Log orthogonal functions: approximation properties and applications
Sheng Chen, Jie Shen
Abstract
Abstract We present two new classes of orthogonal functions, log orthogonal functions and generalized log orthogonal functions, which are constructed by applying a $\log $ mapping to Laguerre polynomials. We develop basic approximation theory for these new orthogonal functions, and apply them to solve several typical fractional differential equations whose solutions exhibit weak singularities. Our error analysis and numerical results show that our methods based on the new orthogonal functions are particularly suitable for functions that have weak singularities at one endpoint and can lead to exponential convergence rate, as opposed to low algebraic rates if usual orthogonal polynomials are used.
Topics & Concepts
MathematicsOrthogonal polynomialsOrthogonal functionsLaguerre polynomialsGravitational singularityRate of convergenceExponential functionApplied mathematicsOrthogonal basisJacobi polynomialsMathematical analysisEngineeringElectrical engineeringPhysicsChannel (broadcasting)Quantum mechanicsFractional Differential Equations SolutionsMathematical functions and polynomialsIterative Methods for Nonlinear Equations