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Item New perturbation-iteration solutions of singular Emden-Fowler equations(Cambridge Scientific Publishers, 2022) Dolapci I.T.; Pakdemirli M.The new perturbation-iteration algorithm which has been applied previously to first order equations and Bratu type equations is implemented for solving singular Emden-Fowler differential equations. The simplest iteration algorithm is used and it is shown that the series solutions of this method produces compatible solutions with the exact solutions © CSP - Cambridge, UK; I&S - Florida, USA, 2022Item Strategies for treating equations with multiple perturbation parameters(Cambridge Scientific Publishers, 2023) Pakdemirli M.Differential equations having more than one small parameter are considered. One of the widespread methods is to express all the small parameters in terms of one small parameter and construct a perturbation expansion in terms of this single parameter. The other approach is to employ expansions containing several small parameters. Both approaches are discussed on example problems and some specific guidelines to follow are given depending on the nature of the problem. A third option which is rarely employed is also discussed in which one parameter is enough to simplify the equation, the other small parameter(s) are assumed to be not small although they are small and hence a single perturbation expansion is sufficient to construct the solution. Example equations from nonlinear dynamics as well as boundary layer type equations are treated to exploit the ideas. © CSP - Cambridge, UK; I&S - Florida, USA, 2023Item Perturbation substitution method for ordinary differential equations(Cambridge Scientific Publishers, 2025) Pakdemirli M.; Dolapci I.T.A new perturbation method is proposed. In addition to the perturbation series expansion of the dependent variable, the independent variable is also expanded as arbitrary functions. The arbitrary function expansions gives more flexibility in choosing the specific forms of the functions so that secular terms, small-divisor terms, blow-up terms which limit the validity of expansions can be eliminated. The method has the capability to produce a number of solutions ranging from regular perturbation solutions to even exact solutions if available. Several linear and nonlinear ordinary differential equation problems are treated with the new method. A boundary layer type problems is treated as well. The link between the new method and the other perturbation methods are outlined in the examples considered. The advantage of the new method is that it inherits more arbitrariness in the expansions so that many different approximate solutions including the regular solutions and even exact solutions can be constructed. The disadvantage is that the selection of the independent expansion functions is not straightforward and the solutions depend on the specific choices. © CSP - Cambridge, UK