In this paper, a numerical alogorthm for solving high-dimensional heat conduction equations is proposed. Based on Shifted-Legendre orthonormal polynomial and $\epsilon -$best approximate solution, we extend the algorithm from low-dimensional space to high-dimensional space, and prove the convergence of the algorithm. Compared with other numerical methods, the proposed algorithm has the advantages of easy expansion and high convergence order, and we prove that the algorithm has $\alpha$-Order convergence. The validity and accuracy of this method are verified by some numerical experiments.

In this paper, we discuss a novel numerical algorithm for solving boundary value problems. We introduce an orthonormal basis generated from compressed Legendre polynomials. This basis can avoid Runge phenomenon caused by high-order polynomial approximation. Based on the new basis, a numerical algorithm of two-point boundary value problems is established. The convergence and stability of the method are proved. The whole analysis is also applicable to higher order equations or equations with more complex boundary conditions. Four numerical examples are tested to illustrate the accuracy and efficiency of the algorithm. The results show that our algorithm have higher accuracy for solving linear and nonlinear problems.