Có 100+ tài liệu thuộc chủ đề "quy luật trong toán học"
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Lyapunov function. In the cases where the theorems of stability and instability by first approximation fail to resolve the issue of stability for a specific system of nonlinear differential equations, more subtle methods must be used. A Lyapunov function for system of equations is a differentiable function V = V (x 1. The derivative with respect to t in the...
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Cauchy Problem. Cauchy problem.. Consider two formulations of the Cauchy problem.. Generalized Cauchy problem. Find a solution w = w(x, y) of equation satisfying the initial conditions. Classical Cauchy problem. Find a solution w = w(x, y) of equation satisfying the initial condition. It is convenient to represent the classical Cauchy problem as a generalized Cauchy problem by rewriting condition...
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Figure 13.8 illustrates the formation of the shock wave described by the generalized solution of Hopf’s equation with f(w. 13.8 (for x and 0.25) are obtained from the smooth (but many-valued) curves shown in Fig. 13.5 by means of Whitham’s rule of equal areas.. (13.1.3.9) Here, D is an arbitrary rectangle in the yx-plane, ψ = ψ(x, y) is any...
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The stable generalized solution of the Cauchy problem is given by w(x, y – 0. x, x, y, ξ – (x, y. x, x, y, ξ + (x, y). (13.1.4.13). where ξ – (x, y) and ξ + (x, y) denote, respectively, the greatest lower bound and the least upper bound of the set of values {ξ = ξ n...
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This results in the three functions. Under the adopted assumptions, there exists a unique, twice continuously differentiable solution of equation satisfying the initial conditions and . This theorem has a local character, i.e., the existence of a unique solution of the Cauchy problem is merely guaranteed in some neighborhood of the line defined by the initial data together with the...
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is the viscosity solution of problem . The function ϕ ∗ is the conjugate of ϕ, i.e., ϕ ∗ (q. (L – x) H ∗ (t) is the viscosity solution of problem . qt – H(q) is the conjugate of the Hamiltonian.. C., Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equa- tions, Birkh¨auser Verlag, Boston, 1998.. C., and Lions, P.-L.,...
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many particular solutions. The specific solution that describes the physical phenomenon under study is separated from the set of particular solutions of the given differential equation by means of the initial and boundary conditions.. Initial and boundary conditions.. In general, a linear second-order partial differential equation of the parabolic type with n independent variables can be written as. Equation is...
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Therefore the formal series solution has the form w(x, t. f (x) and ∂ t w(x, 0. Nonhomogeneous Linear Equations and Their Particular Solutions. For brevity, we write a nonhomogeneous linear partial differential equation in the form L[w. Below are the simplest properties of particular solutions of the nonhomogeneous equa- tion . If w 2 Φ (x, t) is a...
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To facilitate the further analysis, we represent equation in the form. It follows from the adopted assumptions (see the end of Paragraph 14.4.1-1) that p(x), p x (x), q(x), and ρ(x) are continuous functions, with p(x) >. The procedure of constructing solutions to nonstationary boundary value problems is further different for parabolic and hyperbolic equations. see Subsections 14.4.2 and 14.4.3...
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Given the transform f(p), the function 2 f (t) can be found by means of the inverse Laplace transform. In order to solve nonstationary boundary value problems, the following Laplace trans- form formulas for derivatives will be required:. More details on the properties of the Laplace transform and the inverse Laplace transform can be found in Section 11.2. The Laplace...
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(14.7.1.2) Consider the nonstationary boundary value problem for equation with an initial condition of general form. and arbitrary nonhomogeneous linear boundary conditions α 1 ∂w. (14.7.1.5) By appropriately choosing the coefficients α 1 , α 2 , β 1 , and β 2 in and we obtain the first, second, third, and mixed boundary value problems for equation . Representation...
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General relations to solve nonhomogeneous boundary value problems.. Consider the generalized telegraph equation of the form. (14.8.3.1) It is assumed that the functions p, p x , and q are continuous and p >. The solution of equation under the general initial conditions and the arbitrary linear nonhomogeneous boundary conditions can be represented as the sum. (14.8.3.2) Here, the modified...
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Consider a boundary value problem for the Laplace equation. and the expression for Ψ n from the first row in Table 14.8, we obtain the Green’s function in the form. G(x, y, ξ, η. Representation of Solutions to Boundary Value Problems via the Green’s Functions. First boundary value problem.. The solution of the first boundary value problem for equation with...
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TABLE 14.12. The Green’s functions of boundary value problems for equations of various types in bounded domains. boundary conditions Green’s function. g(x, t) for x S G(x, y, t. G(x, y, t. 14.11.1-2. Suppose the equations given in the first column of Table 14.12 contain –L x [w. Then the λ k in the expressions of the Green’s function in...
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14.12.1-2. Consider the problem for the homogeneous linear hyperbolic equation. ∂x + c(x)w with the homogeneous initial conditions. and the boundary conditions and . The solution of problem with the non- stationary boundary condition at x = x 1 can be expressed by formula in terms of the solution u(x, t) of the auxiliary problem for equation with the initial...
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(15.1.2.2) Let us select a specific solution w = w(x, y) of equation and calculate a, b, and c by formulas at some point (x, y), and substitute the resulting expressions into (15.1.1.2). Depending on the sign of the discriminant δ, the type of nonlinear equation at the point (x, y) is determined according to if δ = 0, the...
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in the form F. (15.2.3.8) Sometimes equation may be simpler than . Let u = u(ξ, η) be a solution of equation (15.2.3.8). Then the formulas define the corresponding solution of equation in parametric form.. is reduced by the Legendre transformation to the linear equation with variable coefficients aξ ∂ 2 u. ∂y 2 = 0 to the following linear...
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where λ/k plays the role of the wave propagation velocity (the sign of λ can be arbitrary, the value λ = 0 corresponds to a stationary solution, and the value k = 0 corresponds to a space-homogeneous solution). Traveling-wave solutions are characterized by the fact that the profiles of these solutions at different time* instants are obtained from one another...
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Exponential self-similar solutions exist if equation is invariant under trans- formations of the form. (15.3.4.2) where C >. Let w = Φ (x, t) be a solution of equation (15.3.3.3). t) is a solution of equation (15.3.3.4). In view of the explicit form of solution we have. In practice, exponential self-similar solutions are sought using the above existence criterion: if...
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Linear separable equations of mathematical physics admit exact solutions in the form w(x, y. Many nonlinear partial differential equations with quadratic or power nonlinearities, f 1 (x)g 1 (y) Π 1 [w. also have exact solutions of the form (15.5.1.1). In the Π i [w] are differential forms that are the products of nonnegative integer powers of the function w...