How to simplify dirac_delta?

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Consider the following code:

def simplify_dirac_delta(linear_combination, t):
    r"""
    Simplify a ``linear combination`` of the form

    a_0(t) + a_1(t)*d_1(t) + a_2(t)*d_2(t) + ... + a_n(t)*d_n(t)

    where each a_i(t) is a symbolic expression depending
    on the symbolic variable ``t`` and each d_i(t) is
    either dirac_delta(t) or diff(dirac_delta(t), t).

    Simplification is done by applying the following 
    substitution rules:

    (Rule 1)   f(t) * diff(dirac_delta(t), t) 
                    --> diff(f(t), t) * dirac_delta(t)
    (Rule 2)   f(t) * dirac_delta(t)
                    --> f(0) * dirac_delta(t)
    """
    # Write the linear combination in the form
    #         c_0(t) + c_1(t) * diff(dirac_delta(t),t)
    # and apply Rule 1 
    var("dd")
    expression = linear_combination.collect(diff(dirac_delta(t),t))
    expr = expression.subs({diff(dirac_delta(t),t): dd})
    c_0 = expr.coefficient(dd,0)
    c_1 = expr.coefficient(dd,1)
    dc_1 = diff(c_1,t)
    expression = c_0 + dc_1*dirac_delta(t)
    # Write now the resulting linear combination in the form
    #         c_0(t) + c_1(t) * dirac_delta(t),t)
    # and apply Rule 2 
    expression = expression.collect(dirac_delta(t))
    expr = expression.subs({dirac_delta(t): dd})
    c_0 = expr.coefficient(dd,0)
    c_1 = expr.coefficient(dd,1)
    return c_0 + c_1.subs({t:0})*dirac_delta(t)

The docstring and comments explain quite explicitly what simplify_dirac_delta does. Let us test this function. To this end, let us consider $$\cos(x)\, \delta(x)+e^{2x}\,\delta'(x)+3\sin(4x)\,\delta'(x)+9e^{3x}.$$ The simplification rules transform this expression into $$\begin{aligned} &\cos(x)\,\delta(x)+2e^{2x}\delta(x)+12\cos(4x)\,\delta(x)+9e^{3x} \\ &\qquad=(1+2+12)\delta(x)+9e^{3x}=15\delta(x)+9e^{3x}. \end{aligned}$$ Now we apply simplify_dirac_delta:

sage: DD = dirac_delta(x)
sage: dDD = diff(dirac_delta(x),x)
sage: expr = cos(x)*DD + exp(2*x)*dDD + 3*sin(4*x)*dDD + 9*exp(3*x)
sage: simplify_dirac_delta(expr,x)
15*dirac_delta(x) + 9*e^(3*x)

We get the expected result. Now, let us consider your example:

sage: var('x,k')
(x, k)
sage: G = heaviside(x)*sin(k*x)/k
sage: expr = simplify(k**2*G + diff(G, x, x)); expr
2*cos(k*x)*dirac_delta(x) + sin(k*x)*diff(dirac_delta(x), x)/k
sage: simplify_dirac_delta(expr,x)
3*dirac_delta(x)

Once again we obtain the expected result.

The above code and the examples can be run in this SageMath Cell

Here is a more generic code, making use of wildcards. It should work in many more cases than just linear combinations. It also makes it easier to add new rules.

def simplify_dirac_delta(E, t):
"""
Apply some rules on expression E as long as it changes. The rules depend on a parameter t.
"""
    w = SR.wild(0)

    rules = {w*dirac_delta(t): lambda f: f.subs({t: 0})*dirac_delta(t),
             w*diff(dirac_delta(t), t): lambda f: diff(f, t)*dirac_delta(t)}

    def apply_rules():
        nonlocal E
        for r in rules:
            for e in E.find(r):
                f = e.match(r)[w]
                E = E.substitute({e: rules[r](f)})
        return E

    while(E!=apply_rules()):
        pass

    return E

it can be used like this:

sage: simplify_dirac_delta(k**2*G + diff(G, x, x), x)
3*dirac_delta(x)

Simplifications rules are given as a dictionary. The function will try to apply these rules as long as the result changes, so you have to make sure that your rules can't loop back (alternatively you can delete the loop, and only apply the rules a fixed number of times).

Unfortunately, diff is not a symbolic function and does not accept wildcards, so I don't know how to match a derivative, and it is necessary to pass x as an argument.