scholarly journals Analytic Solution of the Langevin Differential Equations Dominated by a Multibrot Fractal Set

2021 ◽  
Vol 5 (2) ◽  
pp. 50
Author(s):  
Rabha W. Ibrahim ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equations ◽  
Optimal Solution ◽  
Special Kind ◽  
Analytic Solutions ◽  
Fractal Function ◽  
Fractal Set ◽  
Geometric Function ◽  
Fractal Functions ◽  
Subordination Theory

We present an analytic solvability of a class of Langevin differential equations (LDEs) in the asset of geometric function theory. The analytic solutions of the LDEs are presented by utilizing a special kind of fractal function in a complex domain, linked with the subordination theory. The fractal functions are suggested for the multi-parametric coefficients type motorboat fractal set. We obtain different formulas of fractal analytic solutions of LDEs. Moreover, we determine the maximum value of the fractal coefficients to obtain the optimal solution. Through the subordination inequality, we determined the upper boundary determination of a class of fractal functions holding multibrot function ϑ(z)=1+3κz+z3.

scholarly journals A new analytic solution of complex Langevin differential equations

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Rabha Ibrahim
Keyword(s):  
Differential Equations ◽  
Analytic Solution ◽  
Function Theory ◽  
Special Functions ◽  
Analytic Solutions ◽  
Geometric Function Theory ◽  
Content Type ◽  
Complex Differential Equation ◽  
Geometric Function ◽  
Carathéodory Functions

PurposeIn this study, the authors introduce a solvability of special type of Langevin differential equations (LDEs) in virtue of geometric function theory. The analytic solutions of the LDEs are considered by utilizing the Caratheodory functions joining the subordination concept. A class of Caratheodory functions involving special functions gives the upper bound solution.Design/methodology/approachThe methodology is based on the geometric function theory.FindingsThe authors present a new analytic function for a class of complex LDEs.Originality/valueThe authors introduced a new class of complex differential equation, presented a new technique to indicate the analytic solution and used some special functions.

MODELING OF NONLINEAR VIBRATION PROBLEMS WITH FLUID FLOWS THROUGH PIPELINES

2018 ◽  
pp. 44-47
Author(s):  
F.J. Тurayev
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibration ◽  
Viscoelastic Properties ◽  
Construction Material ◽  
Fluid Flows ◽  
Variable Coefficients ◽  
Algebraic Equations ◽  
Flowing Liquid ◽  
Vibration Problems

In this paper, mathematical model of nonlinear vibration problems with fluid flows through pipelines have been developed. Using the Bubnov–Galerkin method for the boundary conditions, the resulting nonlinear integro-differential equations with partial derivatives are reduced to solving systems of nonlinear ordinary integro-differential equations with both constant and variable coefficients as functions of time.A system of algebraic equations is obtained according to numerical method for the unknowns. The influence of the singularity of heredity kernels on the vibrations of structures possessing viscoelastic properties is numerically investigated.It was found that the determination of the effect of viscoelastic properties of the construction material on vibrations of the pipeline with a flowing liquid requires applying weakly singular hereditary kernels with an Abel type singularity.

Optimal Trajectories of Open-Chain Robot Systems: A New Solution Procedure Without Lagrange Multipliers

1998 ◽  
Vol 120 (1) ◽  
pp. 134-136 ◽  
Author(s):  
Sunil K. Agrawal ◽  
Pana Claewplodtook ◽  
Brian C. Fabien
Keyword(s):  
Differential Equations ◽  
Lagrange Multipliers ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Optimal Solution ◽  
Solution Procedure ◽  
Optimal Trajectories ◽  
Point Boundary ◽  
Open Chain ◽  
Robot Systems

For an n d.o.f. robot system, optimal trajectories using Lagrange multipliers are characterized by 4n first-order nonlinear differential equations with 4n boundary conditions at the two end time. Numerical solution of such two-point boundary value problems with shooting techniques is hard since Lagrange multipliers can not be guessed. In this paper, a new procedure is proposed where the dynamic equations are embedded into the cost functional. It is shown that the optimal solution satisfies n fourth-order differential equations. Due to absence of Lagrange multipliers, the two-point boundary-value problem can be solved efficiently and accurately using classical weighted residual methods.

scholarly journals Determination of Avalanche Dynamics Friction Coefficients from Measured Speeds

1983 ◽  
Vol 4 ◽  
pp. 170-173 ◽  
Author(s):  
D. M. McClung ◽  
P. A. Schaerer
Keyword(s):  
Complex Terrain ◽  
Sliding Friction ◽  
Numerical Solutions ◽  
Analytic Solutions ◽  
Dynamics Model ◽  
Friction Coefficients ◽  
Turbulent Friction ◽  
Constant Friction ◽  
Avalanche Dynamics

An avalanche dynamics model, appropriate for complex terrain, for real avalanche paths was developed by Perla, Cheng and McClung in 1980. The model has two friction terms, one for sliding friction which is independent of speed, and one for turbulent friction which is proportional to V2, where V is the centre-of-mass speed along the incline. By introducing speed maxima for avalanches, along with start and stop reference positions, it is possible to determine the the two constant friction coefficients for the model. When this is done, it is found that speed data often exceed a model speed limit implied by the application of V = 0 at the start and stop positions. This effect is illustrated by analytic solutions of the relevant equations, as well as numerical solutions for actual avalanche paths. Some limitations and properties of the fundamental modelling are outlined and suggestions given for future use of such models.

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