A New Attractive Analytic Approach for Solutions of Linear and Nonlinear Neutral Fractional Pantograph Equations

•In this analysis, a successful attempt was made to introduce an analytical approximate solution to the linear and nonlinear neutral fractional pantograph differential equations.•The fractional derivative is considered in the Caputo sense.•Laplace Transform is used to convert neutral fractional pantograph differential equations to Laplace Space.•A new fractional expansion is proposed in the Laplace space as well as a new form of Taylor's formula is introduced.•A new analytical method we called the Laplace-Residual power series method, is introduced and used to create solutions for the Laplace transform of the linear and nonlinear neutral fractional pantograph differential equations.•The proposed expansion is used to create series solutions for the target equations in the newly introduced method.•The inverse Laplace transform is applied to obtain the solution of the original equations.•Three applications are considered to verify the effectiveness of presented method.•Numerical and graphical results are also provided to clarify the required solutions and to illustrate the efficiency of the new method.•Numerical results and comparisons indicate that the Laplace-Residual power series method is very easy and effective to solve such a class of differential equations and can be used to solve other types of differential equations. In this paper, we present analytical solutions for linear and nonlinear neutral Caputo-fractional pantograph differential equations. An attractive new method we called the Laplace-Residual power series method, is introduced and used to create series solutions for the target equations. This method is an efficient simple technique for finding exact and approximate series solutions to the linear and nonlinear neutral fractional differential equations. In addition, numerical and graphical results are also addressed at different values of α to show the behaviors of the Laplace-Residual power series solutions compared with other methods such as Two-stage order-one Runge-Kutta, one-legθ, variational iterative, Chebyshev polynomials, Laguerre wavelet, Bernoulli wavelet, Boubaker polynomials, Hermit wavelet, Proposed and Pricewise fractional-order Taylor methods. Finally, several examples are also considered and solved based on this method to show that our new approach is simple, accurate, and applicable. Maple software is used to calculate the numerical and symbolic quantities in the paper.