Variational Approaches as Fractional Differential Equations Along Theoretical and Numerical Examples

This paper deals along the solutions numerical and presentence estimates As fractional equation as differential class, while the problem nonlinear part admits some distinct hypotheses. In particular, As precise parameter localization, the non-0 solution presentence is recognized requiring the sub linearity nonlinear part at infinity and origin. Furthermore, theoretical and numerical examples of applications are provided.


Introduction
Fractional differential equations (FDEs) are a good tool As modeling of many events in different fields of science and engineering such as chemistry, electrochemistry, electromagnetic, mechanics, electricity, biology, polymer rheology, economics, control theory, regular thermodynamics variation, image and signal processing, aerodynamics, wave propagation, complex medium electrodynamics, biophysics, phenomena of blood flow, damping and visco-elasticity, etc. [5,9,11,12,16,19].
We also cite the papers [6,7,8,17,18] Since systems as fractional was invistigated. As of [17,18], via variation approaches and theory ising critical point the multiple solutions presentence As fractional nonlinear differential equations coupled systems was explored. As of [8], utilizing Principle of Ricceri's Variational, the 1 weak solution presentence As class as fractional differential systems was debated. As of [7], engaging Principle of Ricceri's Variational, the infinite weak solutions numisr presentence As impulsive differential fractional systems class was assured. As of [6], utilizing variation approaches and theory ising critical point, the multiplicity solutions results As an impulsive fractional class as differential systems was explored. In this paper, we are attached in the presentence results and numerical estimates of solutions As the following nonlinear fractional boundary value problem Since ∝∈ (0, 1], > 0, :[0,1] × ℝ → ℝ is an 1 -Caratheodory function. The main result of this paper is the investigate of variational and numerical result As the problem ( ). Also, we direct the reader to [3,4,1] As few associated results at this field.

Preliminaries
At the current part, we focus on many definitions as basic, lemmas, notations, and propositions utilized throughout the current work. For creating appropriate spaces of function and Replace on theory as point of critical for exploring the solutions presentence As the problem ( ) we need the essential following findings and notations that will is utilized in launching our key results.

Solutions as Variational
In this section, we Asmulate our main results. As this we Replace And ≔ √ (∝) .
At this point, we offer an application of Formula 2.2 which will is used later to get problem multiple solutions( ). Thus, from our postulation it is following which ( ̅ ) > 0. Thus, it follows from Formula 2.2 along * = 0 as every >̃, the functional discloses at minimum 1 local minimum 0 ,̃∈ ∝ so ( 0 ,̃) < ̅ , that is just‖ ̅ 0 , ‖ > ̅ 2 . Therefore, the conclusion is gooten.
The result as followe is a straight formula 3.2consequence.