Sentence examples for definition of a solution from inspiring English sources

We first introduce a definition of a solution resembling Stampacchia's definition in the sense of duality and then, using the results of the first part, we prove the existence, uniqueness and regularity of solutions of the problem under mild assumptions on the data.

Now, let us recall the definition of a solution of the fractional boundary value problem (1.1).

We now introduce the definition of a solution to IBVP(1.2).

Now, following [9, 21, 22], let us introduce the definition of a solution of problem (1.1).

Therefore, we give the definition of a solution of system (1.1).

By the definition of a solution of an impulsive stochastic differential equation (ISDE) (see [5]), we find that (x t)) is a solution of system (1.2).

Now, we give the definition of a positive solution and a negative solution of (1.1) and (1.2).

If every (a_{i}(x)) is degenerate on the boundary, by the new definition of a weak solution, the stability of weak solutions can be proved without any boundary value condition.

As it is well known that degenerate and singular equations need not possess classical solutions, we give a precise definition of a weak solution to (1.1)–(1.1).

As it is well known that degenerate equations need not possess classical solutions, we give a precise definition of a weak solution to (1.1).

The above theorem will be proved in Section 3. As is well known that degenerate and singular equations need not possess classical solutions, we give a precise definition of a weak solution to (1.1).