DeWitt equation in quantum field theory
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Abstract:
We take a new look at the DeWitt equation, a defining equation for the effective action functional in quantum field theory. We present a formal solution to this equation and discuss the equation in various contexts, and in particular for models where it can be made completely well defined, such as the Wess-Zumino model in two dimensions.Keywords:
Functional equation
Quantum master equation
Wheeler–DeWitt equation
The symmetry analysis of the Cheng Equation is performed. The Cheng Equation is reduced to a first-order equation of either Abel's Equations, the analytic solution of which is given in terms of special functions. Moreover, for a particular symmetry the system is reduced to the Riccati Equation or to the linear nonhomogeneous equation of Euler type. Henceforth, the general solution of the Cheng Equation with the use of the Lie theory is discussed, as also the application of Lie symmetries in a generalized Cheng equation.
Fisher's equation
Functional equation
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In this paper, the simplest equation method is used to construct exact traveling solutions of the -dimensional KP equation and generalized Fisher equation. We summarize the main steps of the simplest equation method. The Bernoulli and Riccati equation are used as simplest equations. This method is straightforward and concise, and it can be applied to other nonlinear partial differential equations.
Fisher's equation
Fisher equation
Bernoulli differential equation
Independent equation
Universal differential equation
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The general matrix Riccati equation is decomposed into product of two factors where the first one is determined independent of the second factor. Condition for existence of solution of the Riccati equation is given via this decomposition and the existence of solution of the self-adjoint matrix Riccati equation arising in optimal control and Kalman filtering is directly established from that condition.
Algebraic Riccati equation
Matrix (chemical analysis)
Matrix difference equation
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Abstract The symmetry analysis of the Cheng equation is performed. The Cheng equation is reduced to a first-order equation of either Abel’s equations, the analytic solution of which is given in terms of special functions. Moreover, for a particular symmetry the system is reduced to the Riccati equation or to the linear nonhomogeneous equation of Euler type. Henceforth, the general solution of the Cheng equation with the use of the Lie theory is discussed, as also the application of Lie symmetries in a generalized Cheng equation.
Functional equation
Fisher's equation
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The generalized Riccati equation defined as an equation between first order derivative and the cubic polynomial is named Riccati-Abel equation. Unlike solutions of ordinary Riccati equation, the solutions of Riccati-Abel equation do not admit an addition formula. In the present paper we explain a nature of this fault and elaborate a method of solution of this problem. We show that the addition formula for Riccati-Abel equation can be established only for pair of solutions. Furthermore, it is shown that analogously with ordinary Riccati equation, the relationships with linear differential equations and the general complex algebra of third order can be established only for the pair of solutions of Riccati-Abel equation.
Algebraic Riccati equation
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In this paper we consider the functional equation which is a generalization of the generalized D'Alembert equation (see Deeba and Koh[3]) . We will show that equation (0.1) can be reformulated into the following functional equation in distributions: where are operators defined on D′(I) and P is the tensor product operator for distributions. Thus equations (0.2) reduces to equation (0.1) when the solutions are regular distributions, i.e. locally Lebesgue integrable functions.
Functional equation
Operator (biology)
Generalized function
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Generally, the solution of Riccati equation cannot be given through elementary function. The paper here gives several propertions of Riccati equation,and the equation can be simplified by the several propertions. Further, the soluble conditions of the equation have been introduced in the paper, and the condition has been extended with obtaining two conclusions.
Algebraic Riccati equation
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In this research, we study new two techniques that called the extended simple equation method and the novel G′G-expansion method. The extended simple equation method depend on the auxiliary equation dϕdξ=α+λϕ+μϕ2 which has three ways for solving depends on the specific condition on the parameters as follow: When λ=0 this auxiliary equation reduces to Riccati equation, when α=0 this auxiliary equation reduces to Bernoulli equation and when α≠0,λ≠0,μ≠0 we the general solutions of this auxiliary equation while the novel G′G-expansion method depends also on similar auxiliary equation G′G′=μ+λG′G+(v-1)G′G2 which depend also on the value of (λ2-4(v-1)μ) and the specific condition on the parameters as follow: When λ=0 this auxiliary equation reduces to Riccati equation, when μ=0 this auxiliary equation reduces to Bernoulli equation and when (λ2≠4(v-1)μ) we the general solutions of this auxiliary equation. This show how both of these auxiliary equation are special cases of Riccati equation. We apply these methods on two dimensional nonlinear Kadomtsev-Petviashvili Burgers equation in quantum plasma and three-dimensional nonlinear modified Zakharov-Kuznetsov equation of ion-acoustic waves in a magnetized plasma. We obtain the exact traveling wave solutions of these important models and under special condition on the parameters, we get solitary traveling wave solutions. All calculations in this study have been established and verified back with the aid of the Maple package program. The executed method is powerful, effective and straightforward for solving nonlinear partial differential equations to obtain more and new solutions.
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Algebraic Riccati equation
Universal differential equation
Bernoulli differential equation
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The symmetry analysis of the Cheng Equation is performed. The Cheng Equation is reduced to a first-order equation of either Abel's Equations, the analytic solution of which is given in terms of special functions. Moreover, for a particular symmetry the system is reduced to the Riccati Equation or to the linear nonhomogeneous equation of Euler type. Henceforth, the general solution of the Cheng Equation with the use of the Lie theory is discussed, as also the application of Lie symmetries in a generalized Cheng equation.
Fisher's equation
Functional equation
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