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Journal articles on the topic 'Equation'

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Published: 4 June 2021

Last updated: 25 July 2024

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1

Karakostas, George L. "Asymptotic behavior of a certain functional equation via limiting equations." Czechoslovak Mathematical Journal 36, no. 2 (1986): 259–67. http://dx.doi.org/10.21136/cmj.1986.102089.

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2

Parkala, Naresh, and Upender Reddy Gujjula. "Mohand Transform for Solution of Integral Equations and Abel's Equation." International Journal of Science and Research (IJSR) 13, no. 5 (May 5, 2024): 1188–91. http://dx.doi.org/10.21275/sr24512145111.

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3

Domoshnitsky, Alexander, and Roman Koplatadze. "On Asymptotic Behavior of Solutions of Generalized Emden-Fowler Differential Equations with Delay Argument." Abstract and Applied Analysis 2014 (2014): 1–13. http://dx.doi.org/10.1155/2014/168425.

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The following differential equationu(n)(t)+p(t)|u(σ(t))|μ(t) sign u(σ(t))=0is considered. Herep∈Lloc(R+;R+), μ∈C(R+;(0,+∞)), σ∈C(R+;R+), σ(t)≤t, andlimt→+∞⁡σ(t)=+∞. We say that the equation is almost linear if the conditionlimt→+∞⁡μ(t)=1is fulfilled, while iflim⁡supt→+∞⁡μ(t)≠1orlim⁡inft→+∞⁡μ(t)≠1, then the equation is an essentially nonlinear differential equation. In the case of almost linear and essentially nonlinear differential equations with advanced argument, oscillatory properties have been extensively studied, but there are no results on delay equations of this sort. In this paper, new sufficient conditions implying PropertyAfor delay Emden-Fowler equations are obtained.

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4

Becker, Leigh, Theodore Burton, and Ioannis Purnaras. "Complementary equations: a fractional differential equation and a Volterra integral equation." Electronic Journal of Qualitative Theory of Differential Equations, no. 12 (2015): 1–24. http://dx.doi.org/10.14232/ejqtde.2015.1.12.

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5

N O, Onuoha. "Transformation of Parabolic Partial Differential Equations into Heat Equation Using Hopf Cole Transform." International Journal of Science and Research (IJSR) 12, no. 6 (June 5, 2023): 1741–43. http://dx.doi.org/10.21275/sr23612082710.

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6

Zhao, Wenling, Hongkui Li, Xueting Liu, and Fuyi Xu. "Necessary and Sufficient Conditions for the Existence of a Hermitian Positive Definite Solution of a Type of Nonlinear Matrix Equations." Mathematical Problems in Engineering 2009 (2009): 1–13. http://dx.doi.org/10.1155/2009/672695.

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We study the Hermitian positive definite solutions of the nonlinear matrix equationX+A∗X−2A=I, whereAis ann×nnonsingular matrix. Some necessary and sufficient conditions for the existence of a Hermitian positive definite solution of this equation are given. However, based on the necessary and sufficient conditions, some properties and the equivalent equations ofX+A∗X−2A=Iare presented while the matrix equation has a Hermitian positive definite solution.

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7

Yan, Zhenya. "Complex PT -symmetric nonlinear Schrödinger equation and Burgers equation." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 371, no. 1989 (April 28, 2013): 20120059. http://dx.doi.org/10.1098/rsta.2012.0059.

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The complex -symmetric nonlinear wave models have drawn much attention in recent years since the complex -symmetric extensions of the Korteweg–de Vries (KdV) equation were presented in 2007. In this review, we focus on the study of the complex -symmetric nonlinear Schrödinger equation and Burgers equation. First of all, we briefly introduce the basic property of complex symmetry. We then report on exact solutions of one- and two-dimensional nonlinear Schrödinger equations (known as the Gross–Pitaevskii equation in Bose–Einstein condensates) with several complex -symmetric potentials. Finally, some complex -symmetric extension principles are used to generate some complex -symmetric nonlinear wave equations starting from both -symmetric (e.g. the KdV equation) and non- -symmetric (e.g. the Burgers equation) nonlinear wave equations. In particular, we discuss exact solutions of some representative ones of the complex -symmetric Burgers equation in detail.

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8

Prokhorova, M. F. "Factorization of the reaction-diffusion equation, the wave equation, and other equations." Proceedings of the Steklov Institute of Mathematics 287, S1 (November 27, 2014): 156–66. http://dx.doi.org/10.1134/s0081543814090156.

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9

Shi, Yong-Guo, and Xiao-Bing Gong. "Linear functional equations involving Babbage’s equation." Elemente der Mathematik 69, no. 4 (2014): 195–204. http://dx.doi.org/10.4171/em/263.

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10

Mickens, Ronald E. "Difference equation models of differential equations." Mathematical and Computer Modelling 11 (1988): 528–30. http://dx.doi.org/10.1016/0895-7177(88)90549-3.

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11

Cohen, Leon. "Phase-space equation for wave equations." Journal of the Acoustical Society of America 133, no. 5 (May 2013): 3435. http://dx.doi.org/10.1121/1.4806061.

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12

Svinin, Andrei K. "Somos-4 equation and related equations." Advances in Applied Mathematics 153 (February 2024): 102609. http://dx.doi.org/10.1016/j.aam.2023.102609.

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13

Bobylev, Alexander Vasilievich, and Sergei Borisovitch Kuksin. "Boltzmann equation and wave kinetic equations." Keldysh Institute Preprints, no. 31 (2023): 1–20. http://dx.doi.org/10.20948/prepr-2023-31.

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The well-known nonlinear kinetic equations (in particular, the wave kinetic equation and the quantum Nordheim – Uehling – Uhlenbeck equations) are considered as a natural generalization of the classical spatially homogeneous Boltzmann equation. To this goal we introduce the generalized kinetic equation that depends on a function of four real variables F(x1; x2; x3; x4). The function F is assumed to satisfy certain commutation relations. The general properties of this equation are studied. It is shown that the above mentioned kinetic equations correspond to different forms of the function (polynomial) F. Then the problem of discretization of the generalized kinetic equation is considered on the basis of ideas which are similar to those used for construction of discrete models of the Boltzmann equation. The main attention is paid to discrete models of the wave kinetic equation. It is shown that such models possses a monotone functional similar to Boltzmann H-function. The behaviour of solutions of the simplest Broadwell model for the wave kinetic equation is discussed in detail.

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14

Gupta, Rohit, Rakesh Kumar Verma, and Sanjay Kumar Verma. "Solving Wave Equation and Heat Equation by Rohit Transform (RT)." Journal of Physics: Conference Series 2325, no. 1 (August 1, 2022): 012036. http://dx.doi.org/10.1088/1742-6596/2325/1/012036.

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Abstract The wave equation and the heat equation are widely known differential equations coming to light in engineering, basic and material sciences. The differential equations which represent the wave equation and the heat equation are usually solved by the exact technique or by the approximate technique or by the purely numerical technique. Since the implementation of these techniques is very complex, computationally vigorous, and requires elaborate computations, therefore, for finding the solutions of differential equations depicting the wave equation and the heat equation, there is a need to ask for integral transform techniques. Integral transform techniques render productive means for finding the solutions of problems coming to light in engineering, basic and material sciences. The Rohit transform (RT) is a new integral transformation put forward by the author Rohit Gupta recently in the year 2020 and has been utilized for finding the solutions of problems coming to light in engineering, basic and material sciences like other transform techniques. In this study, the RT is brought in for finding the solutions of the heat equation and the wave equation expressed in terms of differential equations which are generally partial in nature.

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15

Bose, A. K. "An integral equation associated with linear homogeneous differential equations." International Journal of Mathematics and Mathematical Sciences 9, no. 2 (1986): 405–8. http://dx.doi.org/10.1155/s0161171286000509.

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Associated with each linear homogeneous differential equationy(n)=∑i=0n−1ai(x)y(i)of ordernon the real line, there is an equivalent integral equationf(x)=f(x0)+∫x0xh(u)du+∫x0x[∫x0uGn−1(u,v)a0(v)f(v)dv]duwhich is satisfied by each solutionf(x)of the differential equation.

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16

Sinkala, Winter, and Tembinkosi F. Nkalashe. "Studying a Tumor Growth Partial Differential Equation via the Black–Scholes Equation." Computation 8, no. 2 (June 16, 2020): 57. http://dx.doi.org/10.3390/computation8020057.

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Two equations are considered in this paper—the Black–Scholes equation and an equation that models the spatial dynamics of a brain tumor under some treatment regime. We shall call the latter equation the tumor equation. The Black–Scholes and tumor equations are partial differential equations that arise in very different contexts. The tumor equation is used to model propagation of brain tumor, while the Black–Scholes equation arises in financial mathematics as a model for the fair price of a European option and other related derivatives. We use Lie symmetry analysis to establish a mapping between them and hence deduce solutions of the tumor equation from solutions of the Black–Scholes equation.

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Čermák, Jan, and Petr Kundrát. "Linear differential equations with unbounded delays and a forcing term." Abstract and Applied Analysis 2004, no. 4 (2004): 337–45. http://dx.doi.org/10.1155/s1085337504306020.

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The paper discusses the asymptotic behaviour of all solutions of the differential equationy˙(t)=−a(t)y(t)+∑i=1nbi(t)y(τi(t))+f(t),t∈I=[t0,∞), with a positive continuous functiona, continuous functionsbi,f, andncontinuously differentiable unbounded lags. We establish conditions under which any solutionyof this equation can be estimated by means of a solution of an auxiliary functional equation with one unbounded lag. Moreover, some related questions concerning functional equations are discussed as well.

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18

Chu, Yu-Ming, Shumaila Javeed, Dumitru Baleanu, Sidra Riaz, and Hadi Rezazadeh. "New Exact Solutions of Kolmogorov Petrovskii Piskunov Equation, Fitzhugh Nagumo Equation, and Newell-Whitehead Equation." Advances in Mathematical Physics 2020 (November 5, 2020): 1–14. http://dx.doi.org/10.1155/2020/5098329.

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This work presents the new exact solutions of nonlinear partial differential equations (PDEs). The solutions are acquired by using an effectual approach, the first integral method (FIM). The suggested technique is implemented to obtain the solutions of space-time Kolmogorov Petrovskii Piskunov (KPP) equation and its derived equations, namely, Fitzhugh Nagumo (FHN) equation and Newell-Whitehead (NW) equation. The considered models are significant in biology. The KPP equation describes genetic model for spread of dominant gene through population. The FHN equation is imperative in the study of intercellular trigger waves. Similarly, the NW equation is applied for chemical reactions, Faraday instability, and Rayleigh-Benard convection. The proposed technique FIM can be applied to find the exact solutions of PDEs.

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19

Hino, Yoshiyuki, and Taro Yoshizawa. "Total stability property in limiting equations for a functional-differential equation with infinite delay." Časopis pro pěstování matematiky 111, no. 1 (1986): 62–69. http://dx.doi.org/10.21136/cpm.1986.118265.

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20

Naher, Hasibun, and Farah Aini Abdullah. "New Traveling Wave Solutions by the Extended Generalized Riccati Equation Mapping Method of the(2+1)-Dimensional Evolution Equation." Journal of Applied Mathematics 2012 (2012): 1–18. http://dx.doi.org/10.1155/2012/486458.

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The generalized Riccati equation mapping is extended with the basic(G′/G)-expansion method which is powerful and straightforward mathematical tool for solving nonlinear partial differential equations. In this paper, we construct twenty-seven traveling wave solutions for the (2+1)-dimensional modified Zakharov-Kuznetsov equation by applying this method. Further, the auxiliary equationG′(η)=w+uG(η)+vG2(η)is executed with arbitrary constant coefficients and called the generalized Riccati equation. The obtained solutions including solitons and periodic solutions are illustrated through the hyperbolic functions, the trigonometric functions, and the rational functions. In addition, it is worth declaring that one of our solutions is identical for special case with already established result which verifies our other solutions. Moreover, some of obtained solutions are depicted in the figures with the aid of Maple.

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21

Chiang, Chun-Yueh. "A Note on the⊤-Stein Matrix Equation." Abstract and Applied Analysis 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/824641.

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This note is concerned with the linear matrix equationX=AX⊤B + C, where the operator(·)⊤denotes the transpose (⊤) of a matrix. The first part of this paper sets forth the necessary and sufficient conditions for the unique solvability of the solutionX. The second part of this paper aims to provide a comprehensive treatment of the relationship between the theory of the generalized eigenvalue problem and the theory of the linear matrix equation. The final part of this paper starts with a brief review of numerical methods for solving the linear matrix equation. In relation to the computed methods, knowledge of the residual is discussed. An expression related to the backward error of an approximate solution is obtained; it shows that a small backward error implies a small residual. Just like the discussion of linear matrix equations, perturbation bounds for solving the linear matrix equation are also proposed in this work.Erratum to “A Note on the⊤-Stein Matrix Equation”

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22

Abdillah, Muhammad Taufik, Berlian Setiawaty, and Sugi Guritman. "The Solution of Generalization of the First and Second Kind of Abel’s Integral Equation." JTAM (Jurnal Teori dan Aplikasi Matematika) 7, no. 3 (July 17, 2023): 631. http://dx.doi.org/10.31764/jtam.v7i3.14193.

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Integral equations are equations in which the unknown function is found to be inside the integral sign. N. H. Abel used the integral equation to analyze the relationship between kinetic energy and potential energy in a falling object, expressed by two integral equations. This integral equation is called Abel's integral equation. Furthermore, these equations are developed to produce generalizations and further generalizations for each equation. This study aims to explain generalizations of the first and second kind of Abel’s integral equations, and to find solution for each equation. The method used to determine the solution of the equation is an analytical method, which includes Laplace transform, fractional calculus, and manipulation of equation. When the analytical approach cannot solve the equation, the solution will be determined by a numerical method, namely successive approximations. The results showed that the generalization of the first kind of Abel’s integral equation solution can be determined using the Laplace transform method, fractional calculus, and manipulation of equation. On the other hand, the generalization of the second kind of Abel’s integral equation solution is obtained from the Laplace transform method. Further generalization of the first kind of Abel’s integral equation solution can be obtained using manipulation of equation method. Further generalization of the second kind of Abel’s integral equation solution cannot be determined by analytical method, so a numerical method (successive approximations) is used.

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23

ZHANG, YUFENG, HONWAH TAM, and JING ZHAO. "GENERALIZED mKdV EQUATION, LIOUVILLE EQUATION, SINE-GORDON EQUATION AND SINH-GORDON EQUATION AS WELL AS A FORMAL BÄCKLUND TRANSFORMATION." International Journal of Modern Physics B 25, no. 18 (July 20, 2011): 2449–60. http://dx.doi.org/10.1142/s0217979211101387.

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A Lie algebra which consists of linear combinations of one basis of the Lie algebra A1 is presented for which an isospectral Lax pair is exhibited. By using the zero curvature equation, the generalized mKdV equation, Liouville equation and sine-Gordon equation, sinh-Gordon equation are generated via polynomial expansions. Finally, we investigate a kind of formal Bäcklund transformation for the generalized sine-Gordon equation. The explicit Bäcklund transformation of the standard sine-Gordon equation is presented. The other equations given in the paper are obtained similarly.

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24

Nikolova, Elena V. "Exact Travelling-Wave Solutions of the Extended Fifth-Order Korteweg–de Vries Equation via Simple Equations Method (SEsM): The Case of Two Simple Equations." Entropy 24, no. 9 (September 13, 2022): 1288. http://dx.doi.org/10.3390/e24091288.

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We apply the Simple Equations Method (SEsM) for obtaining exact travelling-wave solutions of the extended fifth-order Korteweg–deVries (KdV) equation. We present the solution of this equation as a composite function of two functions of two independent variables. The two composing functions are constructed as finite series of the solutions of two simple equations. For our convenience, we express these solutions by special functions V, which are solutions of appropriate ordinary differential equations, containing polynomial non-linearity. Various specific cases of the use of the special functions V are presented depending on the highest degrees of the polynomials of the used simple equations. We choose the simple equations used for this study to be ordinary differential equations of first order. Based on this choice, we obtain various travelling-wave solutions of the studied equation based on the solutions of appropriate ordinary differential equations, such as the Bernoulli equation, the Abel equation of first kind, the Riccati equation, the extended tanh-function equation and the linear equation.

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25

Mohamad- Jawad, Anwar. "The Sine-Cosine Function Method for Exact Solutions of Nonlinear Partial Differential Equations." Journal of Al-Rafidain University College For Sciences ( Print ISSN: 1681-6870 ,Online ISSN: 2790-2293 ), no. 2 (October 17, 2021): 120–39. http://dx.doi.org/10.55562/jrucs.v32i2.327.

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The Sine-Cosine function algorithm is applied for solving nonlinear partial differential equations. The method is used to obtain the exact solutions for different types of nonlinear partial differential equations such as, The K(n + 1, n + 1) equation, Schrödinger-Hirota equation, Gardner equation, the modified KdV equation, perturbed Burgers equation, general Burger’s-Fisher equation, and Cubic modified Boussinesq equation which are the important Soliton equations.Keywords: Nonlinear PDEs, Exact Solutions, Nonlinear Waves, Gardner equation, Sine-Cosine function method, The Schrödinger-Hirota equation.

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26

Wang, Haifeng, and Yufeng Zhang. "Self-Adjointness and Conservation Laws of Frobenius Type Equations." Symmetry 12, no. 12 (December 1, 2020): 1987. http://dx.doi.org/10.3390/sym12121987.

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The Frobenius KDV equation and the Frobenius KP equation are introduced, and the Frobenius Kompaneets equation, Frobenius Burgers equation and Frobenius Harry Dym equation are constructed by taking values in a commutative subalgebra Z2ε in the paper. The five equations are selected as examples to help us study the self-adjointness of Frobenius type equations, and we show that the first two equations are quasi self-adjoint and the last three equations are nonlinear self-adjointness. It follows that we give the symmetries of the Frobenius KDV and the Frobenius KP equation in order to construct the corresponding conservation laws.

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27

Vitanov, Nikolay K., and Zlatinka I. Dimitrova. "Modified Method of Simplest Equation Applied to the Nonlinear Schrödinger Equation." Journal of Theoretical and Applied Mechanics 48, no. 1 (March 1, 2018): 59–68. http://dx.doi.org/10.2478/jtam-2018-0005.

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AbstractWe consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential equations for deep water waves: the nonlinear Schrödinger equation. It is shown that the methodology works also for other equations of the nonlinear Schrödinger kind.

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28

Salatich, A. A., and S. Yu Slavyanov. "Antiquantization of the Double Confluent Heun Equation. The Teukolsky Equation." Nelineinaya Dinamika 15, no. 1 (2019): 79–85. http://dx.doi.org/10.20537/nd190108.

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29

Ndogmo, Jean-Claude, and Fazal Mahomed. "On certain properties of linear iterative equations." Open Mathematics 12, no. 4 (April 1, 2014): 648–57. http://dx.doi.org/10.2478/s11533-013-0364-z.

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Abstract An expression for the coefficients of a linear iterative equation in terms of the parameters of the source equation is given both for equations in standard form and for equations in reduced normal form. The operator that generates an iterative equation of a general order in reduced normal form is also obtained and some other properties of iterative equations are established. An expression for the parameters of the source equation of the transformed equation under equivalence transformations is obtained, and this gives rise to the derivation of important symmetry properties for iterative equations. The transformation mapping a given iterative equation to the canonical form is obtained in terms of the simplest determining equation, and several examples of application are discussed.

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30

ZADOROZHNAYA, Olga V., and Vladimir K. KOCHETKOV. "INTEGRAL REPRESENTATION OF SOLUTIONS OF AN ORDINARY DIFFERENTIAL EQUATION AND THE LOEWNER– KUFAREV EQUATION." Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, no. 67 (2020): 28–39. http://dx.doi.org/10.17223/19988621/67/3.

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The article presents a method of integral representation of solutions of ordinary differential equations and partial differential equations with a polynomial right-hand side part, which is an alternative to the construction of solutions of differential equations in the form of different series. The method is based on the introduction of additional analytical functions establishing the equation of connection between the introduced functions and the constituent components of the original differential equation. The implementation of the coupling equations contributes to the representation of solutions of the differential equation in the integral form, which allows solving some problems of mathematics and mathematical physics. The first part of the article describes the coupling equation for an ordinary differential equation of the first order with a special polynomial part of a higher order. Here, the integral representation of the solution of a differential equation with a second-order polynomial part is indicated in detail. In the second part of the paper, we consider the integral representation of the solution of a partial differential equation with the polynomial second-order part of the Loewner–Kufarev equation, which is an equation for univalent functions.

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31

Iskandarova, Gulistan, and Dogan Kaya. "Symmetry solution on fractional equation." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 7, no. 3 (October 25, 2017): 255–59. http://dx.doi.org/10.11121/ijocta.01.2017.00498.

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As we know nearly all physical, chemical, and biological processes in nature can be described or modeled by dint of a differential equation or a system of differential equations, an integral equation or an integro-differential equation. The differential equations can be ordinary or partial, linear or nonlinear. So, we concentrate our attention in problem that can be presented in terms of a differential equation with fractional derivative. Our research in this work is to use symmetry transformation method and its analysis to search exact solutions to nonlinear fractional partial differential equations.

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Kumar, Anil, and Gaurav Varshney. "IMPLEMENTATION AND ASSESSMENT OF THE SIMPLE EQUATION TECHNIQUE FOR SOLVING NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS." jnanabha 54, no. 01 (2024): 76–82. http://dx.doi.org/10.58250/jnanabha.2024.54109.

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In this paper, the simple equation method is especially used to solve two Nonlinear Partial Differential Partial Equations NLPDEs, the Kodomstev-Petviashvili (KP) equation and the (2+1)-dimensional breaking soliton equation. The modified Benjamin-Bona-Mahony equation and the Klein-Gordon equation in (1+2) dimensions are two illustrations of second order nonlinear equations that can benefit from using this approach. The Bernoulli equation acts as the trial condition and aids in the mathematical a description of the nonlinear wave equation in the simple equation.

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Kumar, Nand Kishor. "Relationship Between Differential Equations and Difference Equation." NUTA Journal 8, no. 1-2 (December 31, 2021): 88–93. http://dx.doi.org/10.3126/nutaj.v8i1-2.44113.

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The study of Differential equations and Difference equations play animportant and significant role in many sciences. These equations are used as mathematical tool used in solving various problems in modeling,physics, chemistry,biology, anthropology, etc. or even in social studies. Differential equations are used to solve real life problems by approximation of numerical methods. Theory of Differential and Difference equations has been taught at all levels in high schools and at the universities for all students, including students majoring in Mathematics. This is a micro –study in which the research is designed in an exploitative, qualitative, descriptive and analytic framework to analyze the differential-difference equations. In this study, theoretical concepts, descriptive, analytical and numerical methods about differential-difference equations which areclarified by related examples. This research article seeks to study the relationship between them. The finite difference method for solving equations leads to difference equationtheory, developing a parallel between difference equations and differential equations. The first differences are related to the first derivatives. Difference equations are discrete versions of differential equations, and similarly differential equations are continuous versions of difference equations.

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34

Bruening, James, and Hao Hao Wang. "An Implicit Equation Given Certain Parametric Equations." Missouri Journal of Mathematical Sciences 18, no. 3 (October 2006): 213–20. http://dx.doi.org/10.35834/2006/1803213.

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35

daiah, P. Red. "Deriving Equations for Energy Equation by Fem." International Journal of Mathematics Trends and Technology 50, no. 2 (October 25, 2017): 121–24. http://dx.doi.org/10.14445/22315373/ijmtt-v50p518.

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36

Zun-Tao, Fu, Liu Shi-Da, and Liu Shi-Kuo. "Solving Nonlinear Wave Equations by Elliptic Equation." Communications in Theoretical Physics 39, no. 5 (May 15, 2003): 531–36. http://dx.doi.org/10.1088/0253-6102/39/5/531.

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37

Wilczyński, Paweł. "Planar nonautonomous polynomial equations: The Riccati equation." Journal of Differential Equations 244, no. 6 (March 2008): 1304–28. http://dx.doi.org/10.1016/j.jde.2007.12.008.

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38

Mach, Andrzej. "On Some Functional Equations Involving Babbage Equation." Results in Mathematics 51, no. 1-2 (November 19, 2007): 97–106. http://dx.doi.org/10.1007/s00025-007-0261-5.

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39

Hongler, M. O., and L. Streit. "Generalized master equations and the telegrapher's equation." Physica A: Statistical Mechanics and its Applications 165, no. 2 (May 1990): 196–206. http://dx.doi.org/10.1016/0378-4371(90)90191-t.

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40

Ostaszewski, A. J. "Homomorphisms from functional equations: the Goldie equation." Aequationes mathematicae 90, no. 2 (June 20, 2015): 427–48. http://dx.doi.org/10.1007/s00010-015-0357-z.

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41

Firsov, Dmitry K., and Joe LoVetri. "FVTD—integral equation hybrid for Maxwell's equations." International Journal of Numerical Modelling: Electronic Networks, Devices and Fields 21, no. 1-2 (2007): 29–42. http://dx.doi.org/10.1002/jnm.662.

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Mancas, S. C., and H. C. Rosu. "Integrable Abel equations and Vein's Abel equation." Mathematical Methods in the Applied Sciences 39, no. 6 (July 28, 2015): 1376–87. http://dx.doi.org/10.1002/mma.3575.

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Ben Adda, Fayçal, and Jacky Cresson. "Fractional differential equations and the Schrödinger equation." Applied Mathematics and Computation 161, no. 1 (February 2005): 323–45. http://dx.doi.org/10.1016/j.amc.2003.12.031.

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Gonzales, R. A., S. Y. Kang, I. Koltracht, and G. Rawitscher. "Integral Equation Method for Coupled Schrödinger Equations." Journal of Computational Physics 153, no. 1 (July 1999): 160–202. http://dx.doi.org/10.1006/jcph.1999.6272.

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Delgado-Vences, Francisco, and Franco Flandoli. "A spectral-based numerical method for Kolmogorov equations in Hilbert spaces." Infinite Dimensional Analysis, Quantum Probability and Related Topics 19, no. 03 (August 31, 2016): 1650020. http://dx.doi.org/10.1142/s021902571650020x.

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We propose a numerical solution for the solution of the Fokker–Planck–Kolmogorov (FPK) equations associated with stochastic partial differential equations in Hilbert spaces. The method is based on the spectral decomposition of the Ornstein–Uhlenbeck semigroup associated to the Kolmogorov equation. This allows us to write the solution of the Kolmogorov equation as a deterministic version of the Wiener–Chaos Expansion. By using this expansion we reformulate the Kolmogorov equation as an infinite system of ordinary differential equations, and by truncating it we set a linear finite system of differential equations. The solution of such system allow us to build an approximation to the solution of the Kolmogorov equations. We test the numerical method with the Kolmogorov equations associated with a stochastic diffusion equation, a Fisher–KPP stochastic equation and a stochastic Burgers equation in dimension 1.

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Kratz, Werner. "Asymptotic behaviour of Riccati's differential equation associated with self-adjoint scalar equations of even order." Czechoslovak Mathematical Journal 38, no. 2 (1988): 351–65. http://dx.doi.org/10.21136/cmj.1988.102230.

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Rajabova, Lutfya Nusratovna, and Farvariddin Mufazalovich Ahmadov. "Solution of a Cauchy type problem for an integral equation of Volterra type with singular kernels, when the roots of the characteristic equations are complex conjugate." BULLETIN OF THE L.N. GUMILYOV EURASIAN NATIONAL UNIVERSITY. Mathematics. Computer science. Mechanics series 146, no. 1 (March 30, 2024): 25–32. http://dx.doi.org/10.32523/bulmathenu.2024/1.3.

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In this paper, we study a two-dimensional Volterra type integral equation with a singularity and a logarithmic singularity for one variable and a strong singularity for another variable. The solution of an integral equation with special kernels in the case when the coefficients of the equation are interconnected is reduced to solving one-dimensional Volterra-type integral equations with special kernels. Using the connection of the considered integral equations with ordinary differential equations with singular coefficients, depending on the sign of the coefficients of the equation, explicit solutions of the studied two-dimensional integral equation are obtained. Depending on the roots of the characteristic equations and the sign of the equation parameters, explicit solutions of this integral equation are obtained. If the characteristic equation has complex conjugate roots, then the integral equation under study with special kernels has a single solution or explicit solutions contain two or four arbitrary functions. In the latter cases, the correct formulation has been clarified and explicit solutions of Cauchy-type problems have been obtained.

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Magalhaes, Pedro, Perrin Neto, and Cristina Magalhães. "New Carré Equation." Metrology and Measurement Systems 17, no. 2 (January 1, 2010): 173–94. http://dx.doi.org/10.2478/v10178-010-0016-6.

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New Carré EquationThe present work offers new equations for phase evaluation in measurements. Several phase-shifting equations with an arbitrary but constant phase-shift between captured intensity signs are proposed. The equations are similarly derived as the so called Carré equation. The idea is to develop a generalization of the Carré equation that is not restricted to four images. Errors and random noise in the images cannot be eliminated, but the uncertainty due to their effects can be reduced by increasing the number of observations. An experimental analysis of the errors of the technique was made, as well as a detailed analysis of errors of the measurement. The advantages of the proposed equation are its precision in the measures taken, speed of processing and the immunity to noise in signs and images.

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Wang, Haifeng, and Chuanzhong Li. "Bäcklund transformation of Frobenius Painlevé equations." Modern Physics Letters B 32, no. 17 (June 18, 2018): 1850181. http://dx.doi.org/10.1142/s0217984918501816.

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In this paper, in order to generalize the Painlevé equations, we give a [Formula: see text]-Painlevé IV equation which can apply Bäcklund transformations to explore. And these Bäcklund transformations can generate new solutions from seed solutions. Similarly, we also introduce a Frobenius Painlevé I equation and Frobenius Painlevé III equation. Then, we find the connection between the Frobenius KP hierarchy and Frobenius Painlevé I equation by the Virasoro constraint. Further, in order to seek different aspects of Painlevé equations, we introduce the Lax pair, Hirota bilinear equation and [Formula: see text] functions. Moreover, some Frobenius Okamoto-like equations and Frobenius Toda-like equations can also help us to explore these equations.

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Buckingham, Michael J. "On the transient solutions of three acoustic wave equations: van Wijngaarden’s equation, Stokes’ equation and the time-dependent diffusion equation." Journal of the Acoustical Society of America 124, no. 4 (October 2008): 1909–20. http://dx.doi.org/10.1121/1.2973231.

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