This paper focuses on the time fractional heat problem with the use of a new fractional derivative. Using Banach?s fixed point theorem and Laplace transforms, we give and prove the integral solution of the problem. In Colombeau?s algebra, The existence and uniqueness of the solution are demonstrated using the Gronwall lemma.

This article is dedicated to establishing the existence and uniqueness of solutions for the following problem: D α x ( t ) = F ( t , x ( t ) ) x ( 0 ) = x 0 , where x 0 is the singular generalized function and F satisfies L ∞ logarithmic type, D α is the Caputo derivative of order m − 1 < α < m with m ∈ N * , which we will confirm to be present in Colombeau algebra. The Gronwall lemma is used in Colombeau’s algebra to establish the main results. To illustrate our theoretical analysis, we ended our work with an example.

This paper focuses on the time fractional wave problem with the use of a new fractional derivative in Colombeau algebra. Using Banach's fixed point theorem and Laplace transforms, we give and prove the integral solution of the problem. In Colombeau's algebra, The existence and uniqueness of the solution are demonstrated using the Gronwall lemma.

This paper is focuses to the existence and uniqueness solution to the following problem: {D(?) ? (t, y) + Af (t, y) = F(t, ? (t, y)) y ? R, t ? 0 f(0, y)= u0(y), D(?)f (0, y)= v0(y)(0.1) where D(?) is the conformable derivation for 1 < ? < 2 which we will prove to be inside Colombeau algebra, u0 and v0 are singular distibution and F provides L? logarithmic type, the operator A is defined in Colombeau?s algebra. Nets of conformable cosine family (??C?) with polynomial development in ? as ? ? 0 are defined for the first time and used for solving irregular fractional problems.