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In the case of differential equations, boundary conditions cannot be satisfied; in algebraic equations, the possible number of solutions is decreased. Singular perturbation theory is a rich and ongoing area of exploration for mathematicians, physicists, and other researchers. The methods used to tackle problems in this field are many.
In perturbation theory, the Poincaré–Lindstedt method or Lindstedt–Poincaré method is a technique for uniformly approximating periodic solutions to ordinary differential equations, when regular perturbation approaches fail. The method removes secular terms—terms growing without bound—arising in the straightforward application of ...
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. [ 1 ] [ 2 ] A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. [ 3 ]
Methods of matched asymptotic expansions have been developed to find approximate solutions to the Smoluchowski convection–diffusion equation, which is a singularly perturbed second-order differential equation. The problem has been studied particularly in the context of colloid particles in linear flow fields, where the variable is given by ...
In mathematics and physics, multiple-scale analysis (also called the method of multiple scales) comprises techniques used to construct uniformly valid approximations to the solutions of perturbation problems, both for small as well as large values of the independent variables. This is done by introducing fast-scale and slow-scale variables for ...
Second, the HAM is a unified method for the Lyapunov artificial small parameter method, the delta expansion method, the Adomian decomposition method, [4] and the homotopy perturbation method. [5] [6] The greater generality of the method often allows for strong convergence of the solution over larger spatial and parameter domains. Third, the HAM ...
Generalized eigenvalue problems are less widespread but are a key in the study of vibrations. They are useful when we use the Galerkin method or Rayleigh-Ritz method to find approximate solutions of partial differential equations modeling vibrations of structures such as strings and plates; the paper of Courant (1943) [2] is
If a system initially rests at its equilibrium position, from where it is acted upon by a unit-impulse at the instance t=0, i.e., p(t) in the equation above is a Dirac delta function δ(t), () = | = =, then by solving the differential equation one can get a fundamental solution (known as a unit-impulse response function)