She cannot stay with you!" It was gentle Abby who formulated the solution: "Put her back in the bog".
Mohanty et al. [32] and Talwar et al. [33] formulated the solution for one-dimensional parabolic equations with singular coefficients and nonlinear terms by using spline in compression approximations.
More recently, Ding et al. [24, 25] and Rashidinia et al. [26, 27] have formulated the solution of second-order telegraph equations and a nonlinear Klein-Gordon, sine-Gordon equation by using parametric spline, respectively.
Different techniques had been used in formulating the solution for specific planning purposes.
The main advantage of the spectral-collocation method versus analytical ones is the ease of formulating the solution, especially for viscous fluids.
Formulating the solution of the integral differential equation for a harmonic pump pulsation one obtains a normalized displacement of the spring mass as a function of three parameters.
In order to formulate the solution of the above parabolic problem in terms of a semigroup, let us use the same variable v ( x, t ) in identity (6), which satisfies the following parabolic problem: (16).
To formulate the solution of the above problem in terms of a semigroup, we need to define a new function v ( x, t ) = u ( x, t ) + ( ψ 0 − ψ 1 ) x x 0 − ψ 0, x ∈ [ 0, x 0 ], (6).
To formulate the solution of the above problem in terms of semigroup, we need to define a new function v ( x, t ) = u ( x, t ) − ψ 2 x − ψ 0, x ∈ [ 0, 1 ], (17).
To formulate the solution of the above problem in terms of semigroup, we need to define a new function v ( x, t ) = u ( x, t ) + ( ψ 0 − ψ 1 ) x x 0 − ψ 0, x ∈ [ 0, 1 ], (6).
To formulate the solution of the above problem in terms of a semigroup, we need to define a new function v ( x, t ) = u ( x, t ) − ψ 2 x − ψ 0, x ∈ [ 0, x 0 ], (15).
