Integral Operators in the Theory of Linear Partial Differential Equations

Bergman, Stefan

doi:10.1007/978-3-642-64985-1

Cited by 96 publications

(19 citation statements)

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“…We shall always assume that the domain in jR 3 , bounded or unbounded, to be considered here has a boundary which satisfies the following Lyapunov conditions (cf. [21]): 1.…”

Section: On the Domainmentioning

confidence: 99%

“…2. For any point x^F there exists a single fixed number J>0 such that the portion of the boundary inside a sphere of radius d with the center of x intersects lines parallel to the normal at x in at most one point, 3. If 0 be the angle between the normals at x\ and x 2 , then where a and oc are positive constants independent of x l9 x z and The sphere mentioned above we shall call by the Lyapunov sphere.…”

Section: On the Domainmentioning

confidence: 99%

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On the First Initial-Boundary Value Problem of Compressible Viscous Fluid Motion

Tani¹

1977

Publ. Res. Inst. Math. Sci.

258

142

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Up io the present day many kinds of mathematical discussions on incompressible viscous fluid motion have fully developed (cf. [32,36] Now in the present paper, we shall show that the first initial-boundary value problem for it can uniquely be solved under suitable assumptions for the initial-boundary data and for the boundary of the domain, from the classical point of view.In § 1 an exact statement and the main theorem (Theorem 1) will be found. In §2 we perform the characteristic transformation and mention the theorem of the transformed problem (Theorem 2). Firstly we prove Theorem 2 and then show that Theorem 2 implies Theorem 1 in the last section § 8. In § § 3-5 linear equations connected with the transformed equations are treated. In more detail, in § 3. 1 we briefly state some basic results for a fundamental solution in the whole space R s due to Eidel'man [9,18] [1, 10-12, 14, 15, 22, 29-31, 54].We estimate the Green matrix in § 4 and the solution in § 5. Making use of these results we give a proof of the existence in § 6 by the method of successive approximation and that of the uniqueness in § 7, therefore the proof of Theorem 2 is completed.Acknowledgement. Deep gratitude is due to Professor N. Itaya

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“…We shall always assume that the domain in jR 3 , bounded or unbounded, to be considered here has a boundary which satisfies the following Lyapunov conditions (cf. [21]): 1.…”

Section: On the Domainmentioning

confidence: 99%

Section: On the Domainmentioning

confidence: 99%

On the First Initial-Boundary Value Problem of Compressible Viscous Fluid Motion

Tani¹

1977

Publ. Res. Inst. Math. Sci.

258

142

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“…Stefan Bergman [l] and Ilya Vekua [4] have given representation formulas for solutions of the partial differential equation (1). We obtain an improvement of their results for the case of two independent variables (namely equation (2) with n set equal to 2).…”

mentioning

confidence: 86%

“…Using Bergman's integral operator of the first kind [l, p. 10], which generates a complete system of solutions for equation (1), namely…”

mentioning

confidence: 99%

A method of ascent for solving boundary value problems

Gilbert¹

1969

Bull. Amer. Math. Soc.

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“…We consider the solution of the inhomogeneous linear elliptic boundary value problem defined by (1.2), in two dimensions. Such two dimensional inhomogeneous linear elliptic boundary value problems can be solved using an integral operator derived by Bergman [2] and Vekua [t 7]. This operator is used by Schryer [141 to solve such problems numerically.…”

Section: T)mentioning

confidence: 99%

Newton's method for convex nonlinear elliptic boundary value problems

Schryer¹

1971

Numer. Math.

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Summary. Newton's method is applied to solving the boundary value problem for the equation L u =/(x, u) where L is a linear second order uniformly elliptic operator and ](x, u) is a convex monotone increasing function of u for each point x in the domain D. The Newton iterates are shown to converge uniformly, quadratically and monotonically downward to the solution of the problem. The convergence is independent of the choice for the initial Newton iterate. Numerical results are presented for several problems of physical interest.

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Integral Operators in the Theory of Linear Partial Differential Equations

Cited by 96 publications

References 0 publications

On the First Initial-Boundary Value Problem of Compressible Viscous Fluid Motion

On the First Initial-Boundary Value Problem of Compressible Viscous Fluid Motion

A method of ascent for solving boundary value problems

Newton's method for convex nonlinear elliptic boundary value problems

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