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3.6: Finite Length Strings
https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/03%3A_Trigonometric_Fourier_Series/3.06%3A_Finite_Length_Strings
We now return to the physical example of wave propagation in a string.
7.3: The Nonhomogeneous Heat Equation
https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/07%3A_Green's_Functions/7.03%3A_The_Nonhomogeneous_Heat_Equation
Boundary value green’s functions do not only arise in the solution of nonhomogeneous ordinary differential equations. They are also important in arriving at the solution of nonhomogeneous partial diff...Boundary value green’s functions do not only arise in the solution of nonhomogeneous ordinary differential equations. They are also important in arriving at the solution of nonhomogeneous partial differential equations. In this section we will show that this is the case by turning to the nonhomogeneous heat equation.
TitlePage
https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/00%3A_Front_Matter/01%3A_TitlePage
Introduction to Partial Differential Equations Russell Herman University of North Carolina Wilmington
11.3: Hyperbolic Functions
https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/11%3A_A_-_Calculus_Review_-_What_Do_I_Need_to_Know_From_Calculus%3F/11.03%3A_Derivatives
In your calculus classes you have also seen that some relations are represented in parametric form. However, there is at least one other set of elementary functions, which you should already know abou...In your calculus classes you have also seen that some relations are represented in parametric form. However, there is at least one other set of elementary functions, which you should already know about. These are the hyperbolic functions. Such functions are useful in representing hanging cables, unbounded orbits, and special traveling waves called solutions. They also play a role in special and general relativity.
5.7: Problems
https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/05%3A_Non-sinusoidal_Harmonics_and_Special_Functions/5.07%3A_Problems
Use the derivative identities of Bessel functions, (5.5.15)-(5.5.5), and integration by parts to show that $\int x^{3} J_{0}(x) d x=x^{3} J_{1}(x)-2 x^{2} J_{2}(x)+C .\nonumber$ by considering \[\i...Use the derivative identities of Bessel functions, (5.5.15)-(5.5.5), and integration by parts to show that $\int x^{3} J_{0}(x) d x=x^{3} J_{1}(x)-2 x^{2} J_{2}(x)+C .\nonumber$ by considering $\int_{0}^{1}\left[J_{p}(\mu x) \frac{d}{d x}\left(x \frac{d}{d x} J_{p}(\lambda x)\right)-J_{p}(\lambda x) \frac{d}{d x}\left(x \frac{d}{d x} J_{p}(\mu x)\right)\right] d x .\nonumber$
7.9: Problems
https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/07%3A_Green's_Functions/7.09%3A_Problems
$x^{2} y^{\prime \prime}-2 x y^{\prime}+2 y=3 x^{2}-x, \quad y(1)=\pi, \quad y^{\prime}(1)=0$ . Find and use the initial value Green’s function to solve \[x^{2} y^{\prime \prime}+3 x y^{\prime}-15 y=... $x^{2} y^{\prime \prime}-2 x y^{\prime}+2 y=3 x^{2}-x, \quad y(1)=\pi, \quad y^{\prime}(1)=0$ . Find and use the initial value Green’s function to solve $x^{2} y^{\prime \prime}+3 x y^{\prime}-15 y=x^{4} e^{x}, \quad y(1)=1, y^{\prime}(1)=0 .\nonumber$ Find the Green’s function for the homogeneous fixed values on the boundary of the quarter plane $x>0, y>0$ , for Poisson’s equation using the infinite plane Green’s function for Poisson’s equation.
2.8: Problems
https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/02%3A_Second_Order_Partial_Differential_Equations/2.08%3A_Problems
Find and sketch the solution of the problem \[\begin{aligned} u_{tt}&=u_{xx},\quad 0\leq x\leq 1,\quad t> o \\ u(x,0)&=\left\{\begin{array}{rr}0,&0\leq x<\frac{1}{4}, \\ 1,&\frac{1}{4}\leq x\leq\frac{...Find and sketch the solution of the problem $\begin{aligned} u_{tt}&=u_{xx},\quad 0\leq x\leq 1,\quad t> o \\ u(x,0)&=\left\{\begin{array}{rr}0,&0\leq x<\frac{1}{4}, \\ 1,&\frac{1}{4}\leq x\leq\frac{3}{4}, \\ 0,&\frac{3}{4}<x\leq 1,\end{array}\right. \\ u_t(x,0)&=0, \\ u(0,t)&=0,\quad t>0, \\ u(1,t)&=0,\quad t>0,\end{aligned} \nonumber$
2: Second Order Partial Differential Equations
https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/02%3A_Second_Order_Partial_Differential_Equations
"Either mathematics is too big for the human mind or the human mind is more than a machine." - Kurt Gödel (1906-1978) Thumbnail: Visualization of heat transfer in a pump casing, created by solving the..."Either mathematics is too big for the human mind or the human mind is more than a machine." - Kurt Gödel (1906-1978) Thumbnail: Visualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution. (CC BY-SA 3.0; via Wikipedia)
9: Transform Techniques in Physics
https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/09%3A_Transform_Techniques_in_Physics
“There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.”, ~ Nikolai Lobatchevsky (1792-1856) Thumbnail: Graph of a shifted Dirac delta f...“There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.”, ~ Nikolai Lobatchevsky (1792-1856) Thumbnail: Graph of a shifted Dirac delta function (1d) (CC By 4.0; Alexander Fufaev via Universaldenker)
9.1: Introduction
https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/09%3A_Transform_Techniques_in_Physics/9.01%3A_Introduction
The solutions of linear partial differential equations can be found by using the method of separation of variables to reduce solving partial differential equations to solving ordinary differential equ...The solutions of linear partial differential equations can be found by using the method of separation of variables to reduce solving partial differential equations to solving ordinary differential equations. We can also use transform methods to transform the given PDE into ODEs or algebraic equations. Solving these equations, we then construct solutions of the PDE using an inverse transform. We will describe in this chapter how one can use Fourier and Laplace transforms to this effect.
9.3: Exponential Fourier Transform
https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/09%3A_Transform_Techniques_in_Physics/9.03%3A_Exponential_Fourier_Transform
Both the trigonometric and complex exponential Fourier series provide us with representations of a class of functions of finite period in terms of sums over a discrete set of frequencies.

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