The Analytical Solutions of Some Conformable Fractional Differential Equationsby Using Effcient Methods

Abstract

This study sought to the analytical solution of some conformable fractional differential equations by using efficient methods” by achieving the following two main objectives: First: Highlighting on a newly emerged operator (conformable fractional operator) and also focus on the relevant rules (properties) of this operator which including some new generalizations and comparisons. Second: Providing the general analytical solutions of several conformable linear and non-linear fractional differential equations by using some new effective methods and demonstrate each case with illustrative examples. This study is organized in four chapters: The first chapter studies the basic concepts of some special functions and some important transformations related to the field of fractional calculus. The second and third chapters contain the basic definitions and rules of three fractional operators which are: The Riemann-Liouville, Caputo and conformable fractional operators. The third chapter highlights on the most important factor in achieving our desired results and it focuses on the conformable fractional operator and their rules (properties). Moreover, some desirable rules and formulas are also derived and generalized related to this operator such as the fractional derivatives of inverse of trigonometric functions, conformable fractional Sumudu transform and conformable Natural transform. In the last chapter, we present the general analytical solutions of several conformable linear and non-linear fractional differential equations by using some new and efficient methods and can be classified as in the following parts: First Part: In this part, we study the general exact solutions of the first, second and higher-order conformable fractional differential equations based on the following well-known methods: Fractional undetermined coefficient method, fractional variation of parameter method and Laplace transform method. Second Part: In this part, we list the general solutions of several conformable fractional differential equations which including: The conformable linear Blasius fractional differential equation, conformable Laguerre fractional differential equation, conformable Van Der Pol Oscillator fractional differential equation, conformable Poisson Boltzmann fractional differential equation and conformable Lane-Emden fractional differential equation by using some new efficient methods which are: Power series method, conformable fractional Laplace transform, conformable fractional Sumudu transform, conformable fractional Natural transform and combined A domain fractional decomposition method. Some illustrated examples and special cases are also given and discussed to show our new approaches. Finally, the study has concluded that the all results that related to the conformable fractional operator were fully consistent with the same results as in the ordinary calculus as a special case when α = 1.