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Numerical solution of Burgers equation with nonlinear damping term has been investigated.We developed new approach based on nonpolynomial cubic tension spline approximation.The proposed approach depends on the parameters involving in tension spline.By choosing suitable values of such parameters the optimal local truncation error of the scheme can be obtained.Convergence analysis of presented method has been discussed in details and we have shown under appropriate condition the method convergence.The method tested on two problems,numerical results have been compared with the exact solution to justify the usefulness and accurate nature of proposed method.