Problem 23 - Applied Partial Differential Equations

Question

Solve $$ \begin{gathered} \frac{d^{4} G}{d x^{4}}=\delta\left(x-x_{0}\right) \\ G\left(0, x_{0}\right)=0 \quad G\left(L, x_{0}\right)=0 \\ \frac{d G}{d x}\left(0, x_{0}\right)=0 \quad \frac{d^{2} G}{d x^{2}}\left(L, x_{0}\right)=0 . \end{gathered} $$

Step-by-Step Solution

Answer

The solution is given by summing over two regions and multiplying each by its corresponding Heaviside step function, resulting in the Green's function \(G(x, x_{0}) = G_1(x)H(x_{0}-x) + G_2(x)H(x-x_{0})\).

Step 1: Setting up Differential Equations

First, it is seen that the fourth order differential form has a \(\delta\left(x-x_{0}\right)\) function, which means the problem can be divided into two parts: \(x < x_{0}\) and \(x > x_{0}\). Therefore, two third order homogeneous differential equations are formed: \[G_1'''+k^4 G_1=0, 0 \le x < x_{0}\] and \[G_2'''+k^4 G_2=0, x_{0}

Step 2: Find General Solutions

Next, find the general solutions for \(G_1\) and \(G_2\). The solutions can be written in the form of: \[G_1(x) = A \sin(kx) + B \cos(kx) + C \sinh(kx) + D \cosh(kx)\] \[G_2(x) = E \sin(k(x-L)) + F \cos(k(x-L)) + G \sinh(k(x-L)) + H \cosh(k(x-L))\]

Step 3: Apply Boundary Values

Apply boundary conditions \(G(0, x_{0})=0\), \(G(L, x_{0})=0\), \(\frac{d G}{d x}(0, x_{0})=0\) and \(\frac{d^{2} G}{d x^{2}}(L, x_{0})=0\) to find the constant coefficients.

Step 4: Solve for Coefficients

Solving the set of equations will give all the exact values of the coefficients A, B, C, D, E, F, G and H. The solution contains the Heaviside step function, as the problem involves the Dirac delta function, which is a derivative of the Heaviside step function.

Step 5: Construct Final Solution

Finally, the Green's function can be constructed by adding the solutions from the two regions together: \[G(x, x_{0}) = G_1(x)H(x_{0}-x) + G_2(x)H(x-x_{0})\], where \(H(x)\) is the Heaviside step function.

Key Concepts

Partial Differential Equations

Partial differential equations (PDEs) are mathematical expressions involving functions and their partial derivatives. They are used to formulate problems involving functions of several variables and are fundamental to understanding phenomena in physics and engineering, such as heat conduction, sound vibration, and fluid dynamics.

PDEs can be first order, second order, or higher, depending on the highest order of the derivative involved. To solve a PDE, one must find a function that satisfies the equation. However, because PDEs involve multiple independent variables, solving them often requires using specific methods such as separation of variables, transform methods, or numerical analysis.

In the given exercise, the PDE is a fourth-order equation involving a fourth derivative with respect to one variable, which adds complexity to the problem.

Boundary Value Problems

A boundary value problem is a type of differential equation along with a set of additional constraints called boundary conditions. These problems require a solution that not only satisfies the differential equation but also meets the conditions specified at the boundaries of the domain.

The significance of boundary conditions lies in their ability to define the physical setup of a problem. For instance, when considering the vibrations of a string, the boundary conditions could specify that the ends of the string are fixed. In our example, the Green's function is subject to specific values at the edges of the domain at x=0 and x=L, which influence how the solution behaves and ensure it reflects the physical context of the problem.

Applying boundary conditions is a crucial step as it determines the unique solution to our PDE. Without them, the PDE can have infinitely many solutions, but the boundary conditions limit these to the ones that have physical relevance.

Dirac Delta Function

The Dirac delta function, denoted as \( \delta(x-x_0) \), is not a function in the traditional sense but rather a 'generalized' function or distribution. It is fundamentally linked to the concept of a 'point source' or an 'impulse.' In the context of PDEs, it's used to model a sudden force or source that is applied at a single point.

The delta function has the unique property that it's zero everywhere except at \( x = x_0 \), where it is undefined but can be thought of as infinitely high, with the integral of the delta function over the entire domain being one. This peculiar behavior makes it a powerful tool for representing point sources in physics and engineering, such as a charge in an electric field or a force in mechanics.

In the exercise, the fourth-order differential equation involves a delta function on the right-hand side, which suggests that the problem is designed to find a Green’s function—an essential concept in the study of linear differential equations which effectively dictates the response of a system to a point source.

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Problem 23

Other exercises in this chapter

Problem 22

Solve $$ \frac{d G}{d x}+G=\delta\left(x-x_{0}\right) \quad \text { with } \quad G\left(0, x_{0}\right)=0 . $$ Show that \(G\left(x, x_{0}\right)\) is not symme

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Problem 23

Determine a particular solution of $$ \nabla^{2} u=f(\mathbf{x}) $$ in infinite two-dimensional space if \(f(\mathbf{x})=g(r)\), where \(r=|\mathbf{x}|\) : (a)

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Problem 22

Use the method of multiple images to obtain the Green's function \(G\left(\mathbf{x}, \mathbf{x}_{0}\right)\) \(\nabla^{2} G=\delta\left(\mathbf{x}-\mathbf{x}_{

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Key Concepts

Partial Differential Equations

Partial differential equations (PDEs) are mathematical expressions involving functions and their partial derivatives. They are used to formulate problems involving functions of several variables and are fundamental to understanding phenomena in physics and engineering, such as heat conduction, sound vibration, and fluid dynamics.

PDEs can be first order, second order, or higher, depending on the highest order of the derivative involved. To solve a PDE, one must find a function that satisfies the equation. However, because PDEs involve multiple independent variables, solving them often requires using specific methods such as separation of variables, transform methods, or numerical analysis.

In the given exercise, the PDE is a fourth-order equation involving a fourth derivative with respect to one variable, which adds complexity to the problem.

Boundary Value Problems

A boundary value problem is a type of differential equation along with a set of additional constraints called boundary conditions. These problems require a solution that not only satisfies the differential equation but also meets the conditions specified at the boundaries of the domain.

The significance of boundary conditions lies in their ability to define the physical setup of a problem. For instance, when considering the vibrations of a string, the boundary conditions could specify that the ends of the string are fixed. In our example, the Green's function is subject to specific values at the edges of the domain at x=0 and x=L, which influence how the solution behaves and ensure it reflects the physical context of the problem.

Applying boundary conditions is a crucial step as it determines the unique solution to our PDE. Without them, the PDE can have infinitely many solutions, but the boundary conditions limit these to the ones that have physical relevance.

Dirac Delta Function

The Dirac delta function, denoted as \( \delta(x-x_0) \), is not a function in the traditional sense but rather a 'generalized' function or distribution. It is fundamentally linked to the concept of a 'point source' or an 'impulse.' In the context of PDEs, it's used to model a sudden force or source that is applied at a single point.

The delta function has the unique property that it's zero everywhere except at \( x = x_0 \), where it is undefined but can be thought of as infinitely high, with the integral of the delta function over the entire domain being one. This peculiar behavior makes it a powerful tool for representing point sources in physics and engineering, such as a charge in an electric field or a force in mechanics.

In the exercise, the fourth-order differential equation involves a delta function on the right-hand side, which suggests that the problem is designed to find a Green’s function—an essential concept in the study of linear differential equations which effectively dictates the response of a system to a point source.