Question

Obtain the solution of each of the Volterra integral equations (a) \(y(t)=t+\int_{0}^{t} \cos (t-\tau) y(\tau) \mathrm{d} \tau\). (b) \(y(t)=e^{t}-\int_{0}^{t}(t-\tau)^{2} y(\tau) \mathrm{d} \tau\).

Short answer

Question: Provide an approximate expression for the unknown functions in the following Volterra integral equations using the successive approximations method: (a) \(y(t) = t + \int_{0}^{t} \cos(t-\tau)y(\tau)d\tau\) (b) \(y(t) = e^t - \int_{0}^{t}(t-\tau)^2y(\tau)d\tau\) Answer: For (a): $$y(t) \approx t+t\sin{t}-t\cos{t}$$ For (b): $$y(t) \approx e^t - \frac{4}{3}te^{t}+\frac{1}{3}t^2e^t + \frac{2}{3}e^t -\frac{1}{3}t^3$$

Step-by-step solution

(Step 1: Initialize the approximation function)

We'll initialize the approximation function for both cases (a) and (b) as follows: $$y_0(\tau) = f(t)$$

(Step 2: Find the first approximation function)

Substitute the initial approximation function into the Volterra integral equation to get the first approximation: For (a): $$y_1(t) = t+\int_{0}^{t} \cos(t-\tau)y_0(\tau)d\tau = t+\int_{0}^{t} \cos(t-\tau)\cdot t\, d\tau$$ For (b): $$y_1(t) = e^t - \int_{0}^{t}(t-\tau)^{2}y_0(\tau)d\tau = e^t-\int_{0}^{t}(t-\tau)^{2} e^{\tau} d\tau$$

(Step 3: Calculate the first approximation for both cases)

Calculate the integral in each case: For (a): $$y_1(t) = t+\int_{0}^{t} \cos(t-\tau)\cdot t\, d\tau = t+t\sin{t}-t\cos{t}$$ For (b): $$y_1(t) = e^t-\int_{0}^{t}(t-\tau)^{2} e^{\tau} d\tau = e^t-\frac{4}{3}te^{t}+\frac{1}{3}t^2e^t + \frac{2}{3}e^t -\frac{1}{3}t^3$$

(Step 4: Repeat the approximation process for both cases)

For better approximation, repeat the steps by plugging the previous approximations back into the equation. It is important to note that higher-order approximations will yield increasingly accurate results. In this answer, we are only providing the first approximations for both cases (a) and (b). It is up to the student to further iterate and find more accurate approximate solutions. Final approximate expressions for the unknown functions: For (a): $$y(t) \approx t+t\sin{t}-t\cos{t}$$ For (b): $$y(t) \approx e^t - \frac{4}{3}te^{t}+\frac{1}{3}t^2e^t + \frac{2}{3}e^t -\frac{1}{3}t^3$$

Key concepts

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are equations that involve some ordinary derivatives (as opposed to partial derivatives) of a function. They arise in many contexts throughout physics, engineering, and other sciences. The solution to an ODE is a function that, when substituted into the equation, satisfies the relationship defined by the ODE. In our exercise, the student encounters a specific type of ODE known as a Volterra integral equation, which incorporates both a predefined function and an integral that involves the unknown function itself. Understanding ODEs is vital for advancing in the field of mathematics and applied sciences, and the method we use in the exercise demonstrates one of the ways to approximate a solution to such equations.

Integral Equations

Integral equations are equations in which an unknown function appears under an integral sign. The Volterra integral equation is one such example, commonly found in various applications such as physics and engineering. The main challenge with integral equations is that traditional differentiation and integration techniques from calculus may not directly apply. Instead, we often employ approximation methods, numerical integration, or iterative techniques to solve these equations. In the provided exercise, the integral equations are solved using an approximation method to find the function that satisfies the given conditions.

Types of Integral Equations

• Volterra integral equations, which have limits of integration dependent on the variable.
• Fredholm integral equations have fixed limits of integration, independent of the variable.

Approximation Methods

Approximation methods are mathematical techniques used when exact solutions to equations are difficult or impossible to find. Ithe iterative process involves making initial guesses and refining them to get closer to the true solution. Methods like these are especially helpful in solving integral and differential equations where exact analytical solutions are elusive. In our example, the first approximation is generated by substituting a basic function into the integral equation, and then the process is repeated for improved accuracy.

Common Approximation Methods

• Successive approximations for refining guesses iteratively.
• Taylor series for expanding about a point to approximate functions.
• Perturbation methods for slight alterations to obtain an approximate solution.

Cosine Function Properties

The cosine function is one of the primary trigonometric functions, and it has several important properties that are utilized in solving equations. It is an even function, meaning that \(\cos(-x) = \cos(x)\), and it has a period of \(2\pi\), which means the function repeats every \(2\pi\) units. These properties are useful when integrating cosine functions, as shown in the first part of the exercise, where the integral of the cosine function is computed as part of the approximation method for the Volterra integral equation.

Key Properties:

• Predictable symmetry and periodicity aids in integration and solving equations.
• Useful for modeling wave patterns and oscillations in physics and engineering.

Exponential Function Properties

The exponential function, commonly denoted as \(e^x\) where \(e\) is Euler's number, is notable for its unique property that it is its own derivative. This function increases rapidly and is used extensively in areas including calculus, complex numbers, and growth/decay problems. In the context of our exercise, understanding the behavior of exponential functions is crucial while solving the Volterra integral equation that includes an exponential function in its expression.

In particular, the exponential function is integral to the second part of the exercise, which requires knowledge of how the function behaves under differentiation and integration to solve the integral equation presented. Properties of the exponential function impact how the approximation is formed and computed.