In Problems 1 and 2, solve the heat equation (1) subject to the given conditions. u(0, y, t)=0, \(u(\pi, y, t)=0\) u(x, 0, t)=0, \(u(x, \pi, t)=0\) \(u(x, y, 0)=u_{0}\)
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Appendix I
Derivative and Integral Formulas
Appendix II
Gamma Function
Appendix III
Table of Laplace Transforms
1
Introduction to Differential Equations
1.1
Definitions and Terminology
1.2
Initial-Value Problems
1.3
Differential Equations as Mathematical Models
2
First-Order Differential Equations
2.1
Solution Curves Without a Solution
2.2
Separable Equations
2.3
Linear Equations
2.4
Exact Equations
2.5
Solutions by Substitutions
2.6
A Numerical Method
2.7
Linear Models
2.8
Nonlinear Models
2.9
Modeling with Systems of First-Order DEs
3
Higher-Order Differential Equations
3.1
Theory of Linear Equations
3.10
Greens Functions
3.11
Nonlinear Models
3.12
Solving Systems of Linear Equations
3.2
Reduction of Order
3.3
Homogeneous Linear Equations with Constant Coefficients
3.4
Undetermined Coefficients
3.5
Variation of Parameters
3.6
CauchyEuler Equations
3.7
Nonlinear Equations
3.8
Linear Models: Initial-Value Problems
3.9
Linear Models: Boundary-Value Problems
4
The Laplace Transform
4.1
Definition of the Laplace Transform
4.2
The Inverse Transform and Transforms of Derivatives
4.3
Translation Theorems
4.4
Additional Operational Properties
4.5
The Dirac Delta Function
4.6
Systems of Linear Differential Equations
5
Series Solutions of Linear Differential Equations
5.1
Solutions about Ordinary Points
5.2
Solutions about Singular Points
5.3
Special Functions
6
Numerical Solutions of Ordinary Differential Equations
6.1
Euler Methods and Error Analysis
6.2
RungeKutta Methods
6.3
Multistep Methods
6.4
Higher-Order Equations and Systems
6.5
Second-Order Boundary-Value Problems
7
Vectors
7.1
Vectors in 2-Space
7.2
Vectors in 3-Space
7.3
Dot Product
7.4
Cross Product
7.5
Lines and Planes in 3-Space
7.6
Vector Spaces
7.7
GramSchmidt Orthogonalization Process
8
Matrices
8.1
Matrix Algebra
8.10
Orthogonal Matrices
8.11
Approximation of Eigenvalues
8.12
Diagonalization
8.13
LU-Factorization
8.14
Cryptography
8.15
An Error-Correcting Code
8.16
Method of Least Squares
8.17
Discrete Compartmental Models
8.2
Systems of Linear Algebraic Equations
8.3
Rank of a Matrix
8.4
Determinants
8.5
Properties of Determinants
8.6
Inverse of a Matrix
8.7
Cramers Rule
8.8
The Eigenvalue Problem
8.9
Powers of Matrices
9
Vector Calculus
9.1
Vector Functions
9.10
Double Integrals
9.11
Double Integrals in Polar Coordinates
9.12
Greens Theorem
9.13
Surface Integrals
9.14
Stokes Theorem
9.15
Triple Integrals
9.16
Divergence Theorem
9.17
Change of Variables in Multiple Integrals
9.2
Motion on a Curve
9.3
Curvature and Components of Acceleration
9.4
Partial Derivatives
9.5
Directional Derivative
9.6
Tangent Planes and Normal Lines
9.7
Curl and Divergence
9.8
Line Integrals
9.9
Independence of the Path
10
Systems of Linear First-Order Differential Equations
10.1
Theory of Linear Systems
10.2
Homogeneous Linear Systems
10.3
Solution by Diagonalization
10.4
Nonhomogeneous Linear Systems
10.5
Matrix Exponential
11
Systems of Nonlinear Differential Equations
11.1
Autonomous Systems
11.2
Stability of Linear Systems
11.3
Linearization and Local Stability
11.4
Autonomous Systems as Mathematical Models
11.5
Periodic Solutions, Limit Cycles, and Global Stability
12
Orthogonal Functions and Fourier Series
12.1
Orthogonal Functions
12.2
Fourier Series
12.3
Fourier Cosine and Sine Series
12.4
Complex Fourier Series
12.5
SturmLiouville Problem
12.6
Bessel and Legendre Series
13
Boundary-Value Problems in Rectangular Coordinates
13.1
Separable Partial Differential Equations
13.2
Classical PDEs and Boundary-Value Problems
13.3
Heat Equation
13.4
Wave Equation
13.5
Laplaces Equation
13.6
Nonhomogeneous Boundary-Value Problems
13.7
Orthogonal Series Expansions
13.8
Fourier Series in Two Variables
14
Boundary-Value Problems in Other Coordinate Systems
14.1
Polar Coordinates
14.2
Cylindrical Coordinates
14.3
Spherical Coordinates
15
Integral Transform Method
15.1
Error Function
15.2
Applications of the Laplace Transform
15.3
Fourier Integral
15.4
Fourier Transforms
15.5
Fast Fourier Transform
16
Numerical Solutions of Partial Differential Equations
16.1
Laplaces Equation
16.2
Heat Equation
16.3
Wave Equation
17
Functions of a Complex Variable
17.1
Complex Numbers
17.2
Powers and Roots
17.3
Sets in the Complex Plane
17.4
Functions of a Complex Variable
17.5
CauchyRiemann Equations
17.6
Exponential and Logarithmic Functions
17.7
Trigonometric and Hyperbolic Functions
17.8
Inverse Trigonometric and Hyperbolic Functions
18
Integration in the Complex Plane
18.1
Contour Integrals
18.2
CauchyGoursat Theorem
18.3
Independence of the Path
18.4
Cauchys Integral Formulas
19
Series and Residues
19.1
Sequences and Series
19.2
Taylor Series
19.3
Laurent Series
19.4
Zeros and Poles
19.5
Residues and Residue Theorem
19.6
Evaluation of Real Integrals
20
Conformal Mappings
20.1
Complex Functions as Mappings
20.2
Conformal Mappings
20.3
Linear Fractional Transformations
20.4
SchwarzChristoffel Transformations
20.5
Poisson Integral Formulas
20.6
Applications
Textbook Solutions for Advanced Engineering Mathematics
Chapter 13.8 Problem 5
Question
In Problems 5–7, solve Laplace’s equation
\(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}}=0\)
for the steady-state temperature u(x, y, z) in the rectangular parallelepiped shown in FIGURE 13.8.2.
The top (z = c) of the parallelepiped is kept at temperature f(x, y) and the remaining sides are kept at temperature zero.
Solution
The first step in solving 13.8 problem number 5 trying to solve the problem we have to refer to the textbook question: In Problems 5–7, solve Laplace’s equation\(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}}=0\)for the steady-state temperature u(x, y, z) in the rectangular parallelepiped shown in FIGURE 13.8.2. The top (z = c) of the parallelepiped is kept at temperature f(x, y) and the remaining sides are kept at temperature zero.
From the textbook chapter Fourier Series in Two Variables you will find a few key concepts needed to solve this.
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full solution
full solution
Title
Advanced Engineering Mathematics 6
Author
Dennis G. Zill
ISBN
9781284105902
In 57, solve Laplaces equation 02 u 0x2 1 02 u 0y2 1 02 u 0z 2 5 0 (14) for the
Chapter 13.8 textbook questions
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Chapter 13: Problem 1 Advanced Engineering Mathematics 6
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Chapter 13: Problem 2 Advanced Engineering Mathematics 6
In Problems 1 and 2, solve the heat equation (1) subject to the given conditions. \(\left.\frac{\partial u}{\partial x}\right|_{x=0}=0\), \(\left.\quad \frac{\partial u}{\partial x}\right|_{x=1}=0\) \(\left.\frac{\partial u}{\partial y}\right|_{y=0}=0\), \(\left.\quad \frac{\partial u}{\partial y}\right|_{y=1}=0\) u(x, y, 0)=xy
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Chapter 13: Problem 3 Advanced Engineering Mathematics 6
In Problems 3 and 4, solve the wave equation (2) subject to the given conditions. u(0, y, t)=0, \(u(\pi, y, t)=0\) u(x, 0, t)=0, \(u(x, \pi, t)=0\) \(u(x, y, 0)=x y(x-\pi)(y-\pi)\) \(\left.\frac{\partial u}{\partial t}\right|_{t=0}=0\)
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Chapter 13: Problem 4 Advanced Engineering Mathematics 6
In Problems 3 and 4, solve the wave equation (2) subject to the given conditions. u(0, y, t)=0, u(b, y, t)=0 u(x, 0, t)=0, u(x, c, t)=0 u(x, y, 0)=f(x, y) \(\left.\frac{\partial u}{\partial t}\right|_{t=0}=g(x, y)\)
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Chapter 13: Problem 5 Advanced Engineering Mathematics 6
In Problems 5–7, solve Laplace’s equation \(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}}=0\) for the steady-state temperature u(x, y, z) in the rectangular parallelepiped shown in FIGURE 13.8.2. The top (z = c) of the parallelepiped is kept at temperature f(x, y) and the remaining sides are kept at temperature zero.
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Chapter 13: Problem 6 Advanced Engineering Mathematics 6
In Problems 5–7, solve Laplace’s equation \(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}}=0\) for the steady-state temperature u(x, y, z) in the rectangular parallelepiped shown in FIGURE 13.8.2. The top (z = 0) of the parallelepiped is kept at temperature f(x, y) and the remaining sides are kept at temperature zero.
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Chapter 13: Problem 7 Advanced Engineering Mathematics 6
In Problems 5–7, solve Laplace’s equation \(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}}=0\) for the steady-state temperature u(x, y, z) in the rectangular parallelepiped shown in FIGURE 13.8.2. The parallelepiped is a unit cube (a = b = c = 1) with the top (z = 1) and bottom (z = 0) kept at constant temperatures \(u_{0}\) and \(-u_{0}\), respectively, and the remaining sides kept at temperature zero.
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