Concept Architecture
Concept
Theoretically, Initial Value Problem (IVP) is a mathematical formulation in which an ordinary differential equation is solved subject to one or more specified initial conditions. The concept provides a unique starting point from which the future behaviour of a dynamic system can be determined. Initial value problems form the theoretical foundation of numerical solution methods for ordinary differential equations and are fundamental to continuous-time modelling in health economics, pharmacokinetics, epidemiology and systems biology.
Mathematically, an initial value problem consists of a differential equation together with an initial condition that specifies the value of the unknown function at a given point. The objective is to determine the function that satisfies both the governing differential equation and the initial condition. When an analytical solution is unavailable, numerical methods such as Euler or Runge-Kutta methods are used to approximate the solution over successive time intervals.
In practice, initial value problems are solved using numerical algorithms implemented within statistical software, scientific computing environments and decision modelling frameworks. In health economics, they underpin dynamic disease models, infectious disease transmission models, pharmacokinetic and pharmacodynamic simulations, and physiological models that generate inputs for economic evaluations and health technology assessments.
Purpose
Used to define and solve dynamic systems in which the initial state of the system is known, allowing future system behaviour to be predicted through analytical or numerical solution of differential equations.
Mathematical Formulae
Primary Formula
dy/dt = f(t, y)
y(t?) = y?
Supporting Formulae
General solution:
y(t) = solution satisfying both dy/dt = f(t, y) and y(t?) = y?
Existence and uniqueness conditions (conceptual):
If f(t, y) is continuous and satisfies a Lipschitz condition in y, then the IVP has a unique local solution.
Related Mathematical Methods
- Euler Method
- Runge-Kutta Method
- Ordinary Differential Equation
- Numerical Integration
- Adaptive Step-Size Methods
- Finite Difference Method
Example
A pharmacokinetic model describes drug elimination using:
dC/dt = ?0.20C
with the initial condition:
C(0) = 150 mg/L
The initial value problem defines both the elimination equation and the starting concentration. A numerical solver computes drug concentration over time, allowing estimation of treatment effectiveness, toxicity and subsequent health economic outcomes.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| LET | =LET(h,0.1,Cprev,B2,Cprev+h*(-0.2*Cprev)) | Performs one numerical integration step from the specified initial value. |
| LAMBDA | User-defined IVP solver | Creates reusable numerical solution functions for dynamic health models. |
| SEQUENCE | =SEQUENCE(101,1,0,0.1) | Generates time intervals for solving the differential equation. |
| EXP | =150*EXP(-0.2*A2) | Calculates the analytical solution when one exists for comparison with numerical methods. |
| ABS | =ABS(B2-C2) | Quantifies numerical approximation error. |
VBA (Optional)
VBA can automate the numerical solution of initial value problems across multiple differential equations and simulation time horizons for dynamic health economic models.
Sources
- Butcher JC. Numerical Methods for Ordinary Differential Equations. Wiley.
- Hairer E, N?rsett SP, Wanner G. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
- Burden RL, Faires JD. Numerical Analysis. Cengage Learning.
Related Concepts (4)
Library
Publications
1
Numerical Analysis of Parameterized Nonlinear Equations — Werner C. Rheinboldt, 1st Edition ed., 1986 (John Wiley & Sons)
A specialist numerical-analysis reference on iterative solution, continuation and computation for parameterized nonlinear systems, linked here to the ontology's numerical-method and optimization branch.
BookView source →
Frequently Asked Questions (6)
What is an initial value problem?
A differential equation together with specified starting conditions used to determine a unique solution.
Source: Butcher JC. Numerical Methods for Ordinary Differential Equations. 3rd ed. Wiley; 2016. doi:10.1002/9781119121534.
What are the components of an initial value problem?
An initial value problem is made up of a differential equation together with specified starting conditions, the two components used to determine a unique solution. The differential equation describes how the quantity changes, while the starting conditions fix its value at the beginning. Combining a governing equation with a known starting point is what constitutes an initial value problem By pairing a rule for change with a fixed starting point, the problem singles out one solution from the many a differential equation alone would permit, which is what makes it well posed.
Source: Butcher JC. Numerical Methods for Ordinary Differential Equations. 3rd ed. Wiley; 2016. doi:10.1002/9781119121534.
Why does an initial value problem specify starting conditions?
An initial value problem specifies starting conditions because a differential equation alone typically admits many solutions, and the starting conditions pin down which one applies. By fixing the value at the beginning, the initial value problem determines a unique solution from the family the equation allows. This role of the starting conditions in selecting one solution is central to what an initial value problem is By pairing a rule for change with a fixed starting point, the problem singles out one solution from the many a differential equation alone would permit, which is what makes it well posed.
Source: Butcher JC. Numerical Methods for Ordinary Differential Equations. 3rd ed. Wiley; 2016. doi:10.1002/9781119121534.
What does an initial value problem determine?
An initial value problem determines a unique solution, since it pairs a differential equation with specified starting conditions precisely so that one particular solution is singled out. Without the starting conditions the equation would leave the solution undetermined. Determining a single solution from the governing equation and the starting point is the purpose of an initial value problem By pairing a rule for change with a fixed starting point, the problem singles out one solution from the many a differential equation alone would permit, which is what makes it well posed.
Source: Butcher JC. Numerical Methods for Ordinary Differential Equations. 3rd ed. Wiley; 2016. doi:10.1002/9781119121534.
What role does the differential equation play in an initial value problem?
The differential equation in an initial value problem describes how the quantity of interest changes, providing the rule that, together with specified starting conditions, determines a unique solution. It sets the relationship the solution must satisfy at every point, while the starting conditions fix where it begins. This governing role of the differential equation is one half of what defines an initial value problem By pairing a rule for change with a fixed starting point, the problem singles out one solution from the many a differential equation alone would permit, which is what makes it well posed.
Source: Butcher JC. Numerical Methods for Ordinary Differential Equations. 3rd ed. Wiley; 2016. doi:10.1002/9781119121534.
How does an initial value problem relate to the Euler method?
An initial value problem is a differential equation with specified starting conditions used to determine a unique solution, while the Euler method is a numerical method for approximating that solution by advancing in small steps from the starting conditions using the local gradient. The Euler method is one way to solve an initial value problem numerically. The two are connected, since the initial value problem supplies the starting point and rule that the Euler method steps forward from By pairing a rule for change with a fixed starting point, the problem singles out one solution from the many a differential equation alone would permit, which is what makes it well posed.
Source: Butcher JC. Numerical Methods for Ordinary Differential Equations. 3rd ed. Wiley; 2016. doi:10.1002/9781119121534.
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British health economist
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Verification date: 2 Apr 2026
Content version: 1.0.0
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