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Partial Differential Equations and Programmatic Tool of the PDE Toolbox
Published in Leonid Burstein, ® Basics, 2022
div – divergence, differential operator that can be defined as div(F) = ∇·F where F is a vector field with Cartesian components Fx, Fy and Fz;
Vector Formulas and Coordinate Systems
Published in Dikshitulu K. Kalluri, Principles of Electromagnetic Waves and Materials, 2017
The differential operators normally include gradient of a scalar (∇ψ), divergence of a vector (∇·A), curl of a vector (∇ × A), Laplacian of a scalar (∇ 2ψ), and Laplacian of a vector (∇ 2A). These will be shown in rectangular, cylindrical, and spherical coordinates as given below.
Numerical differentiation and integration
Published in Victor A. Bloomfield, Using R for Numerical Analysis in Science and Engineering, 2018
The Laplacian is a differential operator given by the divergence of the gradient of a function, often denoted by ∇2 or Δ. In Cartesian coordinates, the Laplacian is given by the sum of second partial derivatives of the function with respect to x, y, and z.
Nonlinear vibrations of fractional nonlocal viscoelastic nanotube resting on a Kelvin–Voigt foundation
Published in Mechanics of Advanced Materials and Structures, 2022
In the last decades, the concept of the fractional order calculus has been developed for the dynamic analysis of the viscoelastic nanobeams [18, 19]. The most important advantage of the fractional derivative approach in applications is their nonlocal property [20–23]. It is well known that the integer order differential operator is a local operator, but the fractional order differential operator is a nonlocal operator. This means that the next state of the system depends not only upon its current state, but also upon all of its historical states. This is more realistic and it is one reason why the fractional calculus has become more and more popular. A more extensive review of the fractional differential equations can be found in Podlubny’s work [24]. It is noted that the CNTs are embedded in a medium in many of their applications in nanotechnology. Such a system can be reasonably treated as a nanobeam resting on a foundation is presented by Cajic et al. [25].
Elasto-thermodiffusive response in a spherical shell subjected to memory-dependent heat transfer
Published in Waves in Random and Complex Media, 2021
Pallabi Purkait, Abhik Sur, M. Kanoria
Diethelm [24] has developed the Caputo [25] and Caputo and Mainardi [26] derivative to be where and indicates the usual mth order derivative of the function. Differential equations of fractional order have been the focus of many studies due to their frequent appearance in various application in fluid mechanics, viscoelasticity, biology, physics, and engineering. The most important advantage of using fractional differential equations in these and other applications is their non-local property [27–30]. It is well known that the integer order differential operator is a local operator but the fractional order differential operator is non-local [31,32].
Influence of moving heat source on skin tissue in the context of two-temperature memory-dependent heat transport law
Published in Journal of Thermal Stresses, 2020
Abhik Sur, Sudip Mondal, M. Kanoria
The well-known Caputo derivative [35, 36] is defined as follows: where and indicates the usual th order derivative of the function. Differential equations of fractional order have been the focus of many studies due to their frequent appearance in various applications in fluid mechanics, viscoelasticity, biology, physics, and engineering. The most important advantage of using fractional differential equations in these and other applications is their nonlocal property. It is well known that the integer-order differential operator is a local operator but the fractional order differential operator is nonlocal [37].