Boundary Value Problems | Engineering Mathematics for Mechanical Engineering PDF Download

Introduction

A boundary value problem for a given differential equation consists of finding a solution of the given differential equation subject to a given set of boundary conditions. A boundary condition is a prescription some combinations of values of the unknown solution and its derivatives at more than one point. 

Let I = (a, b) ⊆ R be an interval. Let p, q, r : (a, b) → R be continuous functions.
Throughout this chapter we consider the linear second order equation given by 

Boundary Value Problems | Engineering Mathematics for Mechanical EngineeringCorresponding to ODE (5.1), there are four important kinds of (linear) boundary conditions. They are given by 

Boundary Value Problems | Engineering Mathematics for Mechanical EngineeringRemark 5.1 (On periodic boundary condition) If the coefficients of ODE (5.1) are periodic functions with period l = b − a and if φ is a solution of ODE (5.1) (note that this solution exists on R), then ψ defined by ψ(x) = φ(x + l) is also a solution. If φ satisfies the periodic boundary conditions, then ψ(a) = φ(a) and ψ ′ (a) = φ ′ (a). Since solutions to IVP are unique in the present case, it must be that ψ ≡ φ. In other words, φ is a periodic function of period l.
Boundary Value Problems do not behave as nicely as Initial value problems. For, there are BVPs for which solutions do not exist; and even if a solution exists there might be many more. Thus existence and uniqueness generally fail for BVPs. The following example illustrate all the three possibilities.

Adjoint forms, Lagrange identity

In mathematical physics there are many important boundary value problems corresponding to second order equations. In the studies of vibrations of a membrane, vibrations of a structure one has to solve a homogeneous boundary value problem for real frequencies (eigen values). As is wellknown in the case of symmetric matrices that there are only real eigen values and corresponding eigen vectors form a basis for the underlying vector space and thereby all symmetric matrices are diagonalisable. Self-adjoint problems can be thought of as corresponding ODE versions of symmetric matrices, and they play an important role in mathematical physics.

Let us consider the equation 

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering

Integrating zBoundary Value Problems | Engineering Mathematics for Mechanical Engineering[y] by parts from a to x, we have 

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering

If we define the second order operator Boundary Value Problems | Engineering Mathematics for Mechanical Engineering∗ by 

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering

then the equation (5.4) becomes 

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering

The operator Boundary Value Problems | Engineering Mathematics for Mechanical Engineering is called the adjoint operator corresponding to the operator Boundary Value Problems | Engineering Mathematics for Mechanical Engineering. It can be easily verified that adjoint of Boundary Value Problems | Engineering Mathematics for Mechanical Engineering is L itself. If Boundary Value Problems | Engineering Mathematics for Mechanical Engineering and Boundary Value Problems | Engineering Mathematics for Mechanical Engineering are the same, then L is said to be self-adjoint .Thus, the necessary and sufficient condition for Boundary Value Problems | Engineering Mathematics for Mechanical Engineering to be self-adjoint is thatBoundary Value Problems | Engineering Mathematics for Mechanical Engineeringwhich is satisfied if 

Boundary Value Problems | Engineering Mathematics for Mechanical EngineeringThus if L is self-adjoint, we have
Boundary Value Problems | Engineering Mathematics for Mechanical EngineeringA general operator Boundary Value Problems | Engineering Mathematics for Mechanical Engineering may not be self-adjoint but it can always be converted into a self-adjoint by suitably multiplying Boundary Value Problems | Engineering Mathematics for Mechanical Engineering with a function.

Two-point boundary value problem

In this section we are going to set up the notations that we are going to use through out our discussion of BVPs. We consider the linear nonhomogeneous second order in the self-adjoint form described below:

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering

Note: that the boundary conditions are in the most general form, and they include the first three conditions given at the beginning of our discussion on BVPs as special cases.

Let us introduce some nomenclature here
Assume hypothesis (HBVP). A nonhomogeneous boundary value problem consists of solving 

Boundary Value Problems | Engineering Mathematics for Mechanical Engineeringfor given constants ηand η2, and a given continuous function f on the interval [a, b]. 

The associated homogeneous boundary value problem is then given by

Boundary Value Problems | Engineering Mathematics for Mechanical EngineeringLet us list some properties of the solutions for BVP that are consequences of the linearity of the differential operator Boundary Value Problems | Engineering Mathematics for Mechanical Engineering.

Fundamental solutions, Green’s functions

Fundamental solution of an ODE gives rise to a representation (integral) formula for solution of the nonhomogeneous equation. When we want to take care of boundary conditions, we impose boundary conditions on fundamental solutions and get Green’s functions. Thus Green’s functions give rise to a representation formula for solution of the nonhomogeneous BVP. The concepts of a fundamental solution as well as a Green’s function are defined in terms of the homogeneous BVPvassociated to the nonhomogeneous BVP.
Let Q denote the square Q := [a, b] × [a, b] in the xξ-plane. Let us partition Q by the line x = ξ and call the two resulting triangles Q1 and Q2. LetBoundary Value Problems | Engineering Mathematics for Mechanical EngineeringNote that the diagonal x = ξ belongs to both the triangles.  

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering

Fundamental solution: A function γ(x, ξ) defined in Q is called a fundamental solution of the homogeneous differential equation Boundary Value Problems | Engineering Mathematics for Mechanical Engineering[y] = 0 if it has the following properties:
(i) The function γ(x, ξ) is continuous in Q.
(ii) The first and second order partial derivatives w.r.t. variable x of the function γ(x, ξ) exist and continuous up to the boundary on Q1 and Q2.
(iii) Let ξ ∈ [a, b] be fixed. Then γ(x, ξ), considered as a function of x, satisfies Boundary Value Problems | Engineering Mathematics for Mechanical Engineering[γ(., ξ)] = 0 at every point of the interval [a, b], except at ξ.(iv) The first derivate has a jump across the diagonal x = ξ, of magnitude 1/p, i.e.,

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering

Construction of Green’s functions

In this paragraph we are going to construct Green’s functions under the assumption that the homogeneous BVP Boundary Value Problems | Engineering Mathematics for Mechanical Engineeringhas only trivial solution. Let (λ1, λ2) 6= (0, 0) be such that a1λ1 + a2λ2 = 0 and let φ1 be solution of Boundary Value Problems | Engineering Mathematics for Mechanical Engineering[y] = 0 satisfying φ1(a) = λand φ ′ 1 (a) = λ2. Choose another solution φ2 of Boundary Value Problems | Engineering Mathematics for Mechanical Engineering[y] = 0 similarly. This way of choosing φ1 and φ2 make sure that both are non-trivial solutions.
Note that φand φ2 form a fundamental pair of solutions of Boundary Value Problems | Engineering Mathematics for Mechanical Engineering[y] = 0, since we assumed that homogeneous BVP has only trivial solutions.By Lagrange’s identity (5.20), we get d dx p(φ ′ 1φ2 − φ1φ ′ 2 ) = 0.
This implies p(φ ′ 1φ2 − φ1φ ′ 2 ) ≡ c, a constant and non-zero, (5.44)
as a consequence of (φ ′ 1φ2 − φ1φ ′ 2 ) being the wronskian corresponding to a fundamental pair of solutions.
Green’s function is then given by
Boundary Value Problems | Engineering Mathematics for Mechanical EngineeringVerifying that G(x, ξ) has the required properties for it to be a Green’s function is left as an exercise.

Generalised Green’s function

In Section 5.3 we constructed Green’s function for the homogeneous BVP when the latter problem had only trivial solutions. Now we investigate the case where the homogeneous BVP has non-trivial solutions.
Let us look at what happens in the case of linear system of equations of size k in k variables. Let A be a k × k matrix, and b ∈ R k . Suppose that Ax = 0 has non-trivial solutions. In this case we know that Ax = b does not have solution for every b ∈ R k . We also know that Ax = b has a solution if and only if b belongs to orthogonal complement of a certain subspace of R k , i.e., b must satisfy a compatibility condition. Note that, in this case, if Ax = b has one solution, then there are infinitely many solutions.
We are going to see that there exists an analogue of Green’s function, called Generalised Green’s function. We also prove that a situation similar to that of matricial analogy given above occurs here too; and the compatibility condition for the existence of a solution to nonhomogeneous BVP is a kind of Fredholm alternative.
Let us recall the notations Q1 and Q2.Boundary Value Problems | Engineering Mathematics for Mechanical Engineering

Generalised Green’s function: Let φ0 be a solution of Boundary Value Problems | Engineering Mathematics for Mechanical Engineering[y] = 0 with homogeneous boundary conditions, such that Boundary Value Problems | Engineering Mathematics for Mechanical Engineering. A function Γ(x, ξ) defined in Q is called a generalised Green’s function if Γ has the following properties:
(i) The function Γ(x, ξ) is continuous in Q.
(ii) The first and second order partial derivatives w.r.t. variable x of the function Γ(x, ξ) exist and continuous up to the boundary on Q1 and Q2.
(iii) Let ξ ∈ [a, b] be fixed. Then γ(x, ξ), considered as a function of x, satisfies 

Boundary Value Problems | Engineering Mathematics for Mechanical Engineeringat every point of the interval [a, b], except at ξ, satisfying the homogeneous boundary conditions U1[y] = 0 and U2[y] = 0.
(iv) The first derivate has a jump across the diagonal x = ξ, of magnitude 1/p, i.e., 

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering

(v) The function Γ(x, ξ) satisfies the condition

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering

The following result characterises the class of functions f, for which the nonhomogeneous equation Boundary Value Problems | Engineering Mathematics for Mechanical Engineering[y] = f has a solution satisfying homogeneous boundary conditions.

Solved Numericals

Q1. We have ddx(13x2+1dydx)+𝜆(3x2+1):Boundary Value Problems | Engineering Mathematics for Mechanical Engineering,y = 0, y(0) = 0, y(π) = 0 then the corresponding eigen value and eigen function are:
Solution:

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering

Differentiating both sides we get

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering

Then the ODE becomes

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering

Auxillary equation is  

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering

Solution is

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering

Hence solution is

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering


Q2. Consider the differential equation Boundary Value Problems | Engineering Mathematics for Mechanical Engineering.If x = 0 at t = 0 and x = 1 at t = 1, the value of x at t = 2 is: 
Solution: 

  • Let d/dx = D
  • So, the differential equation becomes : (D2 - 3D + 2)x = 0
  • Now, as cannot be zero, the function in the bracket will be zero.

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering

Substituting the value in the solution: Boundary Value Problems | Engineering Mathematics for Mechanical Engineering𝑥=𝑐1𝑒𝐷1𝑡+𝑐2𝑒𝐷2𝑡

𝑥=𝑐1𝑒𝑡+𝑐2𝑒2𝑡Applying the boundary condition: x = 0 at t = 0: we get : Boundary Value Problems | Engineering Mathematics for Mechanical Engineering𝑐1=𝑐2
Applying the second boundary condition: 𝑥=1,𝑡=1:x = t, t = 1  It is obtained that : 𝑐1=1𝑒𝑒2,𝑐2=1𝑒2𝑒

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering 


Q3. For λ ∈ ℝ, consider the boundary value problem

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering

Which of the following statement is true?
Solution:
Given: Given D.E. is 

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering

Concept used: We will put 𝑥=𝑒𝑧x = ezand then transform it in z
Putting 𝑥=𝑒𝑧 we get x = ez𝑥2𝑑2𝑦𝑑𝑥2=𝑑2𝑦𝑑𝑧2𝑑𝑦𝑑𝑧, we get

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering𝑑𝑦𝑑𝑥=𝑑𝑦𝑑𝑧

Now or D.E. reduced to

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering(𝑑2𝑑𝑧2+𝑑𝑑𝑧+λ)𝑦=0
Now, using the notation 𝑑𝑑𝑧=𝐷 Boundary Value Problems | Engineering Mathematics for Mechanical Engineering our equation reduced to
Boundary Value Problems | Engineering Mathematics for Mechanical Engineering
Auxillary equation
m2 + m + λ = 0

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering

Converting it into the function of x by putting Boundary Value Problems | Engineering Mathematics for Mechanical Engineering

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering

Now using initial values

Boundary Value Problems | Engineering Mathematics for Mechanical Engineering
So (i) and (ii) has a trivial solution for λ = 1/4λ=1/4
Except that it has non-trivial independent solution.

The document Boundary Value Problems | Engineering Mathematics for Mechanical Engineering is a part of the Mechanical Engineering Course Engineering Mathematics for Mechanical Engineering.
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FAQs on Boundary Value Problems - Engineering Mathematics for Mechanical Engineering

1. What are adjoint forms and how are they related to boundary value problems in mechanical engineering?
Ans. Adjoint forms are differential equations that are related to the original differential equations of a system. In the context of boundary value problems in mechanical engineering, adjoint forms play a crucial role in finding solutions by providing additional information about the system.
2. How does the Lagrange identity help in solving boundary value problems in mechanical engineering?
Ans. The Lagrange identity is a powerful tool that allows for the manipulation of expressions involving derivatives. In the context of boundary value problems in mechanical engineering, the Lagrange identity can be used to simplify equations and derive important relationships between different variables.
3. What is the significance of fundamental solutions and Green's functions in the context of boundary value problems in mechanical engineering?
Ans. Fundamental solutions and Green's functions are essential in solving boundary value problems in mechanical engineering as they provide a systematic approach to finding solutions. They help in representing the response of a system to external forces and boundary conditions.
4. How are Green's functions constructed and utilized in the solution of boundary value problems in mechanical engineering?
Ans. Green's functions are constructed by solving the adjoint form of the differential equations governing the system. Once the Green's function is obtained, it can be used to find the solution to the original boundary value problem by convolving it with the forcing function.
5. What is a generalised Green's function and how is it different from a regular Green's function in the context of boundary value problems in mechanical engineering?
Ans. A generalised Green's function is a more versatile form of Green's function that can be used to solve a wider range of boundary value problems. It incorporates additional information about the system and can handle more complex boundary conditions compared to a regular Green's function.
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