A novel numerical model is developed in this paper to solve the one-dimensional hyperbolic partial differential equations using wave
equation as an example. The proposed numerical scheme was formed by
combining the Eulerian-Lagrangian method of fundamental solutions
(ELMFS) and the D’Alembert solution. The ELMFS based on the diffusion
fundamental solution and the Eulerian-Lagrangian method was a truly
meshless and integral-free numerical method. Moreover, the D’Alembert
formulation was introduced to avoid the difficulty of dealing with the Dirac delta function in the Cauchy problem. According to the D’Alembert
solution, the second-order hyperbolic partial differential equation was
reduced to two first-order hyperbolic partial differential equations which are solved by the ELMFS. The two opposite-direction first-order hyperbolic equations are approximated by two advection-diffusion equations with extremely small diffusion effect. The developed numerical scheme, a purely meshless method, can easily transport the solutions between the Eulerian and Lagrangian coordinates. Furthermore there are some numerical tests for the one-dimensional wave propagation problems. Then the problem of vibrating string in a semi-infinite domain is solved by the proposed numerical schemes. After numerical validations and sensitive tests, it is proven that the ELMFS combining with the D’Alembert solution is a promising meshless numerical solver for second-order hyperbolic partial differential equations.