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Purpose -The purpose of this paper is to apply, for the first time, the authors' newly developed perturbation iteration method to heat transfer problems. The effectiveness of the new method in nonlinear heat transfer problems will be tested. Design/methodology/approach -Nonlinear heat transfer problems are solved by perturbation iteration method. They are also solved by the well-known technique variational iteration method in the literature. Findings -It is found that perturbation iteration solutions converge faster to the numerical solutions. More accurate results can be achieved with this new method for nonlinear heat transfer problems. Research limitations/implications -A few iterations are actually sufficient. Further iterations need symbolic packages to calculate the solutions. Practical implications -This new technique can practically be applied to many heat and flow problems. Originality/value -The new perturbation iteration technique is successfully implemented to nonlinear heat transfer problems. Results show good agreement with the direct numerical simulations and the method performs better than the existing variational iteration method.

Two new perturbation-iteration algorithms for solving differential equations of first order are proposed. Variants of the algorithm are developed depending on the differential order of Taylor series expansions. The iteration algorithms are tested on a number of first order equations. Much better solutions than the regular perturbation solutions are achieved.

The previously developed new perturbation-iteration algorithm has been applied to differential equation systems for the first time. The iteration algorithm for systems is developed first. The algorithm is tested for a single equation, coupled two equations, and coupled three equations. Solutions are compared with those of variational iteration method and numerical solutions, and a good agreement is found. The method can be applied to differential equation systems with success.

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The flow of a non-Newtonian fluid through a porous media in between two parallel plates at different temperatures is considered. The governing momentum equation of third-grade fluid with modified Darcy's law and energy equation have been derived. Approximate analytical solutions of momentum and energy equations are obtained by using perturbation techniques. Constant viscosity, Reynold's model viscosity, and Vogel's model viscosity cases are treated separately. The criteria for validity of approximate solutions are derived. A numerical residual error analysis is performed for the solutions. Within the validity range, analytical and numerical solutions are in good agreement.

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A modified power-law fluid of second grade is considered. The model is a combination of power-law and second grade fluid in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The equations of motion are derived for two dimensional incompressible flows, and from which the boundary layer equations are derived. Symmetries of the boundary layer equations are found by using Lie group theory, and then group classification with respect to power-law index is performed. By using one of the symmetries, namely the scaling symmetry, the partial differential system is transformed into an ordinary differential system, which is numerically integrated under the classical boundary layer conditions. Effects of power-law index and second grade coefficient on the boundary layers are shown and solutions are contrasted with the usual second grade fluid solutions.