Complex potential equations I. A technique for solution

1976 ◽  
Vol 80 (1) ◽  
pp. 165-187 ◽  
Author(s):  
C. B. Collins
Keyword(s):  
Differential Equation ◽  
Close Association ◽  
Dimensional Euclidean Space ◽  
3 Dimensional ◽  
Novel Technique ◽  
Linear Partial Differential Equations ◽  
Classical Differential Geometry ◽  
Equipotential Surfaces ◽  
Geometric Technique ◽  
Image Position

AbstractThis work amalgamates and solves certain problems arising in differential equation theory and in classical differential geometry.It describes a novel technique for solving systems of non-linear partial differential equations of the formwhere f(γ) and g(γ) are arbitrarily assigned functions. The circumstances are determined under which compatible solutions exist, not only when γ is real, but also when γ is complex, and all of the corresponding solutions are found. This is done by using a geometric technique that incorporates the equipotential surfaces of constant γ. In general, these surfaces are imaginary, and a fairly extensive treatment of such surfaces in (complexified) 3-dimensional Euclidean space is included. A close association is discovered between the set of equipotential surfaces and the class of surfaces of constant radii of principal normal curvature.

scholarly journals A cubic surface of revolution

2014 ◽  
Vol 98 (542) ◽  
pp. 281-290 ◽  
Author(s):  
Mark B. Villarino
Keyword(s):  
Differential Equation ◽  
Geometric Property ◽  
A Priori ◽  
Cubic Surface ◽  
Surface Of Revolution ◽  
Cubic Equation ◽  
Classical Differential Geometry ◽  
Theoretical Investigations ◽  
Fixed Line ◽  
Image Position

A well-known exercise in classical differential geometry [1, 2, 3] is to show that the set of all points (x, y, z) ∈ ℝ3 which satisfy the cubic equationis a surface of revolution.The standard proof ([2], [3, p. 11]), which, in principle, goes back to Lagrange [4] and Monge [5], is to verify that (1) satisfies the partial differential equation (here written as a determinant)which characterises any surface of revolution F (x, y, z) = 0 whose axis of revolution has direction numbers (l, m, n) and goes through the point (a, b, c). This PDE, for its part, expresses the geometric property that the normal line through any point of must intersect the axis of revolution (this is rather subtle; see [6]). All of this, though perfectly correct, seems complicated and rather sophisticated just to show that one can obtain by rotating a suitable curve around a certain fixed line. Moreover, to carry out this proof one needs to know a priori just what this axis is, something not immediately clear from the statement of the problem. Nor does the solution give much of a clue as to which curve one rotates.A search of the literature failed to turn up a treatment of the problem which differs significantly from that sketched above (although see [1]).The polynomial (1) is quite famous and has been the object of numerous algebraical and number theoretical investigations. See the delightful and informative paper [7].

scholarly journals Linear Partial Differential Equations with Constant Coefficients: an Elementary Proof of an Existence Theorem

1959 ◽  
Vol 42 ◽  
pp. 3-5
Author(s):  
D. H. Parsons
Keyword(s):  
Differential Equation ◽  
Elementary Proof ◽  
Linear Partial Differential Equation ◽  
Partial Differential ◽  
Independent Variables ◽  
Linear Partial Differential Equations ◽  
Two Factors ◽  
Constant Coefficients ◽  
Image Position ◽  
The Right

We consider a linear partial differential equation with constant coefficients in one dependent and m independent variables, the right-hand side being zero,P being a symbolic polynomial in D1, …, Dm. If P can be decomposed into two factors, so that the equation can be writtenit is evident that the sum of any solution ofand any solution ofis also a solution of (1).

Nonoscillation Criteria for Elliptic Equations

1969 ◽  
Vol 12 (3) ◽  
pp. 275-280 ◽  
Author(s):  
C.A. Swanson
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Elliptic Equations ◽  
Sufficient Conditions ◽  
Elliptic Partial Differential Equation ◽  
Dimensional Euclidean Space ◽  
Normal Vector ◽  
Unit Normal Vector ◽  
Piecewise Continuous ◽  
Image Position

Sufficient conditions will be derived for the linear elliptic partial differential equation(1)to be nonoscillatory in an unbounded domain R in n-dimensional Euclidean space En. The boundary ∂R of R is supposed to have a piecewise continuous unit normal vector at each point. There is no essential loss of generality in assuming that R contains the origin. Otherwise no special assumptions are needed regarding the shape of R: it is not necessary for R to be quasiconical (as in [2]), quasicylindrical, or quasibounded [1].

Oscillation of Elliptic Equations in General Domains

1975 ◽  
Vol 27 (6) ◽  
pp. 1239-1245 ◽  
Author(s):  
E. S. Noussair
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Differential Operator ◽  
Euclidean Space ◽  
Unbounded Domain ◽  
Elliptic Equations ◽  
General Type ◽  
Elliptic Partial Differential Equation ◽  
Dimensional Euclidean Space ◽  
Image Position

Oscillation criteria will be obtained for the linear elliptic partial differential equationin an unbounded domain G of general type in n-dimensional Euclidean space En. The differential operator D is defined as usual by where each α (i), i = 1, … , n, is a non-negative integer.

Strong Oscillation of Elliptic Equations in General Domains

1973 ◽  
Vol 16 (1) ◽  
pp. 105-110 ◽  
Author(s):  
C. A. Swanson
Keyword(s):  
Differential Equation ◽  
Elliptic Equations ◽  
General Type ◽  
Unbounded Domains ◽  
Elliptic Partial Differential Equation ◽  
Continuous Functions ◽  
Positive Definite ◽  
Dimensional Euclidean Space ◽  
The Matrix ◽  
Image Position

Strong oscillation criteria will be obtained for the linear elliptic partial differential equation(1)in unbounded domains R of general type in n-dimensional Euclidean space En. It will be assumed throughout that B and each Aij are real-valued continuous functions in R, and that the matrix (Aij(x)) is symmetric and positive definite in R.

scholarly journals Linear Partial Differential Equations with Constant Coefficients: an Elementary Proof of an Existence Theorem

1959 ◽  
Vol 42 ◽  
pp. 3-5
Author(s):  
D. H. Parsons
Keyword(s):  
Differential Equation ◽  
Elementary Proof ◽  
Linear Partial Differential Equation ◽  
Partial Differential ◽  
Independent Variables ◽  
Linear Partial Differential Equations ◽  
Two Factors ◽  
Constant Coefficients ◽  
Image Position ◽  
The Right

We consider a linear partial differential equation with constant coefficients in one dependent and m independent variables, the right-hand side being zero,P being a symbolic polynomial in D1, …, Dm. If P can be decomposed into two factors, so that the equation can be writtenit is evident that the sum of any solution ofand any solution ofis also a solution of (1).

scholarly journals Characterization of certain differential operators in the solution of linear partial differential equations

1976 ◽  
Vol 17 (2) ◽  
pp. 83-88 ◽  
Author(s):  
Rudolf Heersink
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Differential Operators ◽  
Holomorphic Functions ◽  
Linear Partial Differential Equations ◽  
Representation Of Solutions ◽  
Number Of Publications ◽  
Image Position ◽  
The Given

In this paper we consider differential equations of the formwhere the coefficients Ai are holomorphic functions in a domain G1 × G2 ⊂ C × C. We restrict our attention to those equations for which it is possible to represent the solutions in the formwhere g1(z1) and g2(z2) are arbitrary holomorphic functions in G1 and G2 respectively. The coefficients a1, k and a2, k depend on the given differential equation. Within the last ten years a number of publications have been devoted to this kind of representation of solutions.

Mathematical Notes by Professor Tait (2.) On the Equipotential Surfaces for a Straight Wire

1875 ◽  
Vol 8 ◽  
pp. 443-443
Keyword(s):  
Natural Philosophy ◽  
Line Density ◽  
Straight Wire ◽  
Equipotential Surfaces ◽  
The Wire ◽  
Image Position

Some results given in Vol. I. of Thomson and Tait's Natural Philosophy may be much more simply obtained by calculating the potential of a wire rather than its attraction. That potential is easily found aswhere c is the length of the wire, ρ its line density, r1 and r2 the distances of its ends from the point at which the potential is to be found.

On a singular eigenvalue problem

1968 ◽  
Vol 64 (2) ◽  
pp. 439-446 ◽  
Author(s):  
D. Naylor ◽  
S. C. R. Dennis
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Bessel Functions ◽  
Asymptotic Condition ◽  
Real Constant ◽  
Limit Circle ◽  
Expansion In Eigenfunctions ◽  
Image Position

Sears and Titchmarsh (1) have formulated an expansion in eigenfunctions which requires a knowledge of the s-zeros of the equationHere ka > 0 is supposed given and β is a real constant such that 0 ≤ β < π. The above equation is encountered when one seeks the eigenfunctions of the differential equationon the interval 0 < α ≤ r < ∞ subject to the condition of vanishing at r = α. Solutions of (2) are the Bessel functions J±is(kr) and every solution w of (2) is such that r−½w(r) belongs to L2 (α, ∞). Since the problem is of the limit circle type at infinity it is necessary to prescribe a suitable asymptotic condition there to make the eigenfunctions determinate. In the present instance this condition is

A property of the asymptotic series for a class of Titchmarsh–Weyl m-functions

1986 ◽  
Vol 102 (3-4) ◽  
pp. 253-257 ◽  
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Linear Differential Equation ◽  
Asymptotic Series ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
The Real ◽  
Linear Differential ◽  
Image Position

SynopsisIn an earlier paper [6] we showed that if q ϵ CN[0, ε) for some ε > 0, then the Titchmarsh–Weyl m(λ) function associated with the second order linear differential equationhas the asymptotic expansionas |A| →∞ in a sector of the form 0 < δ < arg λ < π – δ.We show that if the real valued function q admits the expansionin a neighbourhood of 0, then

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