PENDIDIKAN IPS BERBASIS KEARIFAN LOKAL DENGAN PENDEKATAN ETNOPEDAGOGY DITINJAU DARI PRESPEKTIF KURIKULUM

The personal social issues experienced by students in daily life are important studies in social studies education. The study of the role of the environment is very important in preserving local wisdom that is beginning to be abandoned by the younger generation, especially the students as a generation of gold who will continue the eastafet stick of development. The importance of a learning approach that is relevant to the situation of students, learning in accordance with the real world reality of students with ecological intelligence (Ekopedagogy) by exploring local wisdom that can be developed through social studies with the selection of contextual material, and meaningful for students to develop skills concern for the community, and place empathy as a form of positive attitudes towards environmental conservation based on local wisdom through the ecopedagogy approach. Keywords: social studies, local wisdom, ethnopedagogy

Analysis and Applications of the Optimal Homotopy Asymptotic Method to Nonlinear Initial and Boundary Value Problems

In this thesis, we use the standard optimal homotopy asymptotic method for the solution of nonlinear initial and boundary value problems for ordinary and partial differential equations. The obtained results are compared with the results obtained by application of Adomian Decomposition Method, Variational Iteration Method, Homotopy Analysis Method, Homotopy Perturbation Method, Numerical Methods, Differential Transform Method, Double Optimal Linearization Method etc. The optimal homotopy asymptotic method does not required the assumption of initial guess and linearization like Adomian Decomposition Method, Variational Iteration Method, Homotopy Analysis Method, Homotopy Perturbation Method. The optimal homotopy asymptotic method established better accuracy at low order approximation and its accuracy increases with increase in approximation orders; it uses a flexible auxiliary function that control the convergence of the solution and convergence region can easily be adjusted. Moreover, the procedure of this method is simple, well defined, and can easily be used the recursive relations explicitly. The numerical results obtained by optimal homotopy asymptotic method revealed high accuracy and excellent agreement with the exact solutions. Except from the application of optimal homotopy asymptotic method, we have extended the formulation of optimal homotopy asymptotic method to a generalized system of ordinary and partial differential equations. These extended formulations have been implemented for a system of two, three and five equations. The results obtained from extended formulations are compared with the results of known methods like Adomian Decomposition Method, Variational Iteration Method, Homotopy Analysis Method, Homotopy Perturbation Method etc, which provides that the extended formulation gives good results than other methods. Like the optimal homotopy asymptotic method the extended formulation of optimal homotopy asymptotic method provides better accuracy at low order approximation and the accuracy increases with the increase of approximation orders. Also we have developed a new scheme for differential-difference equations in the optimal homotopy asymptotic method. The implementation of this scheme is almost simple as the optimal homotopy asymptotic method. To shows its effectiveness we have used it to different bench mark problems from literature and compare the results with those obtained by other method. This new scheme is extended for coupled differentialdifference equations and implemented which provides better results than Adomian Decomposition Method, Variational Iteration Method, Homotopy Analysis Method, Homotopy Perturbation Method etc and excellent agreement with the exact solutions. We used well known methods, Method of least square, Galerkin’s Method and Collocation method for finding the convergence control parameters of the auxiliary function. For determination of optimal values of constants we use method of least square and collocation method. We use Mathematica 7 for symbolic computation. Most of the work presented in chapters 2, 3,4,5 and 6 of this thesis has been published in different well reputed international journals and the remaining are submitted for possible publications. The details of published/accepted/submitted are included in the list of publications.