Research Article
Boundary Value Problem of Nonlinear Fractional Differential Equations of Mixed Volterra-Fredohlm Integral Equations in Banach Space
Ala Eddine Taier*,
Ranchao Wu,
Naveed Iqbal
Issue:
Volume 12, Issue 1, February 2024
Pages:
1-8
Received:
6 January 2024
Accepted:
29 January 2024
Published:
28 February 2024
Abstract: This paper is dedicated to study the existence and uniqueness of solutions for a boundary value problem of nonlinear fractional differential equations of mixed Volterra-Fredohlm integral equations in Banach space, the recent researches considered the study of differential equations of mixed Volterra-Fredholm integral equations with classical order and the study of existence and uniqueness of solutions using approched numerical methodes, the objective of this paper is the study of the existence and uniqueness of fractional order of differential equations with mixed Volterra-Fredholm integral equations using fixed point theory. This work have two important results, the first result was the discussion of the existence of solutions using the Krasnoselskii fixed point theorem after transforming the problem into integral equation firstly then into operator problem suitable for the fixed point theory. The second result will be the existence and uniqueness of solution, this result was obtained by the use of Banach fixed point theorem after the same transformation used in the first one. This work give as conclusion that the boundary value problem of nonlinear fractional differential equations of mixed Volterra-Fredholm integral equation has a unique solution in Banach space. Finally, this work was ended with an example to illustrate the results obtained.
Abstract: This paper is dedicated to study the existence and uniqueness of solutions for a boundary value problem of nonlinear fractional differential equations of mixed Volterra-Fredohlm integral equations in Banach space, the recent researches considered the study of differential equations of mixed Volterra-Fredholm integral equations with classical order ...
Show More
Research Article
New Theorems Solving Fifth Degree Polynomial Equation in Complete Forms by Proposing New Five Roots Composed of Radical Expressions
Yassine Larbaoui
Issue:
Volume 12, Issue 1, February 2024
Pages:
9-23
Received:
20 February 2024
Accepted:
6 March 2024
Published:
2 April 2024
Abstract: This paper is presenting new theorems and formulas to solve fifth degree polynomial equation in general forms by proposing five formulary solutions that we can calculate nearly simultaneously. The proposed roots for fifth degree polynomial in this paper are composed of radical expressions distributed in a specific architecture that allows them to neutralize each other during multiplication by eliminating radicality and reducing degrees of terms. The proposed architecture for each solution of quantic polynomial is composed of at least six distributed terms, where each term is a multiplication of a pair of radical expressions, whereas multiplications among proposed terms is allowing the reduction of degrees in order to obtain a quartic form at maximum. Furthermore, the proposed architectures of radical roots in this paper are allowing also to solve polynomial equations with degrees higher than five by reducing their expressions. As a result, this paper is presenting two new theorems to solve quantic equation in general forms, where one theorem is eliminating the expression of fourth degree then reducing the whole fifth degree equation into a simplified form, whereas the other new theorem is solving the quantic equation without eliminating the expression of fourth degree. All proposed theorems in this paper are built on a scaling logic concretized by engineering the structure of solutions then calculating the precise formulas of roots for equations, which allow to calculate all roots nearly simultaneously and forward the used engineering methodology to solve polynomial equations with degrees higher than five.
Abstract: This paper is presenting new theorems and formulas to solve fifth degree polynomial equation in general forms by proposing five formulary solutions that we can calculate nearly simultaneously. The proposed roots for fifth degree polynomial in this paper are composed of radical expressions distributed in a specific architecture that allows them to n...
Show More
