PENERAPAN TEORI KONSTRUKTIVIS DALAM PEMBELAJARAN

Constructivism is the basis for thinking of a contextual approach, namely that knowledge is built not a set of facts, concepts, or rules that are ready to be remembered. Students must construct that knowledge and give meaning through real experience. Students need to be accustomed to solving problems, finding something useful for themselves, and struggling with ideas. The teacher will not be able to give all knowledge to students. Students must construct knowledge in their own minds. Knowledge is not static, but is constantly evolving and changing as students construct new experiences that force them to base themselves and modify previous knowledge. Learning must be packaged into the process of constructing knowledge rather than receiving knowledge. In the learning process, students build their own knowledge through active involvement in the learning and teaching process. Students become the center of activities, not teachers. Critical thinking is an attempt by someone to check the truth of information using the availability of evidence, logic, and awareness of bias. Critical thinking skills are the cognitive processes of students in analyzing systematically and specifically the problems faced, distinguishing these problems carefully and thoroughly, as well as identifying and reviewing information to plan problem solving strategies.

Inverse Problems for Some Fractional Differential Equations

Inverse Problems for Some Fractional Differential Equations Fractional Calculus(FC) is about the investigation of arbitrary order derivatives, integrals, special functions and equations involving these operators. This subject has its roots back to late seventeenth century. In recent years scientists and engineers are using it rigorously as it provides an efficient method to model many well known physical phenomenon when compared with their counterpart (integer order calculus). For example, fractional order diffusion/transport equation has been used to explain anomalies in diffusion/transport process which occurs in many physical situations such as transport in heterogenous or porous media. For a physical process scientists are interested in the investigations of causes and effectsofthatphysicalprocess. Theeffectsofanyphysicalprocess(usuallyknown as direct problems) are easier to study then the causes that forces the system to behave in a particular manner. The mathematical models in which we study the causes are termed as inverse problems(IPs). The field of IPs investigates how to convert measurements into information about a physical process. The field of IPs is of great interest as it has many applications just to mention a few are in medical imaging, acoustic, heat conduction, source identification in a stream, shape optimization etc. In this thesis, we have studied time, space as well as space-time fractional differential equations. Through out our research investigation we have used fractional derivatives defined in the sense of Hilfer and Caputo. It is to be noted that Hilfer fractional derivative (HFD) interpolates both Riemann-Liouville(RLFD) and Caputo fractional derivatives(CFD) for particular choices of parameters. For a fourth order time fractional differential equation(TFDE) with nonlocal boundary conditions(knownasSmaraskii-Ionkinboundaryconditions)involvingHilferfractionalderivative(HFD),twoinversesourceproblems(ISPs)areconsidered. ISPof determining a space dependent source term for a TFDE in two space dimensions is also considered. Existence, uniqueness and stability results for the ISPs are proved under certain regularity conditions on the given data. For a multi-term TFDE involving HFDs ISP of recovering a time dependent source term is studied by using Heaviside-Mikusinski’s operational calculus approach. The spectral problem is non-self-adjoint and a bi-orthogonal system of functions(BSFs) is used toconstructtheseriessolutionoftheISPs. Foraspace-timefractionaldifferential equation(STFDE)withDirichletzeroboundaryconditionsalongwithappropriate over-specified conditions two ISPs of recovering space and time dependent sources are considered. In the last research problem of this thesis inverse coefficient problem(ICP)foraspacefractionaldifferentialequation(SFDE)isstudied. Weproved existence, uniqueness and stability results for the solution of the considered IPs by imposing certain regularity conditions on the given datum.