Difference Methods For Time Fractional Telegraph Equation

Telegraph equation are a class of hyperbolic partial differential equation which describes vibrations within objects and how waves are propagated,they has been utilized in many diversified fields such as propagation of electrical signals,heat transfer theory in thermodynamics,random walk theory,the modeling of the reaction diffusion and so on.The fractional equation have recently been considered by many authors,because the use of fractional order telegraph equation for the modelling is more effective than integer order derivatives which can easily explain the hereditary properties and memory.This paper proposes a space-compact finite difference method for the onedimensional time-fractional telegraph equation,and a space-compact ADI difference method for the two-dimensional time fractional telegraph equation.The compact finite difference method is attracting increasing attention because of its low number of grid nodes and high accuracy.The compact ADI difference method is applied to high-dimensional problems,which effectively reduces the calculation time and improves the calculation efficiency.This paper is divided into four chapters.The first chapter is the introduction,which introduces the physical background of the time fractional telegraph equation and domestic and foreign research status.In the second chapter,a compact fourthorder difference scheme in space is constructed for one-dimensional time fractional telegraph equation,the time fractional derivative is approximated by L1 interpolation.Finally,numerical experiment verified the accuracy and efficiency of our schemes.In chapter three,a compact ADI difference scheme is constructed for onedimensional time fractional telegraph equation,also known as alternating direction implicit method.Finally,numerical experiment verified the accuracy and efficiency of our schemes.