| In this dissertation are several key general perturbation theorems having a base in complex analysis which address initial value problems (IVPs), with second order linear differential equations (DEs) with variable coefficients. The DEs may be regular or singular, with or without turning points. These theorems establish the uniform validity of two zeroth order one-term N + 1-time scale expansions with remainders, {dollar}{lcub}cal R{rcub}sp0{dollar} and {dollar}{lcub}cal R{rcub}sp0sb{lcub}rm N{rcub}{dollar}, of order {dollar}O(epsilon){dollar} and {dollar}O(epsilonsp{lcub}N{rcub}){dollar}, respectively for N = 1,2,3,{dollar}....{dollar} Uniform validity was proven with the introduction of a special EMER norm. With the use of the EMER norm constructed from the time scales, the growth of the remainder in the general case, which includes the case of exponential growing solutions, can be shown to be uniform (parallel to) the growth in the approximate solutions.; The one-term multi-time scale approximation with remainder {dollar}O(epsilon){dollar} is called the Basic Multi-time EMER Approximation (BMEA) expansion or {dollar}Wsp0{dollar}. The one-term multi-time scale approximation with remainder {dollar}O(epsilonsp{lcub}N{rcub}){dollar} is called the Enhanced One-Term EMER Approximation (EOTEA) expansion or {dollar}Wsbsp{lcub}N{rcub}{lcub}0{rcub}. Wsp0{dollar} and {dollar}Wsbsp{lcub}N{rcub}{lcub}0{rcub}{dollar} are simple in structure when compared to the cylindrical functions of the exact solutions, thus, increasing their appeal. By using the EOTEA Approximation, we eliminate the need to create higher order terms to obtain higher orders of accuracy. An auxiliary term in the EOTEA Approximation corrects for the lower order ICs in the Remainder Problem (RP) obtained in a multi-term with remainder approach.; The EMER Technique is a generalization of (1) a technique developed by L. E. Levine and S. L. Tuohy (1978) for the slightly damped harmonic oscillator IVP and (2) the Regular Diminishing Error Approach (RDEA) developed by L. E. Levine and W. C. Obi (1975) for regular IVPs with slowly varying variable coefficients. Initial work in the multi-term-multi-time scale with remainder perturbation area was done by E. L. Reiss (1971) for the slightly damped harmonic oscillator IVP.; Many IVPs in physics, engineering, and applied mathematics have DEs with variable coefficients, turning points, or singularities, and hence are candidates for the EMER Approach. Singular problems, by their very nature, generally preclude the use of direct analytical methods to obtain exact solution and must be estimated by approximation techniques. The EMER Technique fully addresses IVPs that have finite singularities, regular, or have at worst an irregular singular point of finite rank at {dollar}spinfty{dollar}. This approach can be compared to existing techniques for singular or turning point problems, such as the popular WKBJ technique and the work by O'Malley. Additional research will be required to explore the EMER Technique's full potential as a practical approach for use by the scientific community. |
