Mathematics and Computation 157, no. 2 (October 2004): 503–14. http://dx.doi.org/10.1016/j.amc.2003.08.049.

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42

Hua, Ni. "Transformation Method for Generating Periodic Solutions of Abel’s Differential Equation." Advances in Mathematical Physics 2019 (April 2, 2019): 1–10. http://dx.doi.org/10.1155/2019/3582142.

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This paper deals with Abel’s differential equation. We suppose that r=r(t) is a periodic particular solution of Abel’s differential equation and, then, by means of the transformation method and the fixed point theory, present an alternative method of generating the other periodic solutions of Abel’s differential equation.

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43

P, Govindaraju, and Senthil Kumar. "A study on stochastic differential equation." Journal of Computational Mathematica 5, no. 2 (December 20, 2021): 68–75. http://dx.doi.org/10.26524/cm109.

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In this paper we study solutions to stochastic differential equations (SDEs) with discontinuous drift. In this paper we discussed The Euler-Maruyama method and this shows that a candidate density function based on the Euler-Maruyama method. The point of departure for this work is a particular SDE with discontinuous drift.

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44

Gülsu, Mustafa, Yalçın Öztürk, and Ayşe Anapali. "A Collocation Method for Solving Fractional Riccati Differential Equation." Advances in Applied Mathematics and Mechanics 5, no. 06 (December 2013): 872–84. http://dx.doi.org/10.4208/aamm.12-m12118.

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AbstractIn this article, we have introduced a Taylor collocation method, which is based on collocation method for solving fractional Riccati differential equation. The fractional derivatives are described in the Caputo sense. This method is based on first taking the truncated Taylor expansions of the solution function in the fractional Riccati differential equation and then substituting their matrix forms into the equation. Using collocation points, the systems of nonlinear algebraic equation is derived. We further solve the system of nonlinear algebraic equation using Maple 13 and thus obtain the coefficients of the generalized Taylor expansion. Illustrative examples are presented to demonstrate the effectiveness of the proposed method.

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45

Navickas, Zenonas, Minvydas Ragulskis, and Liepa Bikulčienė. "SPECIAL SOLUTIONS OF HUXLEY DIFFERENTIAL EQUATION." Mathematical Modelling and Analysis 16, no. 1 (June 24, 2011): 248–59. http://dx.doi.org/10.3846/13926292.2011.579627.

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The conditions when solutions of Huxley equation can be expressed in special form and the procedure of finding exact solutions are presented in this paper. Huxley equation is an evolution equation that describes the nerve propagation in biology. It is often useful to obtain a generalized solitary solution for fully understanding its physical meanings. It is shown that the solution produced by the Exp-function method may not hold for all initial conditions. It is proven that the analytical condition describing the existence of the produced solution in the space of initial conditions (or even in the space of the system's parameters) can not be derived by the Exp-function method because the question about the existence of that solution is omitted. The proposed operator method, on the contrary, brings the load of symbolic computations before the structure of the solution is identified. The method for the derivation of the solution is based on the concept of the rank of the Hankel matrix constructed from the sequence of coefficients representing formal solution in the series form. Moreover, the structure of the algebraic-analytic solution is generated automatically together with all conditions of the solution's existence. Computational experiments are used to illustrate the properties of derived analytical solutions.

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46

Öztürk, Yalçın, Ayşe Anapalı, Mustafa Gülsu, and Mehmet Sezer. "A Collocation Method for Solving Fractional Riccati Differential Equation." Journal of Applied Mathematics 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/598083.

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We have introduced a Taylor collocation method, which is based on collocation method for solving fractional Riccati differential equation with delay term. This method is based on first taking the truncated Taylor expansions of the solution function in the fractional Riccati differential equation and then substituting their matrix forms into the equation. Using collocation points, we have the system of nonlinear algebraic equation. Then, we solve the system of nonlinear algebraic equation using Maple 13, and we have the coefficients of the truncated Taylor sum. In addition, illustrative examples are presented to demonstrate the effectiveness of the proposed method. Comparing the methodology with some known techniques shows that the present approach is relatively easy and highly accurate.

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47

Yiğider, Muhammed, Khatereh Tabatabaei, and Ercan Çelik. "The Numerical Method for Solving Differential Equations of Lane-Emden Type by Padé Approximation." Discrete Dynamics in Nature and Society 2011 (2011): 1–9. http://dx.doi.org/10.1155/2011/479396.

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Numerical solution differential equation of Lane-Emden type is considered by Padé approximation. We apply these method to two examples. First differential equation of Lane-Emden type has been converted to power series by one-dimensional differential transformation, then the numerical solution of equation was put into Padé series form. Thus, we have obtained numerical solution differential equation of Lane-Emden type.

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48

Altoum, Sami H., Aymen Ettaieb, and Hafedh Rguigui. "Generalized Bernoulli Wick differential equation." Infinite Dimensional Analysis, Quantum Probability and Related Topics 24, no. 01 (March 2021): 2150008. http://dx.doi.org/10.1142/s0219025721500089.

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Based on the distributions space on [Formula: see text] (denoted by [Formula: see text]) which is the topological dual space of the space of entire functions with exponential growth of order [Formula: see text] and of minimal type, we introduce a new type of differential equations using the Wick derivation operator and the Wick product of elements in [Formula: see text]. These equations are called generalized Bernoulli Wick differential equations which are the analogue of the classical Bernoulli differential equations. We solve these generalized Wick differential equations. The present method is exemplified by several examples.

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Kalybay, A. A., and A. O. Baiarystanov. "Differential inequality and non-oscillation of fourth order differential equation." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 104, no. 4 (December 30, 2021): 103–9. http://dx.doi.org/10.31489/2021m4/103-109.

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Abstract:

The oscillatory theory of fourth order differential equations has not yet been developed well enough. The results are known only for the case when the coefficients of differential equations are power functions. This fact can be explained by the absence of simple effective methods for studying such higher order equations. In this paper, the authors investigate the oscillatory properties of a class of fourth order differential equations by the variational method. The presented variational method allows to consider any arbitrary functions as coefficients, and our main results depend on their boundary behavior in neighborhoods of zero and infinity. Moreover, this variational method is based on the validity of a certain weighted differential inequality of Hardy type, which is of independent interest. The authors of the article also find two-sided estimates of the least constant for this inequality, which are especially important for their applications to the main results on the oscillatory properties of these differential equations.

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50

Nyein, Ei Ei, and Aung Khaing Zaw. "A fixed point method to solve differential equation and Fredholm integral equation." Journal of Nonlinear Sciences and Applications 13, no. 04 (February 28, 2020): 205–11. http://dx.doi.org/10.22436/jnsa.013.04.05.

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