Heat-like Equations Research Articles

Application of the Sumudu Variational Iteration Method with Atangana-Baleanu-Caputo Operator for Solving Fractional-Order Heat-Like Equations with Initial Conditions

Fractional calculus techniques are widely utilized across various engineering disciplines and applied sciences. Among these techniques is the Sumudu Variational Iteration Method (SVIM), which has not yet been tested with the Atangana-Baleanu-Caputo fractional derivative in academic literature. This work aims to explore the application of SVIM for solving fractional-order partial differential equations using the Atangana-Baleanu-Caputo derivative. The method integrates the Sumudu transform with the variational iteration method. To demonstrate the effectiveness and validity of SVIM, we apply it to solve one-dimensional (1-D), two-dimensional (2-D), and three-dimensional (3-D) fractional-order heat-like partial differential equations. The results indicate that SVIM is both convergent and efficient for solving these types of fractional partial differential equations.

NUMERICAL SIMULATION OF UNSTEADY SIGNAL TRANSDUCTION DURING THE FORMATION OF INVADOPODIA USING LEVEL SET METHOD

The finger-like protrusions formed by an invasive cancer cell known as invadopodia is actively observed recently because of this formation on the cancer invasion. The signal on the interface which is stimulated upon contact between epidermal growth factor receptor and ligand, is investigated to be the start of invadopodia formation. In this research, a model of invadopodia formation with signal variable is formulated in two dimensions. The plasma membrane is assumed to be free boundary. The signal is in an unsteady state. The equation of signal is represented by heat-like equation with time-dependent boundary condition. Trigonometry types of the boundary condition which are sine and cosine function are tested to identify the most suitable boundary condition represented the free boundary. The plasma membrane is considered as zero level set function. The membrane moves by the velocity of the signal inside the cell. To handle the free boundary problem, the level set method combining features of ghost method, interpolation and extrapolation method is applied to solve the model numerically. Our result shows that the free boundary is moved at different positions and seen to move inward meaning that the boundary has shrunk. Cosine function is discovered to fit the boundary conditions since the boundary solutions are stable across the time. The computation of the signal density profiles displayed highest density on the membrane.

Method of directly defining the inverse mapping for nonlinear fuzzy heat-like partial differential equations

Natural phenomena or physical systems can be described using Partial Differential Equations (PDEs), such as wave equations, heat equations, Poisson’s equation, and so on. Consequently, investigations of PDEs have become one of the key areas of modern mathematical analyses, attracting a lot of attention. Many authors have recently expressed an interest in researching the theoretical framework of fuzzy Initial Value Problems (IVPs). The Method of Directly Defining the inverse Mapping (MDDiM) was effectively employed in this research to obtain the second-order approximate fuzzy solution of heatlike equations in one and two dimensions, and the results were compared with exact solutions. In each illustrated example, all the results achieved using Maple 16 were graphically depicted. This is the first time MDDiM was utilized to solve nonlinear Fuzzy Partial Differential Equations (FPDEs).

Null-controllability for weakly dissipative heat-like equations

Null-controllability for weakly dissipative heat-like equations

Solving the heat equation for a perturbed magnetic Laplacian on the complex plane

A Bargmann-like transform associated with the polyanalytic Intissar–Hermite polynomials is efficiently employed to solve the heat-like equations associated with Dirac type operators by giving their explicit solutions. We also provide the integral representation of the heat equation associated with a magnetic-like Laplacian. The constructed Bargmann transform is shown to define a unitary transform from the configuration space on the real line onto a Segal–Bargmann type space. The latter is shown to be realizable as the null space of a special first order partial differential operator involving both the holomorphic and anti-holomorphic derivatives for which the polyanalytic Intissar–Hermite polynomials constitute an orthogonal basis. A closed expression of its reproducing kernel function is also given.

INTUITIONISTIC FUZZY HEAT-LIKE EQUATIONS

In this paper, the variational iteration method (VIM) are used forfind exact intuitionistic fuzzy solution of the intuitionistic fuzzy heat-like equations with variable coefficients in one and two dimensions. Several examplesare given to show the new theorem of the solution.The results obtained in all cases show the reliability and the efficiency of this methods.

Neumann trace tracking of a constant reference input for 1-D boundary controlled heat-like equations with delay

This paper discusses the Proportional Integral (PI) regulation control of the left Neumann trace of a one-dimensional reaction-diffusion equation with a delayed right Dirichlet boundary control. Specifically, a PI controller is designed based on a finite-dimensional truncated model that captures the unstable dynamics of the original infinite-dimensional system. In this setting, the control input delay is handled by resorting to the Artstein transformation. The stability of the resulting infinite-dimensional system, as well as the tracking of a constant reference signal in the presence of a constant distributed perturbation, is assessed based on the introduction of an adequate Lyapunov function. The theoretical results are illustrated with numerical simulations.

Stabilization of a heat-like equation in rectangle domain

In this paper, we discuss the stabilization of the heat-like equation in rectangle domain with boundary control. We firstly transfer the heat-like equation in 2D into series of heat equations in 1D via eigenfunction expansion, then, by using the normal backstepping transformation, we obtain the existence and estimate for series of kernel functions in transformations. Finally, combined with the states and controls estimate of series of heat equations, we finally obtain the exponential and finite time stabilization of the heat-like equation in 2D.

On numerical solutions for the Caputo-Fabrizio fractional heat-like equation

In this article, Laplace homotopy analysis method in order to solve fractional heat-like equation with variable coefficients, are introduced. Laplace homotopy analysis method, founded on combination of homotopy methods and Laplace transform is used to supply a new analytical approximated solutions of the fractional partial differential equations in case of the Caputo-Fabrizio. The solutions obtained are compared with exact solutions of these equations. Reliability of the method is given with graphical consequens and series solutions. The results show that the method is a powerfull and efficient for solving the fractional heat-like equations with variable coefficients.

Application of ANNs approach for wave-like and heat-like equations

AbstractArtificial neural networks are data processing systems which originate from human brain tissue studies. The remarkable abilities of these networks help us to derive desired results from complicated raw data. In this study, we intend to duplicate an efficient iterative method to the numerical solution of two famous partial differential equations, namely the wave-like and heat-like problems. It should be noted that many physical phenomena such as coupling currents in a flat multi-strand two-layer super conducting cable, non-homogeneous elastic waves in soils and earthquake stresses, are described by initial-boundary value wave and heat partial differential equations with variable coefficients. To the numerical solution of these equations, a combination of the power series method and artificial neural networks approach, is used to seek an appropriate bivariate polynomial solution of the mentioned initial-boundary value problem. Finally, several computer simulations confirmed the theoretical results and demonstrating applicability of the method.

Analytical Approximate Solutions of (n + 1)-Dimensional Fractal Heat-Like and Wave-Like Equations

In this paper, we propose a new type (n + 1)-dimensional reduced differential transform method (RDTM) based on a local fractional derivative (LFD) to solve (n + 1)-dimensional local fractional partial differential equations (PDEs) in Cantor sets. The presented method is named the (n + 1)-dimensional local fractional reduced differential transform method (LFRDTM). First the theories, their proofs and also some basic properties of this procedure are given. To understand the introduced method clearly, we apply it on the (n + 1)-dimensional fractal heat-like equations (HLEs) and wave-like equations (WLEs). The applications show that this new technique is efficient, simply applicable and has powerful effects in (n + 1)-dimensional local fractional problems.

Application of Homotopy Perturbation Sumudu Transform Method for Solving Nonlinear Fractional Heat-like Equations

In this paper, we propose an algorithm to find approximate solutions of the proposed system of the fractional heat-like equations. The proposed algorithm basically illustrates how the two powerful algorithms, the homotopy perturbation method (HPM) and the Sumudu transform method (STM) can be combined and used to get exact solutions of fractional partial differential equations. We also present some examples to illustrate the accuracy and the effectiveness of this algorithm.

Analysis of the New Technique to Solution of Fractional Wave- and Heat-like Equation

Analysis of the New Technique to Solution of Fractional Wave- and Heat-like Equation

An analytical solution for a fractional heat-like equation with variable coefficients

The fractional power series method is used to solve a fractional heat-like equations with variable coefficients. The solution process is elucidated, and the results show that the method is simple but effective.

Greedy controllability of finite dimensional linear systems

We analyse the problem of controllability for parameter dependent linear finite-dimensional systems. The goal is to identify the most distinguished realisations of those parameters so to better describe or approximate the whole range of controls. We adapt recent results on greedy and weak greedy algorithms for parameter dependent PDEs or, more generally, abstract equations in Banach spaces. Our results lead to optimal approximation procedures that, in particular, perform better than simply sampling the parameter-space to compute the controls for each of the parameter values. We apply these results for the approximate control of finite-difference approximations of the heat and the wave equation. The numerical experiments confirm the efficiency of the methods and show that the number of weak-greedy samplings that are required is particularly low when dealing with heat-like equations, because of the intrinsic dissipativity that the model introduces for high frequencies.

Variational iteration method with an auxiliary parameter for solving wave-like and heat-like equations in large domains

The present study is an attempt to obtain solutions of wave-like and heat-like equations with high-dimensional in large domains through the well-known variational iteration method (VIM) which is coupled with an auxiliary parameter. The convergence of the proposed method, namely variational iteration method with an auxiliary parameter, is under investigation. In addition, four examples are provided to elucidate the solution procedure and to optimally choose the auxiliary parameter. Comparing the results of the standard variational iteration method, the auxiliary parameter proves very effective in controlling the convergence region of approximate solution.

Exact solutions of fractional heat-like and wave-like equations with variable coefficients

Purpose– This paper aims to apply fractional variational iteration method using He's polynomials (FVIMHP) to obtain exact solutions for variable-coefficient fractional heat-like and wave-like equations with fractional order initial and boundary conditions.Design/methodology/approach– The approach is based on FVIMHP. The authors choose as some examples to illustrate the validity and the advantages of the method.Findings– The results reveal that the FVIMHP method provides a very effective, convenient and powerful mathematical tool for solving fractional differential equations.Originality/value– The variable-coefficient fractional heat-like and wave-like equations with fractional order initial and boundary conditions are solved first. Illustrative examples are included to demonstrate the validity and applicability of the method.

The Use of Sumudu Transform for Solving Certain Nonlinear Fractional Heat-Like Equations

We make use of the properties of the Sumudu transform to solve nonlinear fractional partial differential equations describing heat-like equation with variable coefficients. The method, namely, homotopy perturbation Sumudu transform method, is the combination of the Sumudu transform and the HPM using He’s polynomials. This method is very powerful, and professional techniques for solving different kinds of linear and nonlinear fractional differential equations arising in different fields of science and engineering.

Approximate analytic solution of fractional heat-like and wave-like equations with variable coefficients using the differential transforms method

This paper uses the differential transform method (DTM) to obtain analytical solutions of fractional heat- and wave-like equations with variable coefficients. The time fractional heat-like and wave-like equations with variable coefficients were obtained by replacing a first-order and a second-order time derivative by a fractional derivative of order . The approach mainly rests on the DTM which is one of the approximate methods. The method can easily be applied to many problems and is capable of reducing the size of computational work. Some examples are presented to show the efficiency and simplicity of the method.

Existence of solutions to some classes of partial fractional differential equations

In this paper we obtain the existence of solutions to some classes of partial fractional differential equations. Applications include the existence of solutions to a fractional heat-like equation.