Laplace transform–homotopy perturbation method with arbitrary initial approximation and residual error cancelation

•A modified Laplace transform homotopy perturbation method (MLT–HPM) is presented.•MLT–HPM improves the accuracy of approximate solutions obtained by other methods.•MLT–HPM is employed to study some cases of nonlinear perturbative problems.•MLT–HPM introduces a suitable initial approximation.•MLT–HPM proposes to cancel the residual error in several points of the interval. This paper presents a modified Laplace transform homotopy perturbation method with finite boundary conditions (MLT–HPM) designed to improve the accuracy of the approximate solutions obtained by LT–HPM and other methods. To this purpose, a suitable initial approximation will be introduced, in addition, the residual error in several points of the interest interval (RECP) will be canceled. In order to prove the efficiency of the proposed method a couple of nonlinear ordinary differential equations with mixed boundary conditions, indeed, difficult to approximate, are proposed. The square residual error (S.R.E) of the proposed solutions will result to be of hundredths and tenths, requiring only a first order approximation of MLT–HPM, unlike LT–HPM, which will require more iterations for the same cases study.