Pointwise bounds for a nonlinear heat conduction model of the human head

Duggan, R. C.; Goodman, Aaron

doi:10.1016/s0092-8240(86)80009-x

Cited by 68 publications

(6 citation statements)

References 3 publications

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“…Proof. By using the expansion (8) and formula (7), one can obtain the following linear algebraic system:…”

Section: Some Properties Of First Kind Chebyshev Polynomials and Thei...mentioning

confidence: 99%

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Numerical Solutions for Singular Lane-Emden Equations Using Shifted Chebyshev Polynomials of the First Kind

Ahmed

2023

Contemp. Math.

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This paper describes an algorithm for obtaining approximate solutions to a variety of well-known Lane-Emden type equations. The algorithm expands the desired solution y(x) ≃ yN(x), in terms of shifted Chebyshev polynomials of first kind such that yN(i)(0) = y(i)(0) (i = 0, 1, ..., N). The derivative values y(j)(0) for j = 2, 3, ..., are computed by using the given differential equation and its initial conditions. This makes approximate solutions more consistent with the exact solutions of given differential equations. The explicit expressions of the expansion coefficients of yN(x) are obtained. The suggested method is much simpler compared to any other method for solving this initial value problem. An excellent agreement between the exact and the approximate solutions is found in the given examples. In addition, the error analysis is presented.

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“…Proof. By using the expansion (8) and formula (7), one can obtain the following linear algebraic system:…”

Section: Some Properties Of First Kind Chebyshev Polynomials and Thei...mentioning

confidence: 99%

“…Case 3 ( g( y) = e −y(x) ): This equation arises in the modeling of heat conduction in human head [7,8],…”

Section: Introductionmentioning

confidence: 99%

Numerical Solutions for Singular Lane-Emden Equations Using Shifted Chebyshev Polynomials of the First Kind

Ahmed

2023

Contemp. Math.

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“…has been initiated in [16]. Roughly speaking in this model the right hand side is the heat production rate per unit volume, u(x) is the absolute temperature, x is the radial distance from the centre, a is the inverse of the average thermal conductivity inside the head and b is a heat exchange coefficient.…”

Section: A Nonlinear Heat Conduction Model Of the Human Headmentioning

confidence: 99%

Chebfun Solutions to a Class of 1D Singular and Nonlinear Boundary Value Problems

Gheorghiu

2022

Computation

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The Chebyshev collocation method implemented in Chebfun is used in order to solve a class of second order one-dimensional singular and genuinely nonlinear boundary value problems. Efforts to solve these problems with conventional ChC have generally failed, and the outcomes obtained by finite differences or finite elements are seldom satisfactory. We try to fix this situation using the new Chebfun programming environment. However, for tough problems, we have to loosen the default Chebfun tolerance in Newton’s solver as the ChC runs into trouble with ill-conditioning of the spectral differentiation matrices. Although in such cases the convergence is not quadratic, the Newton updates decrease monotonically. This fact, along with the decreasing behaviour of Chebyshev coefficients of solutions, suggests that the outcomes are trustworthy, i.e., the collocation method has exponential (geometric) rate of convergence or at least an algebraic rate. We consider first a set of problems that have exact solutions or prime integrals and then another set of benchmark problems that do not possess these properties. Actually, for each test problem carried out we have determined how the Chebfun solution converges, its length, the accuracy of the Newton method and especially how well the numerical results overlap with the analytical ones (existence and uniqueness).

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“…Anderson et al [4] presented pointwise upper and lower bounds for the solutions of above SBVP. Heat Conduction Model in a Human Head: The following boundary value problem arises in the study of the distribution of heat sources in a human head (see [5])…”

Section: Physiologymentioning

confidence: 99%

A Review on a Class of Second Order Nonlinear Singular BVPs

et al. 2020

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Several real-life problems are modeled by nonlinear singular differential equations. In this article, we study a class of nonlinear singular differential equations, explore its various aspects, and provide a detailed literature survey. Nonlinear singular differential equations are not easy to solve and their exact solution does not exist in most cases. Since the exact solution does not exist, it is natural to look for the existence of the analytical solution and numerical solution. In this survey, we focus on both aspects of nonlinear singular boundary value problems (SBVPs) and cover different analytical and numerical techniques which are developed to deal with a class of nonlinear singular differential equations − ( p ( x ) y ′ ( x ) ) ′ = q ( x ) f ( x , y , p y ′ ) for x ∈ ( 0 , b ) , subject to suitable initial and boundary conditions. The monotone iterative technique has also been briefed as it gained a lot of attention during the last two decades and it has been merged with most of the other existing techniques. A list of SBVPs is also provided which will be of great help to researchers working in this area.

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Pointwise bounds for a nonlinear heat conduction model of the human head

Cited by 68 publications

References 3 publications

Numerical Solutions for Singular Lane-Emden Equations Using Shifted Chebyshev Polynomials of the First Kind

Numerical Solutions for Singular Lane-Emden Equations Using Shifted Chebyshev Polynomials of the First Kind

Chebfun Solutions to a Class of 1D Singular and Nonlinear Boundary Value Problems

A Review on a Class of Second Order Nonlinear Singular BVPs

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