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Journal articles on the topic 'Ito equation'

Author: Grafiati

Published: 30 May 2022

Last updated: 25 July 2025

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1

SAITO, T., and T. ARIMITSU. "QUANTUM STOCHASTIC LIOUVILLE EQUATION OF ITO TYPE." Modern Physics Letters B 07, no. 29n30 (1993): 1951–59. http://dx.doi.org/10.1142/s0217984993001983.

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The quantum stochastic Liouville equation of Ito type is derived, for the first time, within Nonequilibrium Thermo Field Dynamics (NETFD), a unified canonical formalism for dissipative and/or stochastic fields. With the stochastic time-evolution generator of Ito type, the whole framework is inspected. Since most of the mathematical formulations of noncommutative stochastic variables are based on the equations of Ito type, the construction of the quantum stochastic Liouville equation has been highly desired. It is expected that the unified framework may provide us with a deeper insight for the

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2

Qi, Ziying, and Lianzhong Li. "Lie symmetry analysis, conservation laws and diverse solutions of a new extended (2+1)-dimensional Ito equation." AIMS Mathematics 8, no. 12 (2023): 29797–816. http://dx.doi.org/10.3934/math.20231524.

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<abstract><p>In this paper, a new class of extended (2+1)-dimensional Ito equations is investigated for its group invariant solutions. The Lie symmetry method is employed to transform the nonlinear Ito equation into an ordinary differential equation. The general solution of the solvable linear differential equation with different parameters is obtained, and the plot of the solvable linear differential equation is given. A power series solution for the equation is then derived. Furthermore, a conservation law for the equation is constructed by utilizing a new Ibragimov conservation

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3

Ma, Wen-Xiu, Jie Li, and Chaudry Masood Khalique. "A Study on Lump Solutions to a Generalized Hirota-Satsuma-Ito Equation in (2+1)-Dimensions." Complexity 2018 (December 2, 2018): 1–7. http://dx.doi.org/10.1155/2018/9059858.

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The Hirota-Satsuma-Ito equation in (2+1)-dimensions passes the three-soliton test. This paper aims to generalize this equation to a new one which still has abundant interesting solution structures. Based on the Hirota bilinear formulation, a symbolic computation with a new class of Hirota-Satsuma-Ito type equations involving general second-order derivative terms is conducted to require having lump solutions. Explicit expressions for lump solutions are successfully presented in terms of coefficients in a generalized Hirota-Satsuma-Ito equation. Three-dimensional plots and contour plots of a spe

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4

Niu, Xiaoxing, Mengxia Zhang, and Shuqiang Lv. "A Darboux Transformation for Ito Equation." Zeitschrift für Naturforschung A 71, no. 5 (2016): 427–31. http://dx.doi.org/10.1515/zna-2016-0004.

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AbstractA system proposed by Ito is reconsidered. The corresponding Darboux transformation is presented explicitly. The resulted Bäcklund transformation is shown to be equivalent to the one found by Hirota. Also, a nonlinear superposition formula, which is of differential-algebraic, is obtained.

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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 (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

Tleubergenov, M. I., G. K. Vassilina, and D. T. Azhymbaev. "Construction of the differential equations system of the program motion in Lagrangian variables in the presence of random perturbations." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 105, no. 1 (2022): 118–26. http://dx.doi.org/10.31489/2022m1/118-126.

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The classification of inverse problems of dynamics in the class of ordinary differential equations is given in the Galiullin’s monograph. The problem studied in this paper belongs to the main inverse problem of dynamics, but already in the class of second-order stochastic differential equations of the Ito type. Stochastic equations of the Lagrangian structure are constructed according to the given properties of motion under the assumption that the random perturbing forces belong to the class of processes with independent increments. The problem is solved as follows: First, a second-order Ito d

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7

Ren, Bo, Ji Lin, and Jun Yu. "Supersymmetric Ito equation: Bosonization and exact solutions." AIP Advances 3, no. 4 (2013): 042129. http://dx.doi.org/10.1063/1.4802969.

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8

Cen, Feng-Jie, Yan-Dan Zhao, Shuang-Yun Fang, Huan Meng, and Jun Yu. "Painlevé integrability of the supersymmetric Ito equation." Chinese Physics B 28, no. 9 (2019): 090201. http://dx.doi.org/10.1088/1674-1056/ab38a7.

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9

Yi, Zhang, and Chen Deng-Yuan. "N -Soliton-like Solution of Ito Equation." Communications in Theoretical Physics 42, no. 5 (2004): 641–44. http://dx.doi.org/10.1088/0253-6102/42/5/641.

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10

Carkovs, Jevgeņijs, and Oksana Pavlenko. "Stochastic Modelling for Dynamics of Interacting Populations." Proceedings of the Latvian Academy of Sciences. Section B. Natural, Exact, and Applied Sciences. 73, no. 5 (2019): 455–61. http://dx.doi.org/10.2478/prolas-2019-0070.

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Abstract The paper deals with a mathematical model for two interacting populations. Under the assumption of fast stochastic contacting of populations, we derive stochastic Poisson-type differential equations with a small parameter and propose an approximative algorithm for quantitative analysis of population dynamics that consists of two steps. First, we derive an ordinary differential equation for mean value of each population growth and analyse the average asymptotic population behaviour. Then, applying diffusion approximation procedure, we derive a stochastic Ito differential equation for s

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11

Zhu, Wen-Hui, Jian-Guo Liu, Mohammad Asif Arefin, M. Hafiz Uddin, and Ya-Kui Wu. "Double-Periodic Soliton Solutions of the (2+1)-Dimensional Ito Equation." Advances in Mathematical Physics 2023 (December 28, 2023): 1–9. http://dx.doi.org/10.1155/2023/9321673.

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In this work, a (2 + 1)-dimensional Ito equation is investigated, which represents the generalization of the bilinear KdV equation. Abundant double-periodic soliton solutions to the (2 + 1)-dimensional Ito equation are presented by the Hirota bilinear form and a mixture of exponentials and trigonometric functions. The dynamic properties are described through some 3D graphics and contour graphics.

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12

Inc, Mustafa, E. A. Az-Zo’bi, Adil Jhangeer, Hadi Rezazadeh, Muhammad Nasir Ali, and Mohammed K. A. Kaabar. "New Soliton Solutions for the Higher-Dimensional Non-Local Ito Equation." Nonlinear Engineering 10, no. 1 (2021): 374–84. http://dx.doi.org/10.1515/nleng-2021-0029.

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Abstract In this article, (2+1)-dimensional Ito equation that models waves motion on shallow water surfaces is analyzed for exact analytic solutions. Two reliable techniques involving the simplest equation and modified simplest equation algorithms are utilized to find exact solutions of the considered equation involving bright solitons, singular periodic solitons, and singular bright solitons. These solutions are also described graphically while taking suitable values of free parameters. The applied algorithms are effective and convenient in handling the solution process for Ito equation that

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13

Fan, Xinghua, and Shasha Li. "Bifurcation of Traveling Wave Solutions of the Dual Ito Equation." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/153139.

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The dual Ito equation can be seen as a two-component generalization of the well-known Camassa-Holm equation. By using the theory of planar dynamical system, we study the existence of its traveling wave solutions. We find that the dual Ito equation has smooth solitary wave solutions, smooth periodic wave solutions, and periodic cusp solutions. Parameter conditions are given to guarantee the existence.

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14

Rezazadeh, Hadi, Sharanjeet Dhawan, Savaïssou Nestor, Ahmet Bekir, and Alper Korkmaz. "Computational solutions of the generalized Ito equation in nonlinear dispersive systems." International Journal of Modern Physics B 35, no. 13 (2021): 2150172. http://dx.doi.org/10.1142/s0217979221501721.

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This papers presents new exact analytical solutions of a generalized Ito equation having three nonlinear terms, third- and fifth-order derivative forms that model the dynamics of traveling waves in nonlinear dispersive systems. With the help of Riccati equation method, we obtain different kinds of exact traveling wave solutions containing dark, singular, trigonometric, rational and other form of waves solutions that are more general than classical ones existing in the literature. Despite the originality of the new results obtained, the method used here is very efficient, powerful and can be ex

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15

Salas, Alvaro H., Cesar A. Gómez S, and Bernardo Acevedo Frias. "Computing Exact Solutions to a Generalized Lax-Sawada-Kotera-Ito Seventh-Order KdV Equation." Mathematical Problems in Engineering 2010 (2010): 1–7. http://dx.doi.org/10.1155/2010/524567.

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The Cole-Hopf transform is used to construct exact solutions to a generalization of both the seventh-order Lax KdV equation (Lax KdV7) and the seventh-order Sawada-Kotera-Ito KdV equation (Sawada-Kotera-Ito KdV7).

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16

Anonwa, Ijeoma Donatus, Augustine Omoghaghare Atonuje, and John Nwabueze Igabari. "Exponential almost Sure Stabilization of Nonlinear Delay Differential Systems under Stochastic Optimal Control Driven by Ito Brownian Noise." Asian Research Journal of Mathematics 21, no. 1 (2025): 78–86. https://doi.org/10.9734/arjom/2025/v21i1884.

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This study investigates the role of Brownian white noise in stabilizing nonlinear optimal control delay differential equations (OCDDES) that are typically unstable in their deterministic form. The technique applied involves the use of Lyapunov sample exponent and a specialized partial differential equation suggested by Mao, (1997). It is demonstrated that if the noise scaling parameters of the stochastically perturbed equation is finite, then the new stochastic optimal control delay differential equation (SOCDDES ) is self - stabilized in an almost sure exponential sense. This phenomenon does

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17

Ma, Hongcai, Xiangmin Meng, Hanfang Wu, and Aiping Deng. "A class of lump solutions for ito equation." Thermal Science 23, no. 4 (2019): 2205–10. http://dx.doi.org/10.2298/tsci1904205m.

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In this paper, we investigate the exact solutions for the (1+1)-D Ito equation. Some lump solutions are obtained by using Hirota?s bilinear method, and the conditions to guarantee analytical and rational localization of the lump solutions are presented. Suitable choices of the involved parameters guaranteeing analyticity of the solution are given. The 3-D plots with particular choices of the involved parameters are illustrated.

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18

V. Krishnan, E. "On the Ito-Type Coupled Nonlinear Wave Equation." Journal of the Physical Society of Japan 55, no. 11 (1986): 3753–55. http://dx.doi.org/10.1143/jpsj.55.3753.

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19

Zhou, Yuan, and Solomon Manukure. "Complexiton solutions to the Hirota‐Satsuma‐Ito equation." Mathematical Methods in the Applied Sciences 42, no. 7 (2019): 2344–51. http://dx.doi.org/10.1002/mma.5512.

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20

SAITO, T., and T. ARIMITSU. "QUANTUM STOCHASTIC EQUATIONS FOR A NON-LINEAR DAMPED OSCILLATOR." Modern Physics Letters B 07, no. 09 (1993): 623–31. http://dx.doi.org/10.1142/s0217984993000606.

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A unified framework of stochastic differential equations for quantum systems, formulated within Non-Equilibrium Thermo Field Dynamics (NETFD), is applied to a model of non-linear damped oscillator. The quantum stochastic Liouville equation and the quantum Langevin equations (both of Ito and Stratonovich type), which are consistent with the corresponding master equation, are written down explicitly in the case of a non-conventional treatment. This solves Kubo's third problem: how one can obtain the correlations of random force operators for the Langevin equation compatible with the master equat

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21

Zhang, Lijun, and Chaudry Masood Khalique. "Exact Solitary Wave and Periodic Wave Solutions of a Class of Higher-Order Nonlinear Wave Equations." Mathematical Problems in Engineering 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/548606.

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We study the exact traveling wave solutions of a general fifth-order nonlinear wave equation and a generalized sixth-order KdV equation. We find the solvable lower-order subequations of a general related fourth-order ordinary differential equation involving only even order derivatives and polynomial functions of the dependent variable. It is shown that the exact solitary wave and periodic wave solutions of some high-order nonlinear wave equations can be obtained easily by using this algorithm. As examples, we derive some solitary wave and periodic wave solutions of the Lax equation, the Ito eq

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22

Zhang, Yingnan, Xingbiao Hu, and Jianqing Sun. "Numerical calculation of N -periodic wave solutions to coupled KdV–Toda-type equations." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477, no. 2245 (2021): 20200752. http://dx.doi.org/10.1098/rspa.2020.0752.

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In this paper, we study the N -periodic wave solutions of coupled Korteweg–de Vries (KdV)–Toda-type equations. We present a numerical process to calculate the N -periodic waves based on the direct method of calculating periodic wave solutions proposed by Akira Nakamura. Particularly, in the case of N = 3, we give some detailed examples to show the N -periodic wave solutions to the coupled Ramani equation, the Hirota–Satsuma coupled KdV equation, the coupled Ito equation, the Blaszak–Marciniak lattice, the semi-discrete KdV equation, the Leznov lattice and a relativistic Toda lattice.

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23

Aibarakati, Wafaa, Aly Seadaw, and Noufe Aljahdaly. "Application of mathematical methods for the non-linear seventh order Sawada-Kotera-Ito dynamical wave equation." Thermal Science 23, Suppl. 6 (2019): 2081–93. http://dx.doi.org/10.2298/tsci190705373a.

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This article deal with finding travelling wave solutions for the 7th order Sawada-Kotera-Ito dynamical wave equation which describes the evolution of steeper waves of shorter wavelength than KdV equations using modified extended direct algebraic method. The new solutions derived have various physical structure, we also give graphic representation of the exact and stable solutions.

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24

Grigoriu, M. "A Monte Carlo Solution of Heat Conduction and Poisson Equations." Journal of Heat Transfer 122, no. 1 (1999): 40–45. http://dx.doi.org/10.1115/1.521435.

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A Monte Carlo method is developed for solving the heat conduction, Poisson, and Laplace equations. The method is based on properties of Brownian motion and Ito^ processes, the Ito^ formula for differentiable functions of these processes, and the similarities between the generator of Ito^ processes and the differential operators of these equations. The proposed method is similar to current Monte Carlo solutions, such as the fixed random walk, exodus, and floating walk methods, in the sense that it is local, that is, it determines the solution at a single point or a small set of points of the do

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25

Khalid, T. A. "Application of Elzaki Transform Decomposition Method in Solving Time-Fractional Sawada Kotera Ito Equation." Malaysian Journal of Mathematical Sciences 19, no. 2 (2025): 691–706. https://doi.org/10.47836/mjms.19.2.17.

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The research paper primarily focuses on the theoretical framework and mathematical methodology for solving the Sawada Kotera Ito with time fraction (SKI) equation using the Elzaki transform decomposition technique, including the Caputo-Fabrizio derivative and the Atangana-Baleanu derivative, which are crucial for comprehending the fractional SKI (1). Various numerical examples are given to illustrate the application of the Elzaki transform in solving the fractional SKI equation. We utilize efficient basis functions, namely fractional Lagrange functions, for the interpolation of temporal variab

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26

Hanapi, M. S. M., A. B. M. A. Ibrahim, R. Julius, and P. K. Choudhury. "Quantum antibunching in nonlinear couplers: a phase space approach using the positive P-representation." Mathematical Modeling and Computing 12, no. 2 (2025): 540–48. https://doi.org/10.23939/mmc2025.02.540.

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Quantum antibunching, vital for single-photon sources, enables applications in quantum technology, such as secure communication and quantum computing. This study investigates photon antibunching generation in a Nonlinear coupler (NLC) comprising two coupled waveguides: one linear and one with the second-order nonlinearity. Each waveguide is excited by a coherent laser source, with the second harmonic generation enhancing antibunching. Using the Schrödinger picture, the system's Hamiltonian is transformed into a master equation via the Liouville–von Neumann equation. The master equation is furt

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27

Zou, Li, Zong-Bing Yu, Shou-Fu Tian, Lian-Li Feng, and Jin Li. "Lump solutions with interaction phenomena in the (2+1)-dimensional Ito equation." Modern Physics Letters B 32, no. 07 (2018): 1850104. http://dx.doi.org/10.1142/s021798491850104x.

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In this paper, we consider the (2+1)-dimensional Ito equation, which was introduced by Ito. By considering the Hirota’s bilinear method, and using the positive quadratic function, we obtain some lump solutions of the Ito equation. In order to ensure rational localization and analyticity of these lump solutions, some sufficient and necessary conditions are provided on the parameters that appeared in the solutions. Furthermore, the interaction solutions between lump solutions and the stripe solitons are discussed by combining positive quadratic function with exponential function. Finally, the dy

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28

Ding, Liyuan, Wen-Xiu Ma, and Yehui Huang. "Lump solutions to a generalized Kadomtsev–Petviashvili–Ito equation." Modern Physics Letters B 35, no. 26 (2021): 2150437. http://dx.doi.org/10.1142/s0217984921504376.

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A (2+1)-dimensional generalized Kadomtsev–Petviashvili–Ito equation is introduced. Upon adding some second-order derivative terms, its various lump solutions are explicitly constructed by utilizing the Hirota bilinear method and calculated through the symbolic computation system Maple. Furthermore, two specific lump solutions are obtained with particular choices of the parameters and their dynamical behaviors are analyzed through three-dimensional plots and contour plots.

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29

Hu, Xiaorui, Shuning Lin, and Shoufeng Shen. "New interaction solutions to (1+1)-dimensional Ito equation." Applied Mathematics Letters 101 (March 2020): 106071. http://dx.doi.org/10.1016/j.aml.2019.106071.

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30

Zhang, Hai-Qiang, Xia Gao, Zhi-jie Pei, and Fa Chen. "Rogue periodic waves in the fifth-order Ito equation." Applied Mathematics Letters 107 (September 2020): 106464. http://dx.doi.org/10.1016/j.aml.2020.106464.

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31

Tang, Yaning, Jinli Ma, Wenxian Xie, and Lijun Zhang. "Interaction solutions for the (2+1)-dimensional Ito equation." Modern Physics Letters B 33, no. 13 (2019): 1950167. http://dx.doi.org/10.1142/s0217984919501677.

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In this paper, two classes of interaction solutions of the (2[Formula: see text]+[Formula: see text]1)-dimensional Ito equation are studied in the case of Hirota bilinear form. As the results, the interaction solutions between the rational function and a periodic function as well as the interaction solution between the hyperbolic function and a periodic function are obtained. Based on the interaction solutions, a new transformation is proposed to analyze and discuss the influence of parameters. Furthermore, two kinds of lump solutions can be obtained via the limit behavior of the interaction s

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32

Tleubergenov, M. I., G. K. Vassilina, and A. A. Abdrakhmanova. "Representing a second-order Ito equation as an equation with a given force structure." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 112, no. 4 (2023): 119–29. http://dx.doi.org/10.31489/2023m4/119-129.

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The problem of constructing equivalent equations with a given structure of forces by the given system of stochastic equations is considered. The equivalence of equations in the sense of almost surely is investigated. The paper determines the conditions under which a given system of second-order Ito stochastic differential equations is represented in the form of stochastic Lagrange equations with non-potential forces of a certain structure. Necessary and sufficient conditions for the representability of stochastic equations in the form of stochastic equations with non-potential forces admitting

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33

Wang, Jiao, Tian-Zhou Xu, and Gang-Wei Wang. "Numerical algorithm for time-fractional Sawada-Kotera equation and Ito equation with Bernstein polynomials." Applied Mathematics and Computation 338 (December 2018): 1–11. http://dx.doi.org/10.1016/j.amc.2018.06.001.

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34

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 (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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35

Yong-Yan, Fan, Jalil Manafian, Syed Maqsood Zia, Dinh Tran Ngoc Huy, and Trung-Hieu Le. "Analytical Treatment of the Generalized Hirota-Satsuma-Ito Equation Arising in Shallow Water Wave." Advances in Mathematical Physics 2021 (October 12, 2021): 1–26. http://dx.doi.org/10.1155/2021/1164838.

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In the current study, an analytical treatment is studied starting from the 2 + 1 -dimensional generalized Hirota-Satsuma-Ito (HSI) equation. Based on the equation, we first establish the evolution equation and obtain rational function solutions by means of the bilinear form with the help of the Hirota bilinear operator. Then, by the suggested method, the periodic, cross-kink wave solutions are also obtained. Also, the semi-inverse variational principle (SIVP) will be utilized for the generalized HSI equation. Two major cases were investigated from two different techniques. Moreover, the improv

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36

Tu, Jian Xin, Zhi Ren Wang, Han Zhu, and Ping Wang. "The Nonlinear Random Vibration of a Clamped Rectangular Thin Plate in Magnetic Field." Applied Mechanics and Materials 628 (September 2014): 127–32. http://dx.doi.org/10.4028/www.scientific.net/amm.628.127.

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In this paper, the magneto-elastic nonlinear random vibration of a clamped rectangular thin plate in magnetic field is studied. According to the magneto-elastic theory of plates and shells and the theory of structural random vibration, the magneto-elastic nonlinear random vibration equation of a clamped rectangular thin plate in a magnetic field is derived. Then the nonlinear random vibration equation is transferred into the Ito differential equation, and the Ito differential equation is solved using FPK equation method. Thus the numerical characteristics of displacement response and velocity

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37

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 (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 asc

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38

Chebotarev, Alexander M. "Quantum Stochastic Differential Equation is Unitarily Equivalent to a Symmetric Boundary Value Problem in Fock Space." Infinite Dimensional Analysis, Quantum Probability and Related Topics 01, no. 02 (1998): 175–99. http://dx.doi.org/10.1142/s0219025798000120.

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We show a new remarkable connection between the symmetric form of a quantum stochastic differential equation (QSDE) and the strong resolvent limit of the Schrödinger equations in Fock space: the strong resolvent limit is unitarily equivalent to QSDE in the adapted (or Ito) form, and the weak limit is unitarily equivalent to the symmetric (or Stratonovich) form of QSDE. We also prove that QSDE is unitarily equivalent to a symmetric boundary value problem for the Schrödinger equation in Fock space. The boundary condition describes standard jumps in phase and amplitude of components of Fock vecto

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39

Oduselu–Hassan, Emmanuel Oladayo, Ignatius N. Njoseh, and Jonathan Tsetimi. "A Numerical Approximation of the Stochastic Ito-Volterra Integral Equation." Asian Research Journal of Mathematics 19, no. 11 (2023): 61–68. http://dx.doi.org/10.9734/arjom/2023/v19i11753.

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A stochastic differential equation (SDE) is a differential equation in which one or more of the terms and the solution are stochastic processes. Numerous studies have employed orthogonal polynomials, however most of them focus on deterministic rather than stochastic systems. This is the reason why in this study, we looked into a numerical solution for the stochastic Ito-Volterra integral equation using the explicit finite difference scheme and Bernstein polynomials as trial functions. The equidistant collocation procedure was used to calculate the unknown constant parameters in between and rea

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40

Hu, Xing-Biao, and Yong Li. "Nonlinear superposition formulae of the Ito equation and a model equation for shallow water waves." Journal of Physics A: Mathematical and General 24, no. 9 (1991): 1979–85. http://dx.doi.org/10.1088/0305-4470/24/9/010.

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41

Meng, Xiangmin, and Hongcai Ma. "The Lump Solutions of the (1 + 1)-Dimensional Ito-Equation." Open Journal of Applied Sciences 09, no. 03 (2019): 121–25. http://dx.doi.org/10.4236/ojapps.2019.93011.

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42

Li, Chunxia, and Yunbo Zeng. "Soliton solutions to a higher order Ito equation: Pfaffian technique." Physics Letters A 363, no. 1-2 (2007): 1–4. http://dx.doi.org/10.1016/j.physleta.2006.10.080.

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43

Feng, Binlu, Bo Han, and Huanhe Dong. "Abundant new travelling wave solutions for the coupled Ito equation." Chaos, Solitons & Fractals 39, no. 1 (2009): 393–98. http://dx.doi.org/10.1016/j.chaos.2007.04.012.

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44

Yakup; YAŞAR, YILDIRIM. "WRONSKIAN SOLUTIONS OF (2+1) DIMENSIONAL NON-LOCAL ITO EQUATION." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 67, no. 2 (2018): 126–38. http://dx.doi.org/10.1501/commua1_0000000867.

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Van Gorder, Robert A. "Solutions to a Novel Casimir Equation for the Ito System." Communications in Theoretical Physics 56, no. 5 (2011): 801–4. http://dx.doi.org/10.1088/0253-6102/56/5/02.

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Zhang, Yi, Yun-Cheng You, Wen-Xiu Ma, and Hai-Qiong Zhao. "Resonance of solitons in a coupled higher-order Ito equation." Journal of Mathematical Analysis and Applications 394, no. 1 (2012): 121–28. http://dx.doi.org/10.1016/j.jmaa.2012.03.030.

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Chen, Yaming, Songhe Song, and Huajun Zhu. "Multi-symplectic methods for the Ito-type coupled KdV equation." Applied Mathematics and Computation 218, no. 9 (2012): 5552–61. http://dx.doi.org/10.1016/j.amc.2011.11.045.

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Zhao, Hai-qiong. "Soliton solution of a multi-component higher-order Ito equation." Applied Mathematics Letters 26, no. 7 (2013): 681–86. http://dx.doi.org/10.1016/j.aml.2013.01.008.

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Egorov, A. D. "On composite formulas for mathematical expectation of functionals of solution of the Ito equation in Hilbert space." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 55, no. 2 (2019): 158–68. http://dx.doi.org/10.29235/1561-2430-2019-55-2-158-168.

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This article is devoted to constructing composite approximate formulas for calculation of mathematical expectation of nonlinear functionals of solution of the linear Ito equation in Hilbert space with additive noise. As the leading process, the Wiener process taking values in Hilbert space is examined. The formulas are a sum of the approximations of the nonlinear functionals obtained by expanding the leading random process into a series of independent Gaussian random variables and correcting approximating functional quadrature formulas that ensure an approximate accuracy of compound formulas f

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Saleem, Sidra, Malik Zawwar Hussain, and Imran Aziz. "A reliable algorithm to compute the approximate solution of KdV-type partial differential equations of order seven." PLOS ONE 16, no. 1 (2021): e0244027. http://dx.doi.org/10.1371/journal.pone.0244027.

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Abstract:

The approximate solution of KdV-type partial differential equations of order seven is presented. The algorithm based on one-dimensional Haar wavelet collocation method is adapted for this purpose. One-dimensional Haar wavelet collocation method is verified on Lax equation, Sawada-Kotera-Ito equation and Kaup-Kuperschmidt equation of order seven. The approximated results are displayed by means of tables (consisting point wise errors and maximum absolute errors) to measure the accuracy and proficiency of the scheme in a few number of grid points. Moreover, the approximate solutions and exact sol

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