Optimal control for a phase field model of melting arising from inductive heating

<abstract><p>Due to its unique performance of high efficiency, fast heating speed and low power consumption, induction heating is widely and commonly used in many applications. In this paper, we study an optimal control problem arising from a metal melting process by using a induction heating method. Metal melting phenomena can be modeled by phase field equations. The aim of optimization is to approximate a desired temperature evolution and melting process. The controlled system is obtained by coupling Maxwell's equations, heat equation and phase field equation. The control variable of the system is the external electric field on the local boundary. The existence and uniqueness of the solution of the controlled system are showed by using Galerkin's method and Leray-Schauder's fixed point theorem. By proving that the control-to-state operator $ P $ is weakly sequentially continuous and Fréchet differentiable, we establish an existence result of optimal control and derive the first-order necessary optimality conditions. This work improves the limitation of the previous control system which only contains heat equation and phase field equation.</p></abstract>

Einstein’s Notational Equation of Electro-Magnetic Field Equation in Rindler spacetime

We find Einstein’s notational equation of the electro-magnetic field equation and the electromagneticfield in Rindler space-time. Because, electromagnetic fields of the accelerated frame include in general relativity theory.

Spherical Solution of Classical Quantum Gravity

In the general relativity theory, using Einstein’s gravity field equation, we discover the spherical solution of the classical quantum gravity. The careful point is that this theory is different from the other quantum theory. This theory is made by the Einstein’s classical field equation.

New Gravity Field Equation is derived by Einstein Field Equation in General Relativity Theory

We found the 4-order curvature term satisfied the co-variant derivative. Einstein gravity fieldequation is consist of 2-order curvature terms. Hence, the 4-order curvature term and 2-order curvature termsmake new gravity field equation. In this point, Einstein’s gravity field equation can be modified by new 4-order curvature term because gravity field equation’s term doesn’t have to be 2-order term. Indeed, Einsteinhimself was like that, 0-order term, the cosmological term. Therefore, our theory is based on legitimate facts.

New Coordinate Vacuum Solution in Cosmological General Theory of Relativity

In the general relativity theory, we discover new vacuum solution by Einstein’s gravity field equation. We investigate the new coordinate in cosmological general theory of relativity (CGTR).

PMBH Theory of Representation of Gravity Field Equation and Solutions, Hawking Radiation in Data General Relativity Theory

In the general relativity theory, we find the representation of the gravity field equation and solutions. We treat the representation of Schwarzschild solution, Reissner-Nodstrom solution, Kerr-Newman solution, Robertson -Walker solution. We found new general relativity theory (we call it Data General Relativity Theory; DGRT). We treat the data of Hawking radiation by Data general relativity theory. This theory has to apply black hole (specially, Primordial Massive Black Hole; PMBH) because black hole(PMBH) is an idealistic structure.

Cosmological General Theory of Relativity

In expanded universe, we found gravity field equation and solution. We found Schwarzschildsoluti on, Kerr-Newman solution in expanded universe. Hence, We found new general relativity theory-Cosmological General Theory of Relativity(CGTR).

Matter coupling in Minimal Massive 3D Gravity and spinor-matter interactions in exterior algebra formalism

In this work, by employing the exterior algebra formalism, we study the matter coupling in Minimal Massive 3D Gravity (MMG) by first considering that the matter Lagrangian is connection-independent and then considering that the matter coupling is connection-dependent. The matter coupling in MMG has been previously investigated in the work \cite{arvanitakis_2} in tensorial notation where the matter Lagrangian is considered to be connection-independent. In the first part of the present paper, we revisit the connection-independent matter coupling by using the language of differential forms. We derive the MMG field equation and construct the related source 2-form. We also obtain the consistency relation within this formalism. Next, we examine the case where the matter Lagrangian is connection-dependent. In particular, we concentrate on the spinor-matter coupling and obtain the MMG field equation by explicitly constructing the source term. We also get the consistency relation that the source term should satisfy in order that spinor-matter coupled MMG equation be consistent.

Relativistic Maxwell Theory and Einstein Field Equation derived from Variational Principle

In this article, I'll be reviewing relativistic mechanisms using the calculus of variation in the classical limit. The variational principle is considered to be one of the most important mechanisms to build a theory. Newton's second law of motion is a consequence of Euler-Lagrange equations which gives the least (or stationary) trajectory of a particle between any two arbitrary points. I'll the use action principle by deriving the relativistic Maxwell's field equation, geodesic equation, and Einstein's field equation.

Linearized General Relativity

A complete theory of weak-field gravity is described: the linearized approximation. This is a form of first-order perturbation theory. The concept of a gauge transformation, as applied to the curvature tensor and the field equation, is explained, and it is shown how to reduce the field equation to a wave equation in the Lorenz gauge (under the linear approximation). Thus a huge variety of gravitational calculations become accessible.