Question

Verify that the given function or functions is a solution of the given partial differential equation. $$ \alpha^{2} u_{x x}=u_{i} ; \quad u_{1}(x, t)=e^{-\alpha^{2} t} \sin x, \quad u_{2}(x, t)=e^{-\alpha^{2} \lambda^{2} t} \sin \lambda x, \quad \lambda \text { a real constant } $$

Short answer

Answer: Yes, both \(u_1(x,t)\) and \(u_2(x,t)\) are solutions of the given partial differential equation.

Step-by-step solution

Differentiate \(u_1(x,t)\) with respect to \(x\) and \(t\)

For the given function \(u_1(x,t) = e^{-\alpha^2 t}\sin{x}\), we need to find its second partial derivative with respect to \(x\) and its first partial derivative with respect to \(t\). Using the Chain Rule, we have: $$ u_{1x}(x,t) = e^{-\alpha^{2} t} \cos{x} $$ and $$ u_{1xx}(x,t) = - e^{-\alpha^{2} t} \sin{x} $$ Now, finding the partial derivative with respect to \(t\), we get: $$ u_{1t}(x,t) = -\alpha^2 e^{-\alpha^{2} t} \sin{x} $$

Check if \(u_1(x,t)\) satisfies the given PDE

Substituting the derivatives of \(u_1(x,t)\) into the partial differential equation \(\alpha^2 u_{xx} = u_t\), we have: $$ \alpha^2 \left(- e^{-\alpha^{2} t} \sin{x}\right) = -\alpha^2 e^{-\alpha^{2} t} \sin{x} $$ Since both sides are equal, \(u_1(x,t)\) satisfies the given PDE.

Differentiate \(u_2(x,t)\) with respect to \(x\) and \(t\)

For the given function \(u_2(x,t) = e^{-\alpha^2 \lambda^2 t}\sin{\lambda x}\), we need to find its second partial derivative with respect to \(x\) and its first partial derivative with respect to \(t\). Let's find these derivatives: $$ u_{2x}(x,t) = e^{-\alpha^{2} \lambda^{2} t} \lambda \cos{\lambda x} $$ and $$ u_{2xx}(x,t) = - e^{-\alpha^{2} \lambda^{2} t} \lambda^2 \sin{\lambda x} $$ Now, finding the partial derivative with respect to \(t\), we get: $$ u_{2t}(x,t) = -\alpha^2 \lambda^2 e^{-\alpha^{2} \lambda^{2} t} \sin{\lambda x} $$

Check if \(u_2(x,t)\) satisfies the given PDE

Substituting the derivatives of \(u_2(x,t)\) into the partial differential equation \(\alpha^2 u_{xx} = u_t\), we have: $$ \alpha^2 \left(- e^{-\alpha^{2} \lambda^{2} t} \lambda^2 \sin{\lambda x}\right) = -\alpha^2 \lambda^2 e^{-\alpha^{2} \lambda^{2} t} \sin{\lambda x} $$ Since both sides are equal, \(u_2(x,t)\) satisfies the given PDE. In conclusion, both \(u_1(x,t)\) and \(u_2(x,t)\) are solutions of the given partial differential equation.

Key concepts

Solution Verification

Understanding solution verification in partial differential equations (PDEs) is crucial for determining whether a proposed solution actually satisfies the equation. In this context, we are checking if the functions \( u_1(x,t) \) and \( u_2(x,t) \) are solutions to the PDE \( \alpha^2 u_{xx} = u_t \). By calculating the necessary derivatives and substituting them back into the PDE, we can confirm if the equation holds true.
Here’s the process step by step:

• Identify the function you will verify.
• Compute the required partial derivatives of the function as specified by the PDE.
• Substitute these derivatives into the PDE.
• Check if both sides of the PDE are equal.

If the equality holds, the function is indeed a solution to the equation.
Verification gives confidence that solutions are correctly addressing the mathematical model described by the PDE, making it a fundamental part of applied mathematics.

Chain Rule

The Chain Rule is a principle in calculus that is essential when dealing with composite functions, particularly in the context of differentiation. In our problem, it is applied to find derivatives of functions like \( u_1(x,t) = e^{-\alpha^2 t} \sin x \) and \( u_2(x,t) = e^{-\alpha^2 \lambda^2 t} \sin \lambda x \).
The chain rule states that to differentiate a composite function, you multiply the derivative of the outer function by the derivative of the inner function. Here’s a simplified step-by-step approach:

• Identify the inner function and the outer function.
• Differentiate the outer function with respect to the inner function.
• Differently the inner function with respect to the variable.
• Multiply these derivatives to find the derivative of the composite function.

The chain rule is indispensable in computing derivatives efficiently and accurately, especially when the situation involves complex expressions, such as those encountered in partial differential equations.

Second Partial Derivative

Partial derivatives represent how a function changes as one of the variables changes, while other variables remain constant. The second partial derivative measures how the first partial derivative itself changes with respect to a variable.
When solving PDEs, the second partial derivative often appears in equations, providing insights into more complex changes in functions. Specifically for \( u_1 \), the second partial derivative with respect to \( x \) is \( u_{1xx} = - e^{-\alpha^2 t} \sin x \), signifying how the slope of the function changes with \( x \).
Here's how to compute it:

• First, find the first partial derivative with respect to the desired variable.
• Then, take the derivative of this first derivative with respect to the same variable.

This process grants deeper insights into the behavior of functions within mathematical models. Such understanding is important when addressing real-world problems using applied mathematics principles.

Applied Mathematics

Applied mathematics is the branch of mathematics that deals with mathematical methods and their application to real-world problems. In verifying solutions to a PDE, we leverage concepts like differentiation and algebra to solve equations that model physical phenomena.
PDEs are equations that involve functions and their partial derivatives, and they are commonly used to describe various phenomena in physics, engineering, and many other fields. Examples include modeling heat distribution over time (heat equation), or wave patterns (wave equation).
Working through these equations requires a strong understanding of:

• Mathematical formulation of problems.
• Techniques for differentiation and integration.
• Logical reasoning to interpret results correctly.

By analyzing solutions through the lens of applied mathematics, one can predict and explain behaviors in complex systems, thereby advancing scientific understanding and technological development.