Is There a Universal Method to Solve All Types of PDEs?

  • Thread starter sid_galt
  • Start date
  • Tags
    Theoretical
In summary, theoretical possibility is the idea that something is possible in theory but not necessarily in reality. It differs from actual possibility, which is based on real-world evidence and practicality. Theoretical possibilities can become actual possibilities through scientific research and advancements in technology, but not all theoretical possibilities are achievable. Probability plays a role in determining the likelihood of a theoretical possibility becoming an actual possibility. In scientific research, theoretical possibility is used to explore new ideas, develop hypotheses, and make predictions about the natural world.
  • #1
sid_galt
502
1
Is it theoretically possible to achieve a general method to solve PDEs of mth-order and nth-degree or is there a theorem which rules out any such general solution?
 
Physics news on Phys.org
  • #2
What do you mean by "a general method"- in terms of what kind of functions? Solving just linear equations with constant coefficients is equivalent to solving their characteristic equation, a polynomial. And it can be shown that there are polynomials that cannot be solved in terms of radicals.
 
  • #3


It is theoretically possible to develop a general method for solving PDEs of mth-order and nth-degree, as there are various techniques and algorithms that can be applied to solve different types of PDEs. However, it is not guaranteed that a single method or algorithm can solve all types of PDEs, as the complexity and characteristics of each individual PDE may vary. Additionally, there may be cases where a PDE does not have a closed-form solution and can only be solved numerically.

As for the existence of a theorem that rules out a general solution for all PDEs, there is no such theorem that has been proven. In fact, there are many theorems and techniques that have been developed specifically for solving PDEs, such as the method of characteristics, separation of variables, and finite difference methods. These theorems provide a framework for solving PDEs, but they do not guarantee a general solution for all cases.

In summary, while it is theoretically possible to develop a general method for solving PDEs, it is unlikely that a single method can solve all types of PDEs. The complexity and uniqueness of each PDE make it challenging to find a universal solution, and therefore, a combination of techniques and theorems is often needed to solve PDEs effectively.
 

1. What is "Theoretical possibility"?

Theoretical possibility refers to the idea that something is possible in theory, but may not necessarily be possible in reality. It is often used in scientific discussions to explore potential ideas or scenarios.

2. How is theoretical possibility different from actual possibility?

Theoretical possibility is based on theory and imagination, while actual possibility is based on real-world evidence and practicality. Theoretical possibilities may not always be achievable, while actual possibilities are more likely to be achieved.

3. Can a theoretical possibility become an actual possibility?

Yes, it is possible for a theoretical possibility to become an actual possibility. This often happens through scientific research and advancements in technology. However, not all theoretical possibilities can be turned into actual possibilities.

4. What role does probability play in theoretical possibility?

Probability is a key factor in determining the likelihood of a theoretical possibility becoming an actual possibility. The higher the probability, the more likely it is that the theoretical possibility can be achieved.

5. How is the concept of theoretical possibility used in scientific research?

Scientists use the concept of theoretical possibility to explore new ideas, develop hypotheses, and make predictions about the natural world. It allows them to think outside the box and consider possibilities that may not have been previously considered.

Similar threads

A Solving System of PDEs: Analytical Methods & Solutions
    • Differential Equations
Replies
9
Views
1K
I Deep Learning the new key to nonlinear PDEs?
    • Differential Equations
Replies
2
Views
1K
Looking for resources to help me understand the basics of PDEs for physics
    • STEM Academic Advising
Replies
5
Views
599
A FEM for Solving PDEs: Beginner Textbooks & Fluid Flow
    • Differential Equations
Replies
5
Views
2K
I Understanding the change of variables for PDEs
    • Differential Equations
Replies
2
Views
2K
MHB Matrix Methods for Difference Problems
    • Differential Equations
Replies
3
Views
2K
Guidance on Pursuing a Master's in Theoretical Particle Physics
    • STEM Academic Advising
2
Replies
35
Views
3K
Applied Bibliography on integration and ODE/PDE solving techniques for physics
    • Science and Math Textbooks
Replies
12
Views
912
Numerical methods that need a guess/approximate solutions
    • Differential Equations
Replies
3
Views
1K
Solving PDEs: Separation of Vars., Method of Characteristics
    • Differential Equations
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top