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- https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/03%3A_Trigonometric_Fourier_Series/3.06%3A_Finite_Length_StringsWe now return to the physical example of wave propagation in a string.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/00%3A_Front_Matter/01%3A_TitlePageIntroduction to Partial Differential Equations Russell Herman University of North Carolina Wilmington
- https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/09%3A_Transform_Techniques_in_Physics/9.03%3A_Exponential_Fourier_TransformBoth 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.
- 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_DerivativesIn 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.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/05%3A_Non-sinusoidal_Harmonics_and_Special_Functions/5.07%3A_ProblemsUse 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 \[\int_{0}^{1}\left[J_{p}(\mu x) \frac{d}{d x}\left(x \frac{d}{d x} J_{p}(\lambda x)\right)-J_{p}(\lambda x) \frac{d}{d x}\left(x \frac{d}{d x} J_{p}(\mu x)\right)\right] d x .\nonumber \]
- https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/02%3A_Second_Order_Partial_Differential_Equations/2.08%3A_ProblemsFind 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 \[\begin{aligned} u_{tt}&=u_{xx},\quad 0\leq x\leq 1,\quad t> o \\ u(x,0)&=\left\{\begin{array}{rr}0,&0\leq x<\frac{1}{4}, \\ 1,&\frac{1}{4}\leq x\leq\frac{3}{4}, \\ 0,&\frac{3}{4}<x\leq 1,\end{array}\right. \\ u_t(x,0)&=0, \\ u(0,t)&=0,\quad t>0, \\ u(1,t)&=0,\quad t>0,\end{aligned} \nonumber \]
- 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)
- 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)
- https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/09%3A_Transform_Techniques_in_Physics/9.01%3A_IntroductionThe 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.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/02%3A_Second_Order_Partial_Differential_Equations/2.07%3A_dAlemberts_Solution_of_the_Wave_EquationA general solution of the one-dimensional wave equation can be found. This solution was first Jean-Baptiste le Rond d’Alembert (1717- 1783) and is referred to as d’Alembert’s formula. In this section ...A general solution of the one-dimensional wave equation can be found. This solution was first Jean-Baptiste le Rond d’Alembert (1717- 1783) and is referred to as d’Alembert’s formula. In this section we will derive d’Alembert’s formula and then use it to arrive at solutions to the wave equation on infinite, semi-infinite, and finite intervals.
- 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.