## Computational and applied mathematics for engineering analysis |

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adjoint allowable coordinate transformation analytic solution applied approximation arbitrary assume asymptotic series base vectors becomes boundary conditions boundary layer boundary value problem cartesian system characteristic functions characteristic values characteristic vectors Christoffel symbols coefficients complex components Consider the equation constant contour contravariant convergent coordinate system curve defined definition determine dx dy eigenvalues Euler evaluate Example expansion expression Figure Find finite difference finite element Fourier Transform Fredholm equation given Green's function grid initial conditions integral surface interval inverse kernel Laplace Transform let us consider linearly independent matrix method metric tensor node obtain operator order equation ordinary differential equations orthogonal parameter partial differential equation path of steepest perturbation procedure relation satisfy scalar second order self-adjoint solve space steepest descent Substituting symmetric Taylor series Theorem tion variable written Wronskian yields zero