Solving a class of boundary value problems and fractional Boussinesq-like equation with β-derivatives by fractional-order exponential trial functions

Abstract

The nonlinear two-point boundary value problem (TPBVP) is converted to a new form of the problem including integral terms. Then the combination of weak-form integral equation method (WFIEM) and special functions create a robust and applicable numerical scheme to solve the problem. To display the accuracy of our method, some examples are investigated. Also, the fractional Boussinesq-like equation involving the β-derivative has been considered that describes the propagation of small amplitude long capillary-gravity waves on the surface of shallow water.