The "Green’s Function" Associated with One- and Two-Dimensional Problems

Green’s Function Associated with one and two dimensional Problem

Sana munir

Department of Mathematics, GC University, Faisalabad.

In mathematics a green’s function is type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions. Green’s functions provide an important tool when we study the boundary value problem. They also have intrinsic value for a mathematician.

Also green’s functions in general are distribution, not necessarily proper function. Green functions are also useful for solving wave equation, diffusion equation and in quantum mechanics, where the green’s function of the Hamiltonian is a key concept, with important links to the concept of density of states. The green’s function as used in physics is usually defined with the opposite sign that is

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This definition does not change significantly any of the properties of the Green’s function in heat conduction we know that the Greens’ function represents that temperature at a field point due to a unit heat source applied at source point. In electro static the green’s function stand for the displacement in the solid due to the application of unit point force.

In this project construction of green’s function in one and two dimension has shown. There are more then one way of constructing greens’ function (if it exist) but the result is always same. Due to this we can say that green’s function for a given linear system is unique.

We start with the brief introduction of the Dirac delta or Dirac’s delta function which is not strictly a function in real sense of functions.

Dirac Delta Function:-

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Green’s Function Associated with one dimensional boundary value problem: Consider the following boundary value problem.

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M is defined by

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From(1)

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Divide by A (x) we will get

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Multiply equation (2) by P(x)

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Consider the self-ad-joint boundary value problem

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Now split the boundary value problem into the following boundary value problems.

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The solution of problem (3) is written as

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is complementary function which satisfy the homogeneous differential equation

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is the particular solution to the inhomogeneous differential Equation

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The general solution of (4) is written as

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