Abstract
In this study the method which was obtained from a combination of the conformable fractional double Laplace transform method and the Adomian decomposition method has been successfully applied to solve linear and nonlinear singular conformable fractional Boussinesq equations. Two examples are given to illustrate our method. Furthermore, the results show that the proposed method is effective and is easy to use for certain problems in physics and engineering.
Highlights
There are numerous scientific, engineering, and technological processes that can be mathematically modeled by linear and nonlinear Boussinesq equations such as model flows of water in unconfined aquifers
The fractional Boussinesq equations are appropriate for discussing the water propagation through heterogeneous porous media
The solution is obtained by taking the inverse conformable fractional double Laplace transform for Eq (4.5)
Summary
There are numerous scientific, engineering, and technological processes that can be mathematically modeled by linear and nonlinear Boussinesq equations such as model flows of water in unconfined aquifers. In [2] conformable fractional derivative was used to obtain the exact analytical solutions for the time fractional variant Boussinesq equations. The space-time fractional Boussinesq equations in Caputo sense derivatives are studied by using the homotopy perturbation method [12]. The new conformable fractional double Laplace transform decomposition method is recommended for developing the solutions of singular Boussinesq equation. Example 2 ([17]) By applying the conformable double Laplace transform for the function f xα α tβ β. Example 3 The conformable double Laplace transform for the function f xα tβ , xα =H. Proof By using the definition of conformable fractional double Laplace transform, we have tβ–1xα–1 dt dx λα α tβ–1xα–1 dt dx xα tβ , tβ–1xα–1 dt dx.
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