Goursat Problem Research Articles

A Goursat problem for the fourth moment equation

The solution of a Goursat problem for the pseudoparabolic equation satisfied by the fourth statistical moment of an initially plane wave propagating in a random medium is presented, using an integro–differential equation technique. Two-dimensional propagations are considered.

Use of the operational method in solving the initial characteristic problem of the plane theory of elasticity: PMM vol. 39, n≗ 3, 1975, pp. 564–567

The dimensional Laplace-Carson transform is used to obtain a solution of the Goursat problem for the equation describing plane deformation of an ideally plastic body. The basic relationships in a field formed by circular arcs are investigated as an example. One of the methods of solving the boundary value problems of a plane flow of an ideally plastic medium consists of linearizing the quasilinear, hyperbolic, partial differential equation with help of the Mikhlin [1] substitution which reduces it to the equation of telegraphy. The linearized equation can be solved using the integral Riemann formula [2, 3], by expanding into a series in metacylindrical functions, or by means of the integral transforms.

On the $C^\infty$-Goursat problem for $2$nd order equations with real constant coefficients

On the $C^\infty$-Goursat problem for $2$nd order equations with real constant coefficients

UNIQUENESS CLASSES FOR THE SOLUTION OF GOURSAT'S PROBLEM

Uniqueness classes for the solution of Goursat's problem, which consists in giving Cauchy initial conditions for each of the variables , , are studied for linear partial differential equations with constant coefficients with two groups of variables: time and space . The results obtained generalize a well-known theorem of Gel'fand and ?ilov on uniqueness classes for the solution of Cauchy's problem.Bibliography: 10 items.

The existence of optimal controls for processes described by a system of hyperbolic equations

THE problem of the optimal control of processes described by Goursat's problem for systems of hyperbolic equations is considered. Theorems on the existence of optimal controls are proved and some necessary optimality conditions are obtained. Let the controlled system be described by the hyperbolic equations (1) u tx = f(t, x, u, u t, u x, v) subject to the conditions (2) u(0, x) = ø(x), x ∈ = D l= [0, l], u(t, 0) = ψ(t), t ∈ D T = [0, T] . Here u ( t, x) = { u 1( t, x), …, u n ( t, x)} describes the state of the controlled system, v( t, x) = { v 1( t, x), …, v r ( t, x)} is the controlling vector function, the vector function f( t, x, u, p, q, v) is continuous with respect to u, p, q, v for almost all ( t, x) ∈ Q = 0 < t < T, 0 < x < l and is measurable with respect to ( t, x) for all u ∈ R n , p ∈ = R n , q ∈ R n , ν ∈ = V, where R n is an n-dimensional space, V is some closed boundary set in R r , and the initial functions ø and ψ are absolutely continuous functions in their domains of definition and satisfy matching conditions. As the class of permissible controls we take the set of functions v( t, x), measurable on Q and assuming values of the set V almost everywhere on Q. We denote by Ω the set of permissible controls, and by C ∗(Q) the set of all functions u( t, x), defined on Q and satisfying the conditions: (1) the function u( t, x) is absolutely continuous on Q, (2) for almost all t of D T the derivative u t , considered as a function of x, is absolutely continuous in D L and for almost all x of D l the derivative u x is absolutely continuous with respect to t in D T , (3) the derivative u tx is summable on Q in the Lebesgue sense. For a given permissible control v( t, x) we understand by B solution of problem (1), (2) a function u( t, x) of the class C ∗(Q) , which satisfies Eq. (1) and the initial conditions (2) in the usual sense, almost everywhere on Q [1]. Among all the permissible controls it is required to find a control v( t, x) such that together with the solution u( t, x) of problem (1), (2) corresponding to it, the functional (3) I(v) = ʃ Q g(t,x,u,u t,u x,v)dx dt will attain its greatest value, where the function g( t, x, u, p, q, v) is continuous with respect to ( u, p, q, v) for almost all ( t, x) ∈ Q and is measurable with respect to ( t, x) for almost all values of ( u, p, q, v). For this problem various theorems on the existence of an optimal control are proved and some necessary conditions are obtained. We considered the scalar case of problem (1)-(3) in [2], where the existence and uniqueness of the optimal control was proved, necessary conditions of optimality were given and the smoothness of the optimal control was studied. However, the existence theorem was not clearly formulated and proved there. Hence in this paper the existence of an optimal control is proved anew.

An analogue of the Goursat problem in a class of generalized functions of infinite order

An analogue of the Goursat problem in a class of generalized functions of infinite order

The problem of a dihedral piston

The problem of the plane nonstationary motion of a gas behind a dihedral piston is considered. The problem is linearized on the assumption that the piston angle α is small. The mixed problems and the Goursat problem are solved for the linearized double-wave equation in the region of hyperbolicity and then the mixed boundary value problem is solved in the region of ellipticity. The solutions are obtained in elementary functions and quadratures.

New proofs and generalisations of theorems of existence and uniqueness for the goursat problem

Abstract theorems of existence and uniqueness are proved for a differential equation whose solution takes its values in a sequence of Banach spaces called a Banach filtration (a notion introduced by F. Treves). The abstract theorems are then applied to obtain existence and uniqueness theorems of a classical nature bearing on that generalization of the Cauchy problem of partial differential equations known as the Goursat problem. All the results so obtained remain true in the case when the equations involve more general operators than partial differential operators (e.g., pseudo-differential operators)

On the numerical solution of the Goursat problem

On the numerical solution of the Goursat problem

The abstract Goursat problem

The abstract Goursat problem

Motion of a gas behind a plane detonation wave orthogonal to the free surface: PMM vol. 35, n≗6, 1971, pp. 1100–1108

The problem of detonation of one quarter of a space filled with explosive and initiated on one of the faces is examined. The finding of the solution in the perturbed region is reduced to the solution of Goursat's problem for a quasi-linear differential equation of second order with two independent variables. This problem is solved by the numerical method of characteristics. An examination of singular points is presented. The solution in the perturbed region and the form of the free surface are obtained. The problem of gas motion behind an expanding detonation wave in a space with a conical cutout was examined in papers [1, 2].

A Note on Trapezoidal Methods for the Solution of Initial Value Problems

The trapezoidal rule for the numerical integration of first-order ordinary differential equations is shown to possess, for a certain type of problem, an undesirable property. The removal of this difficulty is shown to be straightforward, resulting in a modified trapezoidal rule. Whilst this latent difficulty is slight (and probably rare in practice), the fact that the proposed modification involves negligible additional programming effort would suggest that it is worthwhile. A corresponding modification for the trapezoidal rule for the Goursat problem is also included.

A note on trapezoidal methods for the solution of initial value problems

The trapezoidal rule for the numerical integration of first-order ordinary differential equations is shown to possess, for a certain type of problem, an undesirable property. The removal of this difficulty is shown to be straightforward, resulting in a modified trapezoidal rule. Whilst this latent difficulty is slight (and probably rare in practice), the fact that the proposed modification involves negligible additional programming effort would suggest that it is worthwhile. A corresponding modification for the trapezoidal rule for the Goursat problem is also included.

A Correction of the Paper 'Non-Characteristic Cauchy Problems and Genealized Goursat Problems in R^n '

A Correction of the Paper 'Non-Characteristic Cauchy Problems and Genealized Goursat Problems in R^n '

A note on gevrey classes of functions connected with goursat problems

The Gevrey classes G(β, δ, d) have been introduced in connection with non-linear local Goursat problems. A lemma is proved which shows that the definition of the class G(β, δ, d) can be simplified. It then follows that the results in two different papers on Goursat problems are compatible when specialized to formally identical cases.

Uniqueness and Goursat problems

A uniqueness theorem by W. Walter for a second order Goursat problem in to independent variables is generalized. In the resulting theorem there are no restrictions on the order or on the number of indipendent variables.

An existence theorem for a general Goursat problem

An existence theorem for a general Goursat problem

On singular problems of Goursat and Cauchy

Introduction. In this paper the problems of Goursat and Cauchy are investigated for equations of hyperbolic type with singular coefficients. Effective algorithms are presented for constructing Duhamel functions of such singular initial problems, reducing these problems to a recurrent sequence of ordinary linear nonhomogeneous differential equations of the second order. Using established differential properties of the operators of Kummer and Gauss, explicit formulas for inverting them are obtained~ as well as simple estimates characterizing the behavior, in the neighborhood of a singular curve of the equation, of the iterations obtained. Studied also are theorems of uniqueness and extremal properties of the solutions of the singular Goursat and Cauchy problems. w i. By the singular Goursat problem we shall understand the problem of finding, in the rectangle = OPMQO, with ver t ices at O(0, 0), P(x0, 0), M(x0, Y0), Q(0, y0), a solution z(x, y) E C2(~) of the equation: L [z, A, B, C] = xz~y + A (x) z~ + B (x) z~ + C (x) z = O, which assumes along the lines x= 0, y=0 , the values (1.1)

Exponential majorization applied to a non-linear cauchy (goursat) problem for functions of gevrey nature

Let f(y, x, z) be a continuous function defined in some neighbourhood of the origin in R3. The function f is supposed to be of Gevrey class d≥1 in the x- and z-variables. If γ<β, and γ+αd≤β then it is proved that $$D_y^\beta = f(y,x,D_y^\gamma D_x^\alpha u),D_y^i u(0,x) = 0, 0 \leqslant j< \beta,$$ has a unique continuous solution of Gevrey class d in the x-variable.

Exponential majorization and global Goursat problems

Exponential majorization and global Goursat problems