Operational Approach and Solutions of Hyperbolic Heat Conduction Equations
<p>Evolution of the initial <span class="html-italic">δ</span>(<span class="html-italic">x</span>) function as the solution of the Fourier Equation for <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math>, <math display="inline"> <semantics> <mrow> <mi>β</mi> <mo>=</mo> <mi>δ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math>, for the interval of time <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mtext> </mtext> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mrow> </semantics> </math>.</p> "> Figure 2
<p>Solution of the Cattaneo heat equation for <span class="html-italic">μ</span> = 0, α = 1, ε = 1 for the initial function <math display="inline"> <semantics> <mrow> <mi>F</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> </mrow> </msup> <msup> <mrow> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> </mrow> <mn>3</mn> </msup> </mrow> </semantics> </math>.</p> "> Figure 3
<p>Solution of the Cattaneo heat equation for <span class="html-italic">μ</span> = 0, <span class="html-italic">α</span> = 5, <span class="html-italic">ε</span> = 5 for the initial function <math display="inline"> <semantics> <mrow> <mi>F</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> </mrow> </msup> <msup> <mrow> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> </mrow> <mn>3</mn> </msup> </mrow> </semantics> </math>.</p> "> Figure 4
<p>Solution <span class="html-italic">F</span>(<span class="html-italic">x</span>, <span class="html-italic">t</span>) of hyperbolic heat equation with initial <span class="html-italic">δ</span>(<span class="html-italic">x</span>) function for <span class="html-italic">α</span> = 100, <span class="html-italic">ε</span> = 10, <span class="html-italic">κ</span> = 1 in the moments of time <span class="html-italic">t</span> = 0.01 (light blue line), <span class="html-italic">t</span> = 0.1 (lilac line), <span class="html-italic">t</span> = 0.3 (pink line), <span class="html-italic">t</span> = 0.6 (blue line) and <span class="html-italic">t</span> = 1.1 (yellow–green line).</p> "> Figure 5
<p>Schematic diagram of an electric cable line with leakage.</p> "> Figure 6
<p>Space-time distribution of <math display="inline"> <semantics> <mrow> <mi>Re</mi> <mo stretchy="false">[</mo> <mi>u</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">]</mo> </mrow> </semantics> </math> for <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mtext> </mtext> <mn>3</mn> </mrow> </semantics> </math> harmonic <math display="inline"> <semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mi>n</mi> <mi>i</mi> <mi>x</mi> </mrow> </msup> </mrow> </semantics> </math> for <span class="html-italic">R<sub>C</sub></span> = 7, <span class="html-italic">C = L = R<sub>L</sub> =</span> 1.</p> "> Figure 7
<p>Space—time distribution of <math display="inline"> <semantics> <mrow> <mi>Re</mi> <mo stretchy="false">[</mo> <mi>u</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">]</mo> </mrow> </semantics> </math> for <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mtext> </mtext> <mn>5</mn> </mrow> </semantics> </math> harmonic <math display="inline"> <semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mi>n</mi> <mi>i</mi> <mi>x</mi> </mrow> </msup> </mrow> </semantics> </math> for <span class="html-italic">R<sub>C</sub> =</span> 7, <span class="html-italic">C = L = R<sub>L</sub> =</span> 1.</p> "> Figure 8
<p>Solution of the Guyer-Krumhansl equation with initial <span class="html-italic">δ</span>(<span class="html-italic">x</span>) function for <span class="html-italic">α</span> = <span class="html-italic">ε</span> = <span class="html-italic">δ</span> = <span class="html-italic">κ</span> = 1 in the interval of time <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mtext> </mtext> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mrow> </semantics> </math>.</p> "> Figure 9
<p>Heat pulse propagation in the Guyer-Krumhansl model with Knudsen number <span class="html-italic">Kn =</span> 1.</p> "> Figure 10
<p>Heat pulse propagation in the Guyer-Krumhansl (GK) model with Knudsen number <span class="html-italic">Kn =</span> 0.2.</p> "> Figure 11
<p>Solution of GK type equation for heat transport in thin films with Knudsen number <span class="html-italic">Kn</span> = 0.2 for the initial function <math display="inline"> <semantics> <mrow> <mi>F</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> <mo>+</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> </mrow> </msup> </mrow> </semantics> </math>.</p> "> Figure 12
<p>Solution of GK equation with distinct ballistic transport: <span class="html-italic">α</span> = <span class="html-italic">ε</span> = <span class="html-italic">δ</span> = 10, <span class="html-italic">κ</span> = 0 for the initial function <math display="inline"> <semantics> <mrow> <mi>F</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> <mo>+</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> </mrow> </msup> </mrow> </semantics> </math>.</p> "> Figure 13
<p>Solution of GK type Equation: <math display="inline"> <semantics> <mrow> <mi>Re</mi> <mo stretchy="false">[</mo> <mi>F</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">]</mo> </mrow> </semantics> </math> for <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mtext> </mtext> <mi>ε</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mi>κ</mi> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mi>δ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> for <math display="inline"> <semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mn>3</mn> <mi>i</mi> <mi>x</mi> </mrow> </msup> </mrow> </semantics> </math> (<span class="html-italic">n</span> = 3) left, and for <math display="inline"> <semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mn>5</mn> <mi>i</mi> <mi>x</mi> </mrow> </msup> </mrow> </semantics> </math> (<span class="html-italic">n =</span> 5) right.</p> "> Figure 14
<p>The behavior of the 1st harmonic (<span class="html-italic">n</span> = 1) of the GK type Equation (36) solution for <span class="html-italic">Kn</span> = 1.</p> "> Figure 15
<p>The behavior of the 3rd harmonic (<span class="html-italic">n</span> = 3) of the GK type Equation (36) solution for <span class="html-italic">Kn</span> = 1.</p> "> Figure 16
<p>The behavior of the 1st harmonic (<span class="html-italic">n</span> = 1) of the GK type Equation (36) solution for <span class="html-italic">Kn</span> = 0.2.</p> "> Figure 17
<p>The behavior of the 3rd harmonic (<span class="html-italic">n</span> = 3) of the GK type Equation (36) solution for <span class="html-italic">Kn</span> = 0.2.</p> ">
Abstract
:1. Introduction
2. Fourier Heat Equation and Its Operational Solution
3. Propagation of a Heat Surge with Fourier Heat Diffusion Type Equation
4. Operational Solution of the Hyperbolic Heat Conduction Equation
5. Propagation of a Heat Surge with the Hyperbolic Heat Conduction Equation
6. Propagation of a Harmonic Signal with the Telegraph Equation
7. Operational Solution of Guyer-Krumhansl Type Heat Equation and Heat Conduction in Thin Films
8. Propagation of a Heat Surge with the Guyer-Krumhansl Heat Equation
9. Solution of the Guyer-Krumhansl Equation for the Exponential–Polynomial Initial Function
10. Harmonic Solution of Guyer-Krumhansl Equation and Temperature Distribution in Thin Films
11. Conclusions
Conflicts of Interest
References
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Zhukovsky, K. Operational Approach and Solutions of Hyperbolic Heat Conduction Equations. Axioms 2016, 5, 28. https://doi.org/10.3390/axioms5040028
Zhukovsky K. Operational Approach and Solutions of Hyperbolic Heat Conduction Equations. Axioms. 2016; 5(4):28. https://doi.org/10.3390/axioms5040028
Chicago/Turabian StyleZhukovsky, Konstantin. 2016. "Operational Approach and Solutions of Hyperbolic Heat Conduction Equations" Axioms 5, no. 4: 28. https://doi.org/10.3390/axioms5040028
APA StyleZhukovsky, K. (2016). Operational Approach and Solutions of Hyperbolic Heat Conduction Equations. Axioms, 5(4), 28. https://doi.org/10.3390/axioms5040028