Question

determine whether the given boundary value problem is self-adjoint. $$ y^{\prime \prime}+y=\lambda y, \quad y(0)-y^{\prime}(1)=0, \quad y^{\prime}(0)-y(1)=0 $$

Short answer

Answer: Yes, the given boundary value problem is self-adjoint.

Step-by-step solution

Rewrite the given differential equation as an operator

Rewrite the given differential equation, \(y'' + y = \lambda y\), as an operator acting on the function \(y(x)\). We can define the operator \(L\) as follows: $$ L[y](x) = y''(x) + y(x) $$ Now the given differential equation can be written as: $$ L[y](x) = \lambda y(x) $$

Obtain the adjoint operator

To find the adjoint operator \(L^*\), integrate by parts twice and rearrange the terms: First, take the complex conjugate of the defining equation of \(L\): $$ L^*[y](x) = y^{* \prime \prime}(x) + y^*(x) $$ Now integrate by parts twice: $$ \int_0^1 (u L^*[y] * v - u L[v] * y^*) dx = 0 $$ After integrating by parts twice and rearranging the terms, we get: $$ \int_0^1 (L[u] * y^* - L^*[u] * v) dx = y'(1)v(1) - y'(0)v(0) - u'(1)y^*(1) + u'(0)y^*(0) $$

Compare the adjoint operator with the given operator

Now, we need to compare the adjoint operator \(L^*\) with the given operator \(L\). If they are the same, then the problem is self-adjoint. Comparing the expressions for the operators: $$ L[y](x) = y''(x) + y(x) $$ $$ L^*[y](x) = y^{* \prime \prime}(x) + y^*(x) $$ It can be observed that \(L^*[y](x) = L[y^*](x)\), which means the adjoint operator is just the complex conjugate of the given operator acting on the complex conjugate of the function. Since both operators are essentially the same, the problem is self-adjoint. However, the boundary conditions given in the problem statement must also be satisfied. For the problem to be self-adjoint, these two conditions should hold: $$ y'(1)v(1) - y'(0)v(0) = 0 $$ and $$ -u'(1)y^*(1) + u'(0)y^*(0) = 0 $$ Considering the boundary conditions given in the problem statement: $$ y(0)-y'(1)=0, \quad y'(0)-y(1)=0 $$ We can see that these satisfy the conditions provided above, and therefore, the problem is indeed self-adjoint.

Key concepts

Differential Equations

Differential equations are mathematical equations that relate a function to its derivatives. In the world of calculus, derivatives represent how a function changes as its input changes. Specifically, a second-order differential equation involves the second derivative, or the rate of change of the rate of change, of a function. The equation given in the problem, \(y'' + y = \lambda y\), is a second-order linear homogeneous differential equation with a spectral parameter \(\lambda\) which is often associated with the concept of eigenvalues. Solutions to such equations describe a wide range of physical phenomena, from the oscillation of a spring to the propagation of sound waves. Understanding the behavior of the solutions requires finding the conditions, namely the boundary conditions, under which the equation behaves in a predictable, well-defined manner.

Adjoining Operator

An adjoining operator, or adjoint, is a concept from functional analysis, which is part of the study of linear operators in a vector space. For a given linear operator \(L\), the adjoint operator \(L^*\) reflects a certain symmetry and is defined such that it satisfies the inner product relation \((L[u], v) = (u, L^*[v])\) for all functions \(u\) and \(v\) that are in the considered space. In simpler terms, the adjoint operator switches the roles of the function and its conjugate in the context of an integral. It is crucial in understanding the behavior of linear operators and is particularly important in the context of self-adjoint boundary value problems, where the adjoint operator must match the original operator when considering the boundary conditions of the problem.

Eigenvalues

In linear algebra, an eigenvalue is a special number that is associated with a linear operator acting on a function. An eigenvalue \(\lambda\) and its corresponding eigenfunction \(y(x)\) satisfy the equation \(L[y](x) = \lambda y(x)\), where \(L\) is a linear operator. This equation means that when the operator \(L\) acts on the eigenfunction, it simply scales it by the factor of the eigenvalue. Eigenvalues play a pivotal role in understanding the long-term behavior of dynamic systems described by differential equations. They are essential in many areas of physics and engineering, especially in problems involving vibrations and resonance.

Sturm-Liouville Theory

Sturm-Liouville theory is a theory of linear differential operators that arise in the context of solving boundary value problems. These problems include a differential equation and additional constraints called boundary conditions. Sturm-Liouville problems have the form \(L[y](x) = \lambda w(x)y(x)\), where \(L\) is a differential operator, \(\lambda\) is the eigenvalue, and \(w(x)\) is a weight function. This theory is fundamental to understanding the qualitative behavior of the solutions of a self-adjoint boundary value problem, as it offers a systematic way to find eigenvalues and eigenfunctions. The boundary conditions play a crucial role in ensuring that the Sturm-Liouville problem is self-adjoint, meaning the original operator \(L\) and its adjoint \(L^*\) coincide under the given boundary conditions and thus, the solutions have desirable properties like orthogonality and completeness.