Cubic B-spline Method For Numerical Solution Of Time Fractional Differential Equations

Time fractional differential equation is an important branch of the research field of differential equations and have a wide range of applications in various types of diffusion systems.However,since time fractional differential equations have fractional derivatives,it is not easy to solve.From the existing research methods,the main method for solving such differential equations is to discretize them and solve numerical solutions.Because the B-spline function is a symmetric single-peak function,it has the characteristics of good smoothness and tight support,while retaining the advantage that the piece-wise interpolation polynomial is simple,stable,convergent and easy to implement on a computer.Moreover,the B-spline collocation method is simple in structure,high in numerical precision,and easy to handle complex boundary problems.It has become one of the most important numerical methods for solving partial differential equations.Therefore,this paper proposes to solve the numerical solution and source term inversion of time fractional differential equations by using cubic B-spline method,and study its convergence and stability for solving such problems.In order to study whether the cubic B-spline method can solve the numerical solution of the time fractional differential equation better,this paper will use the Caputo fractional differential definition to transform the time fractional differential into the integral form,and use the first-order partial derivative of the time dimension adopts first-order difference format to discretized time fractional differential equations.The experimental results show that the cubic B-spline method can solve the time fractional differential equation effectively and quickly,and can control the error to the order of 10^-5.Subsequently,based on the error of the cubic B-spline interpolation method,the validity of the cubic B-spline method for solving the time-fraction differential equations is verified by theoretical proof.By demonstrating that we obtain the convergence order ??t2-?+?x2? of the time fractional differential equations by the cubic B-spline method,the validity of the cubic B-spline method for solving the time-fraction differential equations is further proved.Source term inversion is also an important part of the numerical solution of differential equations.In practical problems,it is usually necessary to estimate the source term,parameter or initial value by observation.Due to the particularity of time fractional differential equation,the current fractional differential is thefunction values of all previous time are related,so it is difficult to solve by using the method of integer differential equations.Therefore,a new functional is proposed in this paper.The equation is squared and integrated in time space to generate a functional,minimizing The source term obtained by the functional is the source term sought.However,it is ill-posed only for the functional partial differential to be equal to zero.Therefore,the regularization method is introduced here,and then the parametric equation is solved by the idea of least squares.The experimental results show that the improved cubic B-spline source term inversion algorithm increases the number of parameters in the source term because of the segmentation feature of the cubic B-spline function,but the algorithm does not need to be iterated,avoiding complicated calculations.The combined effect is better than the polynomial fitting algorithm as a whole.