Question

Solve the Volterra equation φ(t)+∫t(t−x)φ(x)dx = t, for 0 ≤ t ≤ 1
Using the successive approximation. (Note that the answer should be φ(t) = sin(t) )

Answer 1

Answer on Question #41715 – Math – Other

Question.

Solve the Volterra equation $\varphi(t) + \int (t - x)\varphi(x)dx = t$, for $0 \leq t \leq 1$.

Using the successive approximation. Note that the answer should be $\varphi(t) = \sin(t)$.

Solution.

Consider general solution method of Volterra integral equation:

$$\varphi(t) - \lambda \int_{a}^{t} K(t,x)\varphi(x)dx = g(t)$$ $$a \leq t \leq b$$

We find a solution in the form of a series:

$$\varphi(t) = \varphi_{0}(t) + \varphi_{1}(t) \cdot \lambda + \varphi_{2}(t) \cdot \lambda^{2} + \cdots$$ $$\varphi_{0}(t) = g(t); \varphi_{n}(t) = \int_{a}^{t} K(t,x)\varphi_{n-1}(x)dx; n = 1,2,3 \dots$$

Now, let come back to our case:

$$\varphi(t) - \int (x - t)\varphi(x)dx = t$$ $$0 \leq t \leq 1$$

So,

$$g(t) = t; K(t,x) = (x - t); \lambda = 1; a = 0; b = 1$$

Therefore,

$$\varphi_{0}(t) = g(t) = t$$ $$\varphi_{1}(t) = \int_{a}^{t} K(t,x)\varphi_{0}(x)dx = \int_{0}^{t} (x - t)xdx = \int_{0}^{t} (x^{2} - tx)dx = \frac{t^{3}}{3} - \frac{t^{3}}{2} = -\frac{t^{3}}{6} = -\frac{t^{3}}{3!}$$ $$\varphi_{2}(t) = \int_{a}^{t} K(t,x)\varphi_{1}(x)dx = \int_{0}^{t} -(x - t)\frac{x^{3}}{3!}dx = \int_{0}^{t} \frac{tx^{3}}{3!} - \frac{x^{4}}{3!}dx = \frac{t^{5}}{3!}\left(\frac{1}{4} - \frac{1}{5}\right) = \frac{t^{5}}{120} = \frac{t^{5}}{5!}$$ $$\varphi_3(t) = \int_a^t K(t,x) \varphi_2(x) dx = \int_0^t (x - t) \frac{x^5}{5!} x dx = \int_0^t \frac{x^6}{5!} - \frac{tx^5}{5!} x dx = \frac{t^7}{5!} \left(\frac{1}{7} - \frac{1}{6}\right) = -\frac{t^7}{5! \cdot 6 \cdot 7} = -\frac{t^7}{7!}$$

So,

$$\varphi(t) = \varphi_0(t) + \varphi_1(t) \cdot \lambda + \varphi_2(t) \cdot \lambda^2 + \varphi_3(t) \cdot \lambda^3 + \cdots = t - \frac{t^3}{3!} + \frac{t^5}{5!} - \frac{t^7}{7!} + \cdots = \sin(t)$$

Maclaurin series for $\sin(x)$:

$$\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1}$$

So,

$$\varphi(t) = \sin(t)$$

**Answer.**

$$\varphi(t) = \sin(t)$$

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