| The Benjamin-Bona-Mahony-Burgers equation(BBMB equation)has a wide range of applications in various fields of science and engineering.From the perspective of existing research methods,the primary method to solve such partial differential equations is to discretize.Because the B-spline function is smooth and flexible,it also has the advantages of piecewise interpolation polynomial,such as simple calculation,good stability,guaranteed convergence,easy realization on the computer,and the B-spline collocation method is easy to construct,which can get higher numerical accuracy and has advantages in dealing with complex boundary problems.At present,it is widely used in the numerical solution of partial differential equations.Therefore,this thesis improves the interpolation conditions of cubic B-spline functions at the endpoints,applies them to the numerical solution of integer and fractional order BBMB equations,and uses the Von-Neumann method for stability analysis.To investigate whether the improved cubic B-spline method can better solve the BBMB equation,this thesis numerically solves integer and fractional order BBMB equations.For the integer-order BBMB equation,the Crank-Nicolson scheme is used to discretize it in time,and the improved cubic B-spline method proposed used to discretize it in space.The nonlinear term is linearized using quasi-linearization,and the problem is transformed into solving a system of linear equations.Numerical examples show that the improved cubic B-spline method can quickly and effectively solve integer-order BBMB equations,and the error is controlled within a small range.Then,stability analysis is performed and prove the unconditional stability of the method,demonstrating the effectiveness of the improved cubic B-spline method for solving integer-order BBMB equations.Numerical results show that the improved method has a spatial accuracy of O(h~6).For the fractional order BBMB equation,this thesis uses the Caputo fractional order differential definition to convert the time fractional order differential into an integral form.At the same time,the first-order difference scheme is used for the first-order derivative in time,and the time and space derivatives are discretized using Crank-Nicolson and an improved cubic B-spline method.The problem is transformed into solving a linear equation system problem,and the stability of this method is analyzed,Finally,three numerical examples show that the numerical solution obtained by the method used is in good agreement with the exact solution.The method can effectively solve the fractional order BBMB equation and has a spatial convergence accuracy of O(h~6). |
