weak formulation

USAGE SUMMARY

The phrase "weak formulation" is correct and usable in written English.
It is typically used in mathematical or scientific contexts to refer to a less strict version of a problem or equation, often used in the context of differential equations or variational problems. Example: "In order to solve the boundary value problem, we will first consider its weak formulation."

✓ Grammatically correct

Science

Human-verified examples from authoritative sources

Exact Expressions

60 human-written examples

Consider the following weak formulation of the flow problem (19).

Advances in Difference Equations

The numerical method is derived by discretizing the weak formulation.

Journal of Computational Physics

A coupled non-stationary Stokes-Darcy model and its weak formulation are introduced in Section 2.

Advances in Difference Equations

Using the bilinear mappings and, we construct the weak formulation of problem (1.1) as follows: (3.5).

Journal of Inequalities and Applications

In order to introduce the weak formulation, we first define some additional functions spaces.

Boundary Value Problems

This comprises a discussion of the domain, a strong and weak formulation and the objective functions.

Journal of Mathematics in Industry

which is equivalent to the weak formulation of the nonlinear boundary value problem (1.7).

Journal of Inequalities and Applications

First of all, we deduce the boundary weak formulation with Steklov-Poincaré operator.

Boundary Value Problems

(3.19) The above analysis leads to the following weak formulation of the problem described by (3.1).

Boundary Value Problems

Now, we are ready to introduce the concept of weak formulation.

Boundary Value Problems

For the convergence and stability analysis, the following weak formulation of Eqs.

Mathematical Sciences

Expert writing Tips

Best practice

When using "weak formulation", clearly define the function space and test functions involved to ensure clarity and rigor in your mathematical exposition.

Common error

Avoid using "weak formulation" interchangeably with "strong formulation". The former relaxes differentiability requirements, while the latter requires classical derivatives. Understand the distinction to apply the appropriate method for solving differential equations.

Linguistic Context

The phrase "weak formulation" functions as a noun phrase that refers to a mathematical technique used to reformulate differential equations. It's commonly used in scientific and engineering contexts, as shown in the examples provided by Ludwig. This approach is often used when dealing with problems that lack smooth solutions.

Expression frequency: Very common

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Ludwig's WRAP-UP

The phrase "weak formulation" is a grammatically correct and commonly used term, particularly within scientific and mathematical domains. As Ludwig AI indicates, it refers to a specific mathematical technique used in solving differential equations, especially when dealing with non-smooth solutions. Its usage is primarily confined to formal and scientific contexts, appearing frequently in academic papers and technical reports. When writing about this concept, ensure a clear definition of the function space and test functions to maintain rigor. Alternatives include "variational formulation" or "generalized formulation", depending on the specific nuance you wish to convey.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

variational formulation

Focuses on the variational approach used to derive the weak form.

generalized formulation

Highlights the broader, less restrictive nature of the formulation.

relaxed formulation

Emphasizes the relaxation of constraints in the formulation.

abstract formulation

Suggests a more theoretical or mathematical representation.

finite element formulation

Links the formulation to a specific numerical method.

Galerkin formulation

Specifies the Galerkin method used in the formulation.

weighted residual formulation

Highlights the use of weighted residuals in deriving the formulation.

integral formulation

Focuses on the integral form of the equations.

minimum energy principle formulation

Emphasizes the principle of minimum energy used in the formulation.

principle of virtual work formulation

Highlights the application of virtual work principles.

FAQs

What is the purpose of using a "weak formulation"?

A "weak formulation" allows for the solution of differential equations with less restrictive conditions on the solution's smoothness, enabling the use of numerical methods like the finite element method.

When is it appropriate to use a "weak formulation"?

A "weak formulation" is appropriate when dealing with differential equations where the solutions are not necessarily smooth or when using numerical methods that require a variational form of the equation.

How does a "weak formulation" differ from a strong formulation?

A "weak formulation" reduces the differentiability requirements on the solution compared to a strong formulation, which requires the solution to satisfy the differential equation pointwise.

What are some mathematical techniques used in deriving a "weak formulation"?

Techniques like integration by parts and the application of the divergence theorem are often used to derive a "weak formulation" from a strong formulation of a differential equation.