Computational technique for heat and advection-diffusion equations

In the literature, several techniques are implemented to obtain the approximate solution of heat and advection-diffusion equations. However, each method involves certain drawbacks such as high arithmetic computations, lower accuracy in terms of error, and difficult for computer programming. In the present work, the octic B-spline collocation approach is implemented to incorporate the drawback of the other numerical studies in the literature with with high accuracy in terms of error and MATLAB programming is executed to compute the tedious calculation in an easy way toward the improvement of the approximate solution of heat and advection-diffusion equation. The time derivative is discretized by forward difference technique and the Crank-Nicolson scheme is applied for the remaining terms of the advection-diffusion equation. The stability of the scheme is examined and found that the scheme is unconditionally stable. To test the accuracy and efficiency of the scheme, four test problems are computed. A better approximate solution is obtained as compared to existing methods and a good agreement on analytical solutions by the proposed scheme.