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We have undertaken an investigation of a kind of third-order equation called Agaciro equation within the folder of both integer and fractional order derivative. In the way of deriving the general exact solution of this equation, we employed the philosophy of the Green function together with some integral transform operators and special functions including but not limited to the Laplace, Fourier, and Mellin transform. We presented some examples of exact solution of this class of third-order equations for integer and fractional order derivative. It is important to point out that the value of Agaciro equation can be extended to describe assorted phenomenon in sciences.

An important method used in the field of partial and ordinary linear is Green function method. It is important to remember that the construction of the Green function is an art and difficult exercise because, in the way of construction, one must first make sure of the existence and the uniqueness of this function, which is a whole topic in mathematics. Once the uniqueness and the existence are insured, one needs to have a knowledge of methods of solving either partial differential equations or ordinary differential equations, again this is a whole topic on its own in mathematics. Within the scope of fractional calculus, one further needs a clear knowledge of special function and other useful integral transform operators including but not limited to Laplace transform, Fourier transform, and Mellin transform. This method was recently extended to the scope of partial and ordinary differential equations with noninteger order derivative [

In mathematical sciences and related disciplines, a Green function is the desired answer of an inhomogeneous differential equation defined on a domain, with particular preliminary conditions or boundary conditions. By means of the superposition theory, the convolution of a Green function with a subjective function

In this present work we will present the discussion underpinning the construction of Green function of a novel equation called “Agaciro” equation presented as

With the purpose to provide lodgings to readers that are not in the field of fractional calculus, we dedicate this subdivision to the symposium supporting the fundamental principle of the fractional calculus. But we will much stress on the properties of the Caputo fractional derivative since it will be used throughout the remainder of the paper.

The Riemann-Liouville integral gives the orthodox outward appearance of fractional calculus. The theory for intermittent functions consequently including the “boundary condition” of repeating after a period is the Weyl integral. The Riemann-Liouville integral of order

There is one more option for working out the definition of fractional derivation: Caputo established it in 1967 in his paper [

The Laplace transform is an extensively used integral transform in the midst of numerous applications in physics and engineering. The Laplace transform of the function

Another useful property of the Caputo derivative is the following:

We will devote this section to the symposium supporting the construction of the Green function of the nonhomogeneous Agaciro equation (

We will present the general solution of the Agaciro equation when

Then applying the Laplace transform on both sides of (

The general Green function associated with the Agaciro equation is given as

We will in the next subsection present some examples of exact solution of Agaciro equation.

Let us consider the following nonhomogeneous Agaciro equation:

Let us now consider the nonhomogeneous Agaciro equation given as follows:

with initial condition

Let us consider the following nonlinear Agaciro equation:

We present in this section the numerical results of the nonhomogeneous Agaciro equation as function of time and space. The numerical results have been depicted in Figures

Exact solution of Agaciro equation for

Exact solution of Agaciro equation (

Exact solution of Agaciro equation (

Exact solution of Agaciro equation (

We devote this section to the symposium supporting the construction of the Green function for the space-time fractional Agaciro equation (

Consider the following time-space fractional Agaciro equation:

with initial condition

The aim of this work was to investigate a class of partial differential equations within the concept of integer and fractional order derivative. This class of equations is referred to as Agaciro equations. We presented the general solution together with some examples of this equation within the scope of ordinary and fractional derivation. To achieve this, we made use of the so-called Green function method together with some well-known integral transform operators.

The authors declare that there is no conflict of interests regarding the publication of this paper.