Analytical/Numerical Hybrid Solution for One-Dimensional Ablation Problem

2003 ◽  
Author(s):  
Fa´bio Yukio Kurokawa ◽  
Antonio Joa˜o Diniz ◽  
Joa˜o Batista Campos-Silva
Keyword(s):  
Differential Equations ◽  
Integral Transform ◽  
Thermal Protection ◽  
Linear Partial Differential Equation ◽  
Equation System ◽  
Partial Differential ◽  
One Dimensional ◽  
Plane Plate ◽  
Non Linear ◽  
Non Linear System

Ablation is a thermal protection process with several applications in engineering, mainly in the field of airspace industry. The use of conventional materials must be quite restricted, because they would suffer catastrophic flaws due to thermal degradation of their structures. However, the same materials can be quite suitable once being protected by well-known ablative materials. The process that involves the ablative phenomena is complex, could involve the whole or partial loss of material that is sacrificed for absorption of energy. The analysis of the ablative process in a blunt body with revolution geometry will be made on the stagnation point area that can be simplified as a one-dimensional plane plate problem. In this work the Generalized Integral Transform Technique (GITT) is employed for the solution of the non-linear system of coupled partial differential equations that model the phenomena. The solution of the problem is obtained by transforming the non-linear partial differential equation system to a system of coupled first order ordinary differential equations and then solving it by using well-established numerical routines. The results of interest such as the temperature field, the depth and the rate of removal of the ablative material are presented and compared with those ones available in the open literature.

scholarly journals RADIATION HEAT TRANSFER WITH ABLATION IN A FINITE PLATE

2005 ◽  
Vol 4 (2) ◽  
pp. 190
Author(s):  
F. A. A. Gomes ◽  
J. B. C. Silva ◽  
A. J. Diniz
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Thermal Protection ◽  
Heated Surface ◽  
Phase Changes ◽  
Solution Technique ◽  
Governing Equations ◽  
Radiation Heat ◽  
Partial Differential ◽  
Non Linear System

The phenomenon of ablation is a process of thermal protection with several applications, mainly, in mechanical and aerospace engineering. Ablative thermal protection is applied using special materials (named ablative materials) externally on the surface of a structure in order to isolate it against thermal effects. The ablative phenomenon is a complex process involving phase changes with partial or total loss of the material. So the position of the boundary is initially unknown. The governing equations of the process form a non-linear system of coupled partial differential equations. The one-dimensional analysis of an ablative process on the plate is performed by using the generalized integral transform technique – GITT for solution of the system of governing equations. By application of this solution technique, the system of partial differential equations is transformed into a system of infinite ordinary differential equations that can be solved after the truncation of that system by numerical techniques codes available. The plate of finite thickness at constant properties is subjected to a time-dependent prescribed radiation heat flux at one face, initially with a uniform temperature T0, and insulated on the other face. After an initial heating period, ablation starts at the heated surface through melting and continuous removal of the plate material. The results of interest are the thickness and the loss rate of the ablative material. The obtained results are compared with available results from other solution techniques in the literature.

scholarly journals RADIATION HEAT TRANSFER WITH ABLATION IN A FINITE PLATE

2005 ◽  
Vol 4 (2) ◽  
Author(s):  
F. A. A. Gomes ◽  
J. B. C. Silva ◽  
A. J. Diniz
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Thermal Protection ◽  
Heated Surface ◽  
Phase Changes ◽  
Solution Technique ◽  
Governing Equations ◽  
Radiation Heat ◽  
Partial Differential ◽  
Non Linear System

The phenomenon of ablation is a process of thermal protection with several applications, mainly, in mechanical and aerospace engineering. Ablative thermal protection is applied using special materials (named ablative materials) externally on the surface of a structure in order to isolate it against thermal effects. The ablative phenomenon is a complex process involving phase changes with partial or total loss of the material. So the position of the boundary is initially unknown. The governing equations of the process form a non-linear system of coupled partial differential equations. The one-dimensional analysis of an ablative process on the plate is performed by using the generalized integral transform technique – GITT for solution of the system of governing equations. By application of this solution technique, the system of partial differential equations is transformed into a system of infinite ordinary differential equations that can be solved after the truncation of that system by numerical techniques codes available. The plate of finite thickness at constant properties is subjected to a time-dependent prescribed radiation heat flux at one face, initially with a uniform temperature T0, and insulated on the other face. After an initial heating period, ablation starts at the heated surface through melting and continuous removal of the plate material. The results of interest are the thickness and the loss rate of the ablative material. The obtained results are compared with available results from other solution techniques in the literature.

scholarly journals A solution of linear and non-linear partial differential equation of fractional order

2021 ◽  
Author(s):  
Raju Muneshwar ◽  
K. L. Bondar ◽  
Y. H. Shirole
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Series Solution ◽  
Solution Method ◽  
Linear Partial Differential Equation ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Non Linear ◽  
Time Variables ◽  
Better Than

We know that the solution of partial differential equations by analytical method is better than the solution by approximate or series solution method. In this paper, we discuss the solution of linear and non-linear fractional partial differential equations involving derivatives with respect to time or space variables by converting them into the partial differential equations of integer order. Also we develop an analytical formulation to solve such fractional partial differential equations. Moreover, we discuss the method to solve the fractional partial differential equations in space as well as time variables simultaneously with the help of some examples

scholarly journals Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients

2021 ◽  
Author(s):  
Shohei Nakajima
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Stochastic Partial Differential Equations ◽  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Partial Differential ◽  
Continuity Theorem ◽  
Existence Of Weak Solutions ◽  
One Dimensional

AbstractWe prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.

scholarly journals Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Raheel Kamal ◽  
Kamran ◽  
Gul Rahmat ◽  
Ali Ahmadian ◽  
Noreen Izza Arshad ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Meshless Method ◽  
Inverse Laplace Transform ◽  
Contour Integral ◽  
Partial Differential ◽  
One Dimensional ◽  
The Laplace Transform

AbstractIn this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

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