George, SanthoshKanagaraj, K.2021-08-192021-08-192020https://idr.nitk.ac.in/handle/123456789/16869This thesis is devoted for obtaining a stable approximate solution for ill-posed operator equation F x = y: In the second Chapter we consider a non-linear illposed equation F x = y; where F is monotone operator defined on a Hilbert space. Our analysis in Chapter 2 is in the setting of a Hilbert scale. In the rest of the thesis, we studied weighted or fractional regularization method for linear ill-posed equation. Precisely, in Chapter 3 we studied fractional Tikhonov regularization method and in Chapters 4 and 5 we studied fractional Lavrentiv regularization method for the linear ill-posed equation A x = y; where A is a positive self-adjoint operator. Numerical examples are provided to show the reliability and effectiveness of our methods.enDepartment of Mathematical and Computational SciencesIll-Posed ProblemRegularization parameterDiscrepancy principleFractional Tikhonov regularization methodMonotone OperatorLavrentiev RegularizationHilbert ScalesAdaptive Parameter Choice StrategyThesis