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The word adjoint has a number of related meanings. In linear algebra, it refers to the conjugate transpose and is most commonly denoted A^(H). The analogous concept applied ...

The linear Boussinesq equation is the partial differential equation u_(tt)-alpha^2u_(xx)=beta^2u_(xxtt) (1) (Whitham 1974, p. 9; Zwillinger 1997, p. 129). The nonlinear ...

The complex structure of a point x=x_1,x_2 in the plane is defined by the linear map J:R^2->R^2 J(x_1,x_2)=(-x_2,x_1), (1) and corresponds to a counterclockwise rotation by ...

Also known as the Serret-Frenet formulas, these vector differential equations relate inherent properties of a parametrized curve. In matrix form, they can be written [T^.; ...

If two single-valued continuous functions kappa(s) (curvature) and tau(s) (torsion) are given for s>0, then there exists exactly one space curve, determined except for ...

A partial differential diffusion equation of the form (partialU)/(partialt)=kappadel ^2U. (1) Physically, the equation commonly arises in situations where kappa is the ...

A method which can be used to solve the initial value problem for certain classes of nonlinear partial differential equations. The method reduces the initial value problem to ...

The partial differential equation (u_t)/(u_x)=1/4(u_(xxx))/(u_x)-3/8(u_(xx)^2)/(u_x^2)+3/2(p(u))/(u_x^2), where p(u)=1/4(4u^3-g_2u-g_3). The special cases ...

In bispherical coordinates, Laplace's equation becomes (1) Attempt separation of variables by plugging in the trial solution f(u,v,phi)=sqrt(coshv-cosu)U(u)V(v)Psi(psi), (2) ...

Following the work of Fuchs in classifying first-order ordinary differential equations, Painlevé studied second-order ordinary differential equation of the form ...