Chapter map

1. 1. Why successive approximation matters
2. 2. The integral equation preserves the data
3. 3. Choosing the zeroth function
4. 4. First approximation
5. 5. Second approximation
6. 6. Third approximation
7. 7. Fourth approximation
8. 8. The fixed-point viewpoint
9. 9. Contraction estimates
10. 10. Choosing the interval
11. 11. Existence and uniqueness are different claims
12. 12. Nonlinear equations
13. 13. Picard iteration and numerical solvers
14. 14. Picard and DJM
15. 15. Implementation and quadrature
16. 16. Common misconceptions
17. 17. Student and parent learning route
18. 18. What a responsible report contains

Read in order for a complete derivation, or use the map to move between mechanism, worked arithmetic, limitations, student practice and reporting. The same mathematical name can hide different assumptions, so each chapter states the local claim and its check.

1. Why successive approximation matters

Picard iteration turns solving into a repeatable action on whole functions rather than a guess at isolated values. This is the point where the idea becomes more than terminology: the objects and operations can be named precisely.

How the mathematics works

For y prime=f(x,y) with y(x0)=y0, define T[v](x)=y0 plus the integral from x0 to x of f(t,v(t))dt. Write the equation before simplifying it, preserve exact quantities as long as practical and identify which step depends on an assumption rather than arithmetic alone.

A concrete calculation or interpretation

A solution is precisely a fixed point y=T[y], so iteration and existence theory meet in one equation. Reconstruct the calculation independently, label units or norms where relevant and keep rounded values separate from exact expressions.

Limits, evidence and transfer

The operator must map the declared function set into itself before a convergence theorem applies. Draw the operator as input function, integral evaluation and output function. A careful conclusion reports the domain, iteration or approximation order, comparison method and remaining uncertainty instead of turning one successful example into a universal promise.

2. The integral equation preserves the data

Integrating the ODE from x0 to x embeds the initial value in every candidate produced by T. This is the point where the idea becomes more than terminology: the objects and operations can be named precisely.

How the mathematics works

At x=x0 the integral is zero, so every Picard iterate equals y0 there. Write the equation before simplifying it, preserve exact quantities as long as practical and identify which step depends on an assumption rather than arithmetic alone.

A concrete calculation or interpretation

Differentiating the integral equation recovers the ODE when the required regularity holds. Reconstruct the calculation independently, label units or norms where relevant and keep rounded values separate from exact expressions.

Limits, evidence and transfer

Dropping y0 or moving the lower limit silently changes the problem. Check the initial value before checking any later point. A careful conclusion reports the domain, iteration or approximation order, comparison method and remaining uncertainty instead of turning one successful example into a universal promise.

3. Choosing the zeroth function

The start is often the constant function y0, although other starts can be used when the operator and proof allow them. This is the point where the idea becomes more than terminology: the objects and operations can be named precisely.

How the mathematics works

For y prime=y with y(0)=1, choose phi0(x)=1. Write the equation before simplifying it, preserve exact quantities as long as practical and identify which step depends on an assumption rather than arithmetic alone.

A concrete calculation or interpretation

This start satisfies the data and makes the first integral transparent. Reconstruct the calculation independently, label units or norms where relevant and keep rounded values separate from exact expressions.

Limits, evidence and transfer

A good start does not replace the need to show that all iterates remain in the chosen set. State what the start preserves and why it is convenient. A careful conclusion reports the domain, iteration or approximation order, comparison method and remaining uncertainty instead of turning one successful example into a universal promise.

4. First approximation

Applying T once gives phi1(x)=1 plus the integral of 1, hence 1+x. This is the point where the idea becomes more than terminology: the objects and operations can be named precisely.

How the mathematics works

At x=0.5 the first approximation is 1.5 while exp(0.5) is about 1.6487212707. Write the equation before simplifying it, preserve exact quantities as long as practical and identify which step depends on an assumption rather than arithmetic alone.

A concrete calculation or interpretation

Its residual phi1 prime minus phi1 equals -x. Reconstruct the calculation independently, label units or norms where relevant and keep rounded values separate from exact expressions.

Limits, evidence and transfer

The approximation is locally useful but its error grows away from the expansion point. Compute both value error and residual at three points. A careful conclusion reports the domain, iteration or approximation order, comparison method and remaining uncertainty instead of turning one successful example into a universal promise.

5. Second approximation

Insert phi1 into the operator to obtain phi2=1+x+x squared over 2. This is the point where the idea becomes more than terminology: the objects and operations can be named precisely.

How the mathematics works

Each application integrates the previous polynomial and contributes the next Taylor term for this example. Write the equation before simplifying it, preserve exact quantities as long as practical and identify which step depends on an assumption rather than arithmetic alone.

A concrete calculation or interpretation

At x=0.5, phi2=1.625. Reconstruct the calculation independently, label units or norms where relevant and keep rounded values separate from exact expressions.

Limits, evidence and transfer

The Taylor pattern belongs to this linear benchmark and is not the definition of Picard iteration. Derive the integral rather than quoting the pattern. A careful conclusion reports the domain, iteration or approximation order, comparison method and remaining uncertainty instead of turning one successful example into a universal promise.

6. Third approximation

A third application gives phi3=1+x+x squared over 2+x cubed over 6. This is the point where the idea becomes more than terminology: the objects and operations can be named precisely.

How the mathematics works

At x=0.5, phi3=1.6458333333, with absolute error about 0.0028879374. Write the equation before simplifying it, preserve exact quantities as long as practical and identify which step depends on an assumption rather than arithmetic alone.

A concrete calculation or interpretation

The original ODE residual is minus x cubed over 6, whose magnitude at x=0.5 is about 0.0208333333. Reconstruct the calculation independently, label units or norms where relevant and keep rounded values separate from exact expressions.

Limits, evidence and transfer

A small value error at one point is not a uniform bound on a longer interval. Record the exact fraction before decimal rounding. A careful conclusion reports the domain, iteration or approximation order, comparison method and remaining uncertainty instead of turning one successful example into a universal promise.

7. Fourth approximation

The next iterate adds x fourth over 24 and gives 1.6484375 at x=0.5. This is the point where the idea becomes more than terminology: the objects and operations can be named precisely.

How the mathematics works

Its absolute error is about 0.0002837707, and the residual magnitude there is about 0.0026041667. Write the equation before simplifying it, preserve exact quantities as long as practical and identify which step depends on an assumption rather than arithmetic alone.

A concrete calculation or interpretation

Successive integration explains the shrinking local defect. Reconstruct the calculation independently, label units or norms where relevant and keep rounded values separate from exact expressions.

Limits, evidence and transfer

The improvement depends on interval, equation and regularity, not merely on iteration count. Plot phi1 through phi4 beside exp(x). A careful conclusion reports the domain, iteration or approximation order, comparison method and remaining uncertainty instead of turning one successful example into a universal promise.

8. The fixed-point viewpoint

A fixed point is a function that the operator leaves unchanged. This is the point where the idea becomes more than terminology: the objects and operations can be named precisely.

How the mathematics works

If phi_n approaches phi in a norm and T is continuous in the needed sense, the limit can satisfy phi=T[phi]. Write the equation before simplifying it, preserve exact quantities as long as practical and identify which step depends on an assumption rather than arithmetic alone.

A concrete calculation or interpretation

This converts an iterative pattern into a solution argument. Reconstruct the calculation independently, label units or norms where relevant and keep rounded values separate from exact expressions.

Limits, evidence and transfer

Pointwise-looking plots alone do not justify passing a limit through an integral operator. Name the function space and norm in any theorem statement. A careful conclusion reports the domain, iteration or approximation order, comparison method and remaining uncertainty instead of turning one successful example into a universal promise.

9. Contraction estimates

If f is Lipschitz in y with constant L on a suitable region, the integral operator can satisfy a bound involving L times the interval length. This is the point where the idea becomes more than terminology: the objects and operations can be named precisely.

How the mathematics works

Choosing h with Lh less than one can make T a contraction in the sup norm. Write the equation before simplifying it, preserve exact quantities as long as practical and identify which step depends on an assumption rather than arithmetic alone.

A concrete calculation or interpretation

The Banach fixed-point theorem then supplies a unique fixed point and geometric error control. Reconstruct the calculation independently, label units or norms where relevant and keep rounded values separate from exact expressions.

Limits, evidence and transfer

A global Lipschitz claim should not be inferred from a local derivative sample. Calculate Lh for the proposed interval before invoking contraction. A careful conclusion reports the domain, iteration or approximation order, comparison method and remaining uncertainty instead of turning one successful example into a universal promise.

10. Choosing the interval

The rectangle around the initial data must keep x and candidate y values inside the domain where f is controlled. This is the point where the idea becomes more than terminology: the objects and operations can be named precisely.

How the mathematics works

An existence interval is often limited by both a bound on f and a Lipschitz condition. Write the equation before simplifying it, preserve exact quantities as long as practical and identify which step depends on an assumption rather than arithmetic alone.

A concrete calculation or interpretation

Shortening the interval can turn a failed contraction estimate into a valid local one. Reconstruct the calculation independently, label units or norms where relevant and keep rounded values separate from exact expressions.

Limits, evidence and transfer

This does not prove global continuation or rule out later singularities. Write the interval calculation beside the iteration. A careful conclusion reports the domain, iteration or approximation order, comparison method and remaining uncertainty instead of turning one successful example into a universal promise.

11. Existence and uniqueness are different claims

Continuity of f can support local existence under standard hypotheses, while uniqueness usually needs stronger control such as local Lipschitz continuity in y. This is the point where the idea becomes more than terminology: the objects and operations can be named precisely.

How the mathematics works

Picard-Lindelöf joins the constructive sequence with a uniqueness result. Write the equation before simplifying it, preserve exact quantities as long as practical and identify which step depends on an assumption rather than arithmetic alone.

A concrete calculation or interpretation

A non-Lipschitz equation can have several solutions through the same initial point. Reconstruct the calculation independently, label units or norms where relevant and keep rounded values separate from exact expressions.

Limits, evidence and transfer

Never use convergence of one numerical sequence as proof that no other solution exists. Separate the existence sentence from the uniqueness sentence. A careful conclusion reports the domain, iteration or approximation order, comparison method and remaining uncertainty instead of turning one successful example into a universal promise.

12. Nonlinear equations

For y prime=y squared with y(0)=1, the operator becomes T[v]=1 plus the integral of v squared. This is the point where the idea becomes more than terminology: the objects and operations can be named precisely.

How the mathematics works

The first iterates develop nonlinear cross-terms and approach 1 over 1-x only within a suitable region. Write the equation before simplifying it, preserve exact quantities as long as practical and identify which step depends on an assumption rather than arithmetic alone.

A concrete calculation or interpretation

The exact solution has a singularity at x=1. Reconstruct the calculation independently, label units or norms where relevant and keep rounded values separate from exact expressions.

Limits, evidence and transfer

No local contraction argument can make that singularity disappear. Compare iterates on intervals ending at 0.25, 0.75 and 0.95. A careful conclusion reports the domain, iteration or approximation order, comparison method and remaining uncertainty instead of turning one successful example into a universal promise.

13. Picard iteration and numerical solvers

The theoretical iteration constructs entire functions, whereas practical ODE solvers usually advance values step by step. This is the point where the idea becomes more than terminology: the objects and operations can be named precisely.

How the mathematics works

Successive approximation still motivates integral formulations, deferred corrections and convergence reasoning. Write the equation before simplifying it, preserve exact quantities as long as practical and identify which step depends on an assumption rather than arithmetic alone.

A concrete calculation or interpretation

A symbolic Picard iterate may become expensive even when a Runge-Kutta method is easy to compute. Reconstruct the calculation independently, label units or norms where relevant and keep rounded values separate from exact expressions.

Limits, evidence and transfer

Method choice should match the desired proof, representation and accuracy. Compare computational work as well as final error. A careful conclusion reports the domain, iteration or approximation order, comparison method and remaining uncertainty instead of turning one successful example into a universal promise.

14. Picard and DJM

Both can begin from u=f+N(u), but Picard replaces the whole approximation by f+N(current). This is the point where the idea becomes more than terminology: the objects and operations can be named precisely.

How the mathematics works

DJM records telescoping operator increments as separate components. Write the equation before simplifying it, preserve exact quantities as long as practical and identify which step depends on an assumption rather than arithmetic alone.

A concrete calculation or interpretation

For some formulations their partial sums are closely related. Reconstruct the calculation independently, label units or norms where relevant and keep rounded values separate from exact expressions.

Limits, evidence and transfer

Similar-looking first terms do not make their convergence analyses identical. Write both recurrences on the same integral equation. A careful conclusion reports the domain, iteration or approximation order, comparison method and remaining uncertainty instead of turning one successful example into a universal promise.

15. Implementation and quadrature

When the integral has no closed form, numerical quadrature and interpolation become part of each operator application. This is the point where the idea becomes more than terminology: the objects and operations can be named precisely.

How the mathematics works

Errors then enter from the iteration, the quadrature and the representation of the current function. Write the equation before simplifying it, preserve exact quantities as long as practical and identify which step depends on an assumption rather than arithmetic alone.

A concrete calculation or interpretation

Changing the grid between iterations can make differences hard to interpret. Reconstruct the calculation independently, label units or norms where relevant and keep rounded values separate from exact expressions.

Limits, evidence and transfer

A smooth curve from software is not an independent residual check. Repeat one step with a tighter quadrature rule. A careful conclusion reports the domain, iteration or approximation order, comparison method and remaining uncertainty instead of turning one successful example into a universal promise.

16. Common misconceptions

Picard iteration is not simply repeated substitution into the differential equation and does not guarantee global convergence. This is the point where the idea becomes more than terminology: the objects and operations can be named precisely.

How the mathematics works

The iterates are functions, the interval matters and the norm must be stated. Write the equation before simplifying it, preserve exact quantities as long as practical and identify which step depends on an assumption rather than arithmetic alone.

A concrete calculation or interpretation

A fixed point may be unique only inside the set covered by the assumptions. Reconstruct the calculation independently, label units or norms where relevant and keep rounded values separate from exact expressions.

Limits, evidence and transfer

Do not call a finite iterate exact because it matches several decimals. Keep assumptions beside conclusions. A careful conclusion reports the domain, iteration or approximation order, comparison method and remaining uncertainty instead of turning one successful example into a universal promise.

17. Student and parent learning route

A learner can begin with y prime=y, then try a forced linear ODE and finally a simple nonlinear equation. This is the point where the idea becomes more than terminology: the objects and operations can be named precisely.

How the mathematics works

Parents can ask what function was fed into the operator and how the next one was checked. Write the equation before simplifying it, preserve exact quantities as long as practical and identify which step depends on an assumption rather than arithmetic alone.

A concrete calculation or interpretation

The work connects sequences, functions, integration, inequalities and proof. Reconstruct the calculation independently, label units or norms where relevant and keep rounded values separate from exact expressions.

Limits, evidence and transfer

It opens later options without guaranteeing a pathway or outcome. Keep a table of iterates, residuals and interval assumptions. A careful conclusion reports the domain, iteration or approximation order, comparison method and remaining uncertainty instead of turning one successful example into a universal promise.

18. What a responsible report contains

State the ODE, initial data, integral operator, starting function, function set, interval and norm. This is the point where the idea becomes more than terminology: the objects and operations can be named precisely.

How the mathematics works

List enough iterates to reproduce the arithmetic and report residuals against the original equation. Write the equation before simplifying it, preserve exact quantities as long as practical and identify which step depends on an assumption rather than arithmetic alone.

A concrete calculation or interpretation

If a contraction theorem is used, show the bound and explain where it holds. Reconstruct the calculation independently, label units or norms where relevant and keep rounded values separate from exact expressions.

Limits, evidence and transfer

Do not turn a local theorem into a global claim. Finish with one supported conclusion and one unresolved limit. A careful conclusion reports the domain, iteration or approximation order, comparison method and remaining uncertainty instead of turning one successful example into a universal promise.

A practical seven-day learning plan

• Day 1: Rewrite the target equation and define every symbol in plain language.
• Day 2: Derive the first construction or reduction step by hand.
• Day 3: Complete the central worked example using exact arithmetic first.
• Day 4: Substitute the approximation into the original problem and graph the residual.
• Day 5: Change one interval, start, parameter or order and explain the effect.
• Day 6: Compare one related method on the same problem and error measure.
• Day 7: Present one supported conclusion, one limit and one next experiment.

This plan develops problem-solving skills because it moves repeatedly between definition, calculation, evidence and communication. A parent can ask to see the notebook rather than a perfect-looking graph, while a student learns that corrections are useful only when their effect can be measured.

Frequently asked questions

What is Picard iteration?

It repeatedly applies the integral operator associated with an initial-value problem to construct a sequence of approximate solution functions.

Why rewrite the ODE as an integral equation?

The integral form preserves the initial value directly and creates the fixed-point operator used in the existence and uniqueness argument.

Does Picard iteration always converge?

No. Standard guarantees require conditions such as continuity and a Lipschitz bound on an appropriate rectangle or function set.

Is a fixed point automatically unique?

Uniqueness follows under additional assumptions, commonly a contraction condition, rather than from the notation alone.

How is the method checked numerically?

Compare successive iterates, evaluate the original differential-equation residual and benchmark against a known or high-accuracy solution.

What should a student report?

The integral operator, starting function, interval, iterates, norm, convergence evidence, residual and limitations.

Primary source and current Singapore context

Émile Picard, “Sur la méthode des approximations successives et les équations différentielles linéaires,” Bulletin de la Société Mathématique de France 22, 1894, 52–57.

Open the primary or authoritative source record

The source supports the method’s definition and research context. The worked calculations here are original teaching examples and are checked directly against their stated equations.

For current Singapore terminology, the official SEAB Secondary Education Certificate page states that the SEC begins with the 2027 graduating cohort and reports subjects at G1, G2 and G3 levels. This article does not claim that the named advanced method is an SEC syllabus requirement. It shows how school algebra, functions and calculus can connect to later mathematical sciences.

SEAB: Secondary Education Certificate

NTU Mathematics course information

NUS Applied and Computational Mathematics research