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The Quarterly Journal of Mechanics and Applied Mathematics 1952 5(4):432-440; doi:10.1093/qjmam/5.4.432
© 1952 by Oxford University Press
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ON THE INTEGRATION OF SOME VECTOR DIFFERENTIAL EQUATIONS. I

G. N. WARD

( Department of Mathematics, The University Manchester )

By using a suitable vector identity, which is derived in section 2, it is shown that the system of inhomogeneous first-order vector differential equations (1) given below can be solved directly in vector form for suitable boundary conditions. Such systems of equations occur in a number of branches of mathematical physics.

Two main cases arise, depending on whether the system of equations is elliptic or hyperbolic. The solution in the elliptic case is straightforward, but convergence difficulties occur in the hyperbolic case which are overcome by using Hadamard's finite part technique for dealing with fractionally infinite integrals.

The results are given in invariant vector forms that do not depend upon any particular choice of coordinate axes.


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