A nonlocal problem with integral gluing condition for a third‐order loaded equation with parabolic‐hyperbolic operator involving fractional derivatives

2019 ◽  
Vol 43 (6) ◽  
pp. 3716-3726
Author(s):  
Praveen Agarwal ◽  
Obidjon Kh.Abdullaev
Keyword(s):  
Fractional Derivatives ◽  
Nonlocal Problem ◽  
Hyperbolic Operator ◽  
Third Order ◽  
Loaded Equation

Construction of Green’s functional for a third order ordinary differential equation with general nonlocal conditions and variable principal coefficient

2020 ◽  
Vol 27 (4) ◽  
pp. 593-603 ◽  
Author(s):  
Kemal Özen
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Nonlocal Problem ◽  
Nonlocal Conditions ◽  
Adjoint Problem ◽  
Linear Ordinary Differential Equation ◽  
Third Order ◽  
Order Ordinary Differential Equation ◽  
Order Linear ◽  
Theoretical Results

AbstractIn this work, the solvability of a generally nonlocal problem is investigated for a third order linear ordinary differential equation with variable principal coefficient. A novel adjoint problem and Green’s functional are constructed for a completely nonhomogeneous problem. Several illustrative applications for the theoretical results are provided.

scholarly journals Numerical calculating linear vibrations of third order systems involving fractional operators

2012 ◽  
Vol 34 (2) ◽  
pp. 91-99
Author(s):  
Nguyen Van Khang ◽  
Tran Dinh Son ◽  
Bui Thi Thuy
Keyword(s):  
Dynamic Response ◽  
Numerical Method ◽  
Numerical Algorithm ◽  
Fractional Derivatives ◽  
Integration Method ◽  
Fractional Operators ◽  
Dynamic Calculation ◽  
Third Order ◽  
Linear Vibrations ◽  
Definition Of

This paper presents a numerical method for dynamic calculation of third order systems involving fractional operators. Using the Liouville-Rieman's definition of fractional derivatives, a numerical algorithm is developed on base of the well-known Newmark integration method to calculate dynamic response of third order systems. Then, we apply this method to calculate linear vibrations of viscoelastic systems containing fractional derivatives.

Hamiltonian Formulation for Continuous Third-order Systems Using Fractional Derivatives

2021 ◽  
Vol 14 (1) ◽  
pp. 35-47
Keyword(s):  
Variational Principle ◽  
Fractional Derivatives ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Lagrange Equations ◽  
Third Order ◽  
Derivative Operator ◽  
Hamilton Equations ◽  
Liouville Fractional Derivative ◽  
Continuous Field

Abstract: We constructed the Hamiltonian formulation of continuous field systems with third order. A combined Riemann–Liouville fractional derivative operator is defined and a fractional variational principle under this definition is established. The fractional Euler equations and the fractional Hamilton equations are obtained from the fractional variational principle. Besides, it is shown that the Hamilton equations of motion are in agreement with the Euler-Lagrange equations for these systems. We have examined one example to illustrate the formalism. Keywords: Fractional derivatives, Lagrangian formulation, Hamiltonian formulation, Euler-lagrange equations, Third-order lagrangian.

scholarly journals Nonlocal Problem for a Third-Order Equation with Multiple Characteristics with General Boundary Conditions

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 110
Author(s):  
Abdukomil Risbekovich Khashimov ◽  
Dana Smetanová
Keyword(s):  
Quadratic Forms ◽  
Nonlocal Problem ◽  
General Boundary ◽  
Fredholm Integral Equations ◽  
Third Order ◽  
Existence Of A Solution ◽  
Energy Integrals ◽  
Multiple Characteristics ◽  
Uniqueness Of The Solution ◽  
Non Local

The article considers third-order equations with multiple characteristics with general boundary value conditions and non-local initial data. A regular solution to the problem with known methods is constructed here. The uniqueness of the solution to the problem is proved by the method of energy integrals. This uses the theory of non-negative quadratic forms. The existence of a solution to the problem is proved by reducing the problem to Fredholm integral equations of the second kind. In this case, the method of Green’s function and potential is used.

scholarly journals Generalized trapezium-type inequalities in the settings of fractal sets for functions having generalized convexity property

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Zareen A. Khan ◽  
Saima Rashid ◽  
Rehana Ashraf ◽  
Dumitru Baleanu ◽  
Yu-Ming Chu
Keyword(s):  
Fractional Derivatives ◽  
Generalized Convexity ◽  
Recent Literature ◽  
Convexity Property ◽  
Power Mean ◽  
Third Order ◽  
Trapezium Type ◽  
Fractal Sets ◽  
The Third ◽  
Inequality Theory

AbstractIn the paper, we extend some previous results dealing with the Hermite–Hadamard inequalities with fractal sets and several auxiliary results that vary with local fractional derivatives introduced in the recent literature. We provide new generalizations for the third-order differentiability by employing the local fractional technique for functions whose local fractional derivatives in the absolute values are generalized convex and obtain several bounds and new results applicable to convex functions by using the generalized Hölder and power-mean inequalities.As an application, numerous novel cases can be obtained from our outcomes. To ensure the feasibility of the proposed method, we present two examples to verify the method. It should be pointed out that the investigation of our findings in fractal analysis and inequality theory is vital to our perception of the real world since they are more realistic models of natural and man-made phenomena.

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